Ghent University Faculty of Engineering Department of...

Ghent University Faculty of Engineering

Department of Industrial Management Chairman: Prof. Dr. Ir. R. VAN LANDEGHEM

Polytechnical University of Barcelona (UPC) Technical School for Industrial Engineering of Barcelona

Department of Organization of Enterprises Chairman: Prof. A. SALAMERO

MODELLING AND OPTIMIZING INVENTORY ROUTING PROBLEMS

WHEN SALES-POINTS IMPLEMENT ORDER-UP-TO INVENTORY POLICIES

by Lennert VAN GUYT

Promotor UGENT: Prof.Dr.Ir. E-H. AGHEZZAF Promotor UPC: Prof.Dr.Ir. M. MATEO

Thesis submitted in order to obtain the academic grade of Master in Mechanical Engineering Option Industrial Management

Academy Year 2005-2006

Preface As a foreign Erasmus exchange student at the Polytechnical University of Catalunya (UPC), I

would like to express my thanks to my promoter over there, Prof. Dr. Ir. Manel Mateo Doll.

Special thanks to my Ugent promotor, Prof. Dr. Ir. E-H. Aghezzaf for helping me with all my

questions and supporting me while finishing my master thesis in Ghent; to M.Sc. Ing. Birger

Raa for patiently answering my never-ending stream of questions; to my parents, who gave

me the opportunity to expand my horizons abroad; to Nicolas Libbrecht for helping me out

with both study and non-study related issues; to Bram De Laere, who gave unexpected

support; to all my friends and relatives that supported me the last year.

Lennert Van Guyt

Ghent, Belgium,

August 2006.

“The author gives the permission to make this thesis available for consultation and to

copy parts of the thesis for personal use.

Every other use is restricted by the copyright, in particular with regard to the obligation

of explicitly mentioning the source when quoting results of this thesis.”

Date Signature

Modelling and Optimizing Inventory Routing Problems when Sales-Points Implement Order-Up-To Inventory Policies

by Lennert VAN GUYT

Thesis submitted in order to obtain the academic grade of Master in Mechanical Engineering option Industrial Management Academy Year 2005-2006 Promotor UGENT: Prof. Dr. Ir. E-H. AGHEZZAF VANLANDEGHEM Promotor UPC: Prof.Dr.Ir. M. MATEO Ghent University Faculty of Engineering Department of Industrial Management Chairman: Prof. Dr. Ir. H. VANLANDEGHEM Polytechnical University of Catalunya Technical School for Industrial Engineering of Barcelona Department of Organization of Enterprises Chairman: Prof. Dr. Ir. Antonio SALAMERO

Abstract

Over the past several years, interest in logistics and supply chain management has grown rapidly. One of the issues researched intensively, is the Inventory Routing Problem: this is concerned with the distribution of a single product from one facility (e.g., a warehouse) to a set of n customers over a certain planning horizon. The objective is to determine a distribution plan that minimizes fleet operating and average total distribution and inventory holding cost with reducing the chance of a stock-out at any of the sales-points, during the planning horizon. This integrates inventory and distribution aspects in the same planning process. Aghezzaf et al. (2005) developed a model, solving the IRP, where the retailers experience a constant and deterministic demand. A remarkable property is that vehicles can perform multi-tours; this implies that a vehicle’s travel plan can contain more than one tour. We analyzed the performance of this model when some of the retailers experienced stochastic demand. Simulations yielded that the performance of this model was independent of the size of the logistic problem. However, the percentage of retailers with stochastic demand had an important impact on the performance, especially on the average cost rates per hour. Surprisingly, the fill rate, the percentage of total demand fulfilled, remained quite high in all problems considered. Therefore, a new strategy was developed, serving the two types of retailers independently: this is clearly a sub-optimization. This resulted in a possible smaller cost rate per hour, yet the difference with the result from the IRP model was smaller than expected. We could conclude that usage of the IRP problem for logistic systems with different types of retailers (stochastic and deterministic demand) yielded better results than expected. Keywords: Inventory Routing Problem; Supply Chain Management; EOQ; Order-up-to policy

Table of contents

1 INTRODUCTION .............................................................................................................1

2 INVENTORY MODELS..................................................................................................4 2.1 Deterministic demand .............................................................................................................................4

2.1.1 Infinite horizon (EOQ)..........................................................................................................................4 2.1.2 Finite horizon........................................................................................................................................6

2.2 Varying demands .............................................................................................................................................8 2.2.1 Wagner-Within model...........................................................................................................................8 2.2.2 Models with capacity constraints ........................................................................................................10

2.3 Stochastic inventory models..........................................................................................................................10 2.3.1 Newsvendor problem ..........................................................................................................................10 2.3.2 Finite horizon models..........................................................................................................................11 2.3.3 Infinite horizon models .......................................................................................................................13

2.3.3.1 Description ................................................................................................................................13 2.3.3.2 Algorithm ..................................................................................................................................15 2.3.3.3 Implementation of the algorithm...............................................................................................17

3 IRP MODEL ...................................................................................................................20

3.1 Introduction ...........................................................................................................................................20

3.2 A brief survey of solution approaches for the IRP..............................................................................20

3.3 Model for long-term IRP.......................................................................................................................22 3.3.1 Multi-tours ..........................................................................................................................................22 3.3.2 Mixed integer formulation of the problem and solution approach......................................................24 3.3.3 Performance analysis ..........................................................................................................................25

4 SIMULATION APPROACH.........................................................................................27

4.1 Introduction ...........................................................................................................................................27

4.2 Input of the IRP model ..........................................................................................................................27 4.2.1 Retailers withdeterministic demand (EOQ)........................................................................................29 4.2.2 Retailers with stochastic demand (Order-Up-To) ...............................................................................29

4.3 Output of the IRP model ...............................................................................................................................32

4.4 Performance measurements..........................................................................................................................36 4.4.1 Cost rates.............................................................................................................................................36 4.4.2 Service levels ......................................................................................................................................36

4.4.2.1 Service level based on stockout probability in a cycle (P1).......................................................37 4.4.2.2 Service level based on stockout probability in a cycle (P2) ......................................................37

4.4.3 Average time between stockout ..........................................................................................................37

4.5 Simulation method.........................................................................................................................................37 4.5.1 Parameters ..........................................................................................................................................37 4.5.2 Optimal distribution plan generated by the IRP model.......................................................................38

4.5.3 Retailer-dependent properties .............................................................................................................38 4.5.4 Results for the whole logistic system..................................................................................................42

5 SIMULATION RESULTS AND ANALYSIS ..............................................................45

5.1 Results ......................................................................................................................................................45 5.1.1 Small problems ...................................................................................................................................45

5.1.1.1 n=30 ..........................................................................................................................................45 5.1.1.2 n=39...........................................................................................................................................48 5.1.1.3 n=50 ..........................................................................................................................................49

5.1.2 Medium-sized problems .....................................................................................................................49 5.1.2.1 n= 67..........................................................................................................................................49 5.1.2.2 n= 81 .........................................................................................................................................50 5.1.2.3 n= 93 .........................................................................................................................................50

5.1.3 Large problems ...................................................................................................................................50 5.1.3.1 n=98...........................................................................................................................................51 5.1.3.2 n=110 ........................................................................................................................................51 5.1.3.3 n=120 ........................................................................................................................................52

5.2 Analysis ...................................................................................................................................................52 5.2.1 Cost rates ............................................................................................................................................52

5.2.1.1 Shortage costs............................................................................................................................52 5.2.1.2 Holding cost ..............................................................................................................................53 5.2.1.3 Total cost rate............................................................................................................................54

5.2.2 Servicelevel P1 ...................................................................................................................................55

5.3 Conclusion ..............................................................................................................................................56

6 IMPLEMENTING A DIFFERENT REPLENISHMENT STRATEGY...................57

6.1 Introduction ...........................................................................................................................................57

6.2 Costs associated with the new strategy ................................................................................................57 6.2.1 Deterministic demand.........................................................................................................................57 6.2.2 Stochastic demand...............................................................................................................................58

6.3 Diminishing the used fleet.....................................................................................................................60

7 CONCLUSION ...............................................................................................................62

APPENDIX A. PROPERTIES OF C(⋅,⋅) AND BOUNDS FOR(S*,S*) ............................................................63

APPENDIX B. CALCULATING THE OPTIMAL (S,S)-VALUES ...............................................................66

APPENDIX C. DISCUSSION CONSTRAINTS IRP-MODEL.......................................................................69

APPENDIX D. DATA-FILES FOR THE DIFFERENT LOGISTIC PROBLEMS ......................................70

APPENDIX E. VBA CODE FOR SIMULATION............................................................................................85

SOURCES ............................................................................................................................................................93

1

1 Introduction

Over the past several years, interests in logistics and supply chain management, both in

industry and academics, have grown rapidly. A number of forces have contributed to this

trend. Firstly, it has become clear that many enterprises already have achieved huge savings

by reducing their manufacturing costs as much as practically possible. However, many of

these companies are becoming aware of the magnitude of the extra savings that can be

achieved by planning and managing their supply chain more effectively. A remarkable

example in the US is Wal-Mart: a part of its success is attributed by the implementation of a

new logistics strategy called cross-docking. Another force is the implementation of

information and communication systems, which can provide access to data from all

components of the supply chain. In that perspective, the influence of the Internet (and

consequently E-commerce) on business practice has been immense. For instance, the Direct-

Business-Model utilized by leading companies such as Dell Computers and Amazon.com

enables customers to order products over the internet and thus allows companies to sell their

products without relying on third party distributors or conventional stores. Finally, the

deregulation of the transportation industry has led to the development of a variety of

transportation modes and reduces transportation costs, while significantly increasing the

complexity of logistics systems.

These developments have motivated the academic community to pursue researching supply

chain issues. Indeed, in the last five years, considerable progress has been made in the

analysis and solution of logistics and supply chain problems. One of the issues that has been

researched many times is the Inventory Routing Problem (IRP). This is concerned with the

distribution of a single product from one facility (e.g., a warehouse) to a set of n customers

over a certain planning horizon. The objective is to determine a distribution plan that

minimizes fleet operating and average total distribution and inventory holding cost with

reducing the chance of a stock-out at any of the sales-points, during the planning horizon.

This integrates inventory and distribution aspects in the same planning process.

This thesis will take a look at the performance of the IRP-model, proposed by Aghezzaf et

al.(2005), when some of the retailers implement an order-up-to strategy (OUP). The subject of

this thesis lies thus in the field of logistic management, defined by the Council of Logistics

Management as:

2

The process of planning, implementing and controlling the efficient, effective flow and storage

of goods, services and related information from point of origin to point of consumption for the

purpose of conforming to customer requirements.

The latter leads to several comments. Firstly, the definition suggests that we have to take into

account every facility that has an impact on cost: from suppliers and manufacturing facilities

through warehouses and distribution centres to retailers and stores. In our case we only

assume a two-echelon supply chain with a warehouse and multiple retailers. None of the

production steps will be taken into consideration in the cost calculation.

The second observation made from the definition, is that the objective of logistics

management is to be efficient and cost-effective across the entire system. This implies that not

the separate costs at the different facilities but the total system-wide costs are to be

minimized. Thus the emphasis is not on simply minimizing transportation costs or reducing

inventories, but rather on taking a system approach to logistics management. It means that we

are not only taking into account the costs of the supplier( transportation and handling costs),

but as well the inventory costs of the retailers.

Finally, we can classify logical decisions into three levels, of which the scope differ:

Strategic level: decisions that have a long-lasting effect on the firms, (e.g. the number

of locations and capacities of warehouses and manufacturing plants network

configuration)

Tactical level: decision that are updated anywhere between once every week, month ,

or once every quarter, (e.g. purchasing and production decisions, inventory policies,

and transportations strategies including the frequency with which customers are

visited).

Operational level: day-to-day decisions such as scheduling, routing and loading trucks.

In this thesis we will only be working on the tactical and operational level, as we assume

given network configurations.

This is an outline of the remainder of this thesis. The second chapter shows a brief survey of

relevant inventory models. We describe the IRP-model, developed by Aghezzaf et al., in

chapter three. The objective is then to examine the performance of this model on the long run,

3

when retailers with stochastic demand are implemented as well. This is evaluated by means of

simulation. The fourth chapter gives an insight into the tools that were utilized during

simulation. In chapter 5, we discuss their results. Then, in chapter 6, we try to diminish the

long run average cost by implementing a new replenishment strategy. Finally, we end with a

conclusion.

4

2 Inventory models

This chapter is devoted to a brief survey of the most relevant types of inventory models

utilized in business processes. Only independent-demand inventories are considered: the

demand of these products is only subject to market forces and independent of operations. On

the contrary, dependent demand inventories, such as work-in-process inventory or raw-

material inventories, have dependent demand of operations! These inventories are controlled

by materials requirements planning systems (MRP) which support the planning and control of

inventories and capacity in manufacturing companies. Because the manufacturing processes

lie beyond our scope, we will not consider these type of inventories.

2.1 Deterministic demand In these models demands are assumed to be constant and deterministic. The problem of

finding the optimal order quantity has been reduced to balancing the ordering and holding

costs. 2.1.1 Infinite horizon (EOQ) This model has been introduced in 1915 (by Harris) and it attends to the trade-offs between

ordering and holding costs. (Fig.1)

Fig.1. Trade-off between ordering and holding cost.

We consider a supply chain, containing a warehouse that faces a constant demand for a single

item. The warehouse itself places orders for the item from another facility in the distribution

network, which is assumed to have an unlimited quantity of the product. We make the

following assumptions:

5

The demand is constant: D items/unit time

A fixed order quantity: Q items/order

A fixed set-up cost K: paid every time the warehouse places an order

An inventory holding cost h is being charged for every unit held in inventory per unit

time

The lead time, the time that elapses between the placement of an order an its receipt, is

zero

No initial inventory

The planning horizon is infinite

The objective now is to find the optimal ordering policy minimizing total ordering and

holding cost per unit of time. It is important to notice that this should happen without stock-

outs. In this case, any optimal ordering policy must satisfy the Zero Inventory Ordering

Property, which states that every order is received precisely when the inventory level drops

to zero. In order to determine the optimal ordering policy, we consider the inventory level as a

function of time (see Fig. 2). This type of function is a saw-toothed inventory pattern. We

define the cycle time T as the time between two successive replenishments.

Fig.2. Saw-toothed pattern of the inventory level.

Thus, total inventory cost in a cycle of length T is:

6

2hTQK +

And because Q= TD, the average total cost per unit of time is:

2hQ

QKD

+

Hence , the Economic Order Quantity (EOQ):

hKDQ 2* =

This is the ordering quantity at which inventory set-up cost per unit of time (KD/Q) equals

inventory holding cost per unit of time (hQ/2). Of course, because of all the assumptions, this

model is not very realistic. However, it is still one of the most used inventory policies in

business. The retailers considered in the IRP-model discussed later on, are assumed having an

EOQ-inventory model. To make this model more realistic, we should be able to lose some of

the assumptions made above. It can be shown very easily how some of the assumptions can

be relaxed, without losing any of the simplicity of the model. For instance, consider the initial

inventory to be positive, at level I0; in this case the first order for Q* items is simply delayed

until time I0/D. Further, the assumption of zero lead time can also easily be relaxed. In fact,

the model can handle any deterministic lead time L. To do this, simply place an order for Q*

items when the inventory level is D*L. On the other hand, relaxing the assumptions of fixed

demands and infinite planning horizon requires significant changes to the above solution. We

discuss these cases in the following paragraphs.

2.1.2 Finite horizon In order to make the model from the previous paragraph more realistic, we start working with

a finite horizon, say t. This horizon could, for example, represent the winter season in the

retail industry, in which the demand for the product might be assumed to be constant and

known. Another assumption that we relax, is the one that states that order quantities should be

fixed. The objective is to find an inventory policy that minimizes ordering and holding cost on

the interval [0,t].

In order to do so, consider any policy, say P, that places 1≥m orders in the interval [0,t]. It is

easy to see that the first order should be placed in the beginning and the last must be placed so

that the inventory at time t is zero. Figure 3 illustrates such a P-policy. For any i,

7

11 −≤≤ mi , let Ti be the time between the placement of the ith order and the (i+1)th order and

let Tm be the time between the placement of the last order and t. Thus, by definition,

∑ ==

m

i iTt1

and P places the jth order at time

∑ =

j

t iT1

, for mj ≤≤1 .

Fig. 3. Evolution of the inventory level implementing a P-policy.

Again, it is clear that the policy P must satisfy the Zero Inventory Ordering Property. For the

policy P, let I(τ) be the inventory level at time τ ∈[0,t] . Thus, the total cost per unit of time

for this policy becomes

ττ dhKmt

t

∫ )+0

I([1]

The only thing known about the function I(τ) is that it declines at a rate of D ( slope of D)

between orders and reaches zero exactly m times. Consequently, we can express the total

inventory of the entire horizon as a function of the time between orders {Ti}i=1…m as follows:

∑∑==

=m

ii

m

i

ii TDDTT1

2

1 22.

Thus, if m orders are placed we can find the best times to place them by solving:

Min{ miTtTT i

m

ii

m

ii ,...,2,1,0,

11

2 =∀≥=∑∑==

}

8

The optimal solution is Ti = mt for each i= 1,2,…,m. Hence, an optimal policy must have the

following property:

For a problem with one product over the interval [0,t], the inventory policy with minimum

cost that places m orders is achieved by placing orders of equal size at equally spaced points

in time.

This implies that total purchasing and carrying cost per unit time associated with P is at least

mhDt

tKm

2+

Consequently, by selecting the value of m that minimizes this value we can construct a policy

of minimal cost. Let

KhDt2

=α

And thus the best value of m is either α or α , which yields smaller cost. So in this case of

finite horizon we place orders at regularly spaced intervals of time, and these orders are of

same size each time.

2.2 Varying demands The previous paragraphs have focused on situations where demand was both known in

advance and constant over time. This paragraph will work with retailers where demand is

known in advance, yet varies with time. This could happen, for example, if orders have been

placed in advance. In this case, the horizon is determined by the periods where demands are

known. Just as in the previous paragraphs we will be working with single item models.

2.2.1 Wagner-Within model The problem comes down to planning a sequence of orders over a T period planning horizon.

Thus, every period we should decide about the size of the order. The following assumptions

are made:

Demand during period t is known and is denoted dt>0

The per unit order cost is c and a fixed order cost K is incurred every time an order is

placed; that is, if y units are ordered, the order cost is cy+ Kδ(y)

The holding cost is h>0 per unit per period

9

Initial inventory is zero.

Lead-times are zero; that is, an order arrives as soon as it is placed

All ordering and demand occurs at the start of the period. Inventory is charged on the

amount on hand at the end of the period.

The optimal policy states how much to order in each period so that demands are met without

backlogging and the total cost, including the cost of ordering and holding inventory, is

minimized. This basic model was first analyzed by Wagner and Whitin (1958) and has now

been called the Wagner-Whitin model. The problem can be formulated as follows:

Problem WW: ( )[ ]∑=

+T

ttt hIyKMin

1δ

s.t.

tttt dyII −+= −1 , t=1,2,…,T.

TtyI

I

tt ,...,2,1,0,00

=≥=

With yt the amount to be ordered in period t, and It be the amount of product in inventory at

the end of period T. Wagner and Whiterin stated that any optimal policy is a zero-inventory

ordering policy, that is, a policy in which

.,...,2,1,01 TtIy tt ==−

Consequently, ordering only occurs when inventory is zero. This implies that in an optimal

policy an order is of size equal to satisfy demands for an integer number of subsequent

periods. Using this property, they developed a dynamic programming algorithm to determine

those periods when ordering takes place: by constructing a simple acyclic network with nodes

}1,...,2,1{ += TV , we can view the problem of determining a policy as a shortest path

problem. Formally, let lij, the length of arc (i,j) in this network, be the cost of ordering in

period i to satisfy the demands in periods i,i+1,….,j-1, for all 11 +≤≤≤ Tji .

That is,

∑−

=

−+=1

)(j

ikkij dikhKl

All other arcs have lij = ∞+ . The length of the shortest path from node 1 to node T+1 in this

acyclic network is the minimal cost of satisfying the demands for periods 1 through T. The

optimal policy, that is a specification of the periods in which an order is placed, can be easily

10

reconstructed from the shortest path itself. Again, by relaxing some assumptions, we could

make this model more realistic. For example, if we assume that lead-times are greater than

one and known in advance. In that case order required in period t would be ordered in period

t-L, with L the lead-time.

2.2.2 Models with capacity constraints An important generalization of the Wagner-Within model is the inclusion of upper bounds on

the amount that can be ordered in a given period. This implies adding the following

constraints to problem WW:

.,...,2,1, TtCy tt =≤

with tC 0≥ the maximum amount that can be ordered in period t.

This problem has been approached in different ways, including a dynamic programming

approach by and a branch and bound algorithm.

2.3 Stochastic inventory models Until now all the inventory models considered were deterministic in nature; demand is

assumed to be known. However, for many logistic systems, it is not possible to make this

assumption; typically, demand is a random variable whose distribution may be known.

2.3.1 Newsvendor problem This model only takes into account one period and is mostly used for companies that sell

seasonal products, for example a company in the retailing industry sells summer fashion

items. The decision about the ordering quantities has to be made a few months before the

‘selling’ season. Because they do not know how the response to there new products will be,

these decisions are mostly based on the sales of previous years, the current economic

conditions and professional judgement.

The demand D for each new product is randomly distributed, generated from a distribution

with continuous cumulative distribution function F(⋅). Because of the randomness of the

demand, the quantity of units to order is calculated by using the expected cost z(y), which is a

function of the amount purchased y:

11

z(y)= cy-rE[min(y,D)]-vE[max(0,y-D)] for ,0≥y

where c is the variable purchase cost, r the selling price per unit, v the salvage value per unit

and E() the expectation (with r>c>v). The objective is to choose y so that the expected cost

z(y) would be minimized, which implies that the optimal production quantity S should satisfy

Pr{D ≤ S}=vrcr

−− .

This problem is called the newsboy/newsvendor problem.

In this analysis we have used three assumptions:

No initial inventory

No fixed ordering cost

Backorders are not supplied

If we relax the first two assumptions, we observe that the expected cost of ordering (y-y0)

units is

)(0 yzcyK +−

with y0 the initial inventory and K the fixed ordering cost. Hence, S clearly minimizes this

expected cost if we decide to order. So, there are two cases to consider:

• If y0 S≥ , we should not produce anything

• If Sy ≤0 , the best we can do is to raise the inventory level to S. However, this is

optimal only if –cy0+z(y0), the cost associated with not producing anything, is larger

than or equals K-cy0+z(S), the cost associated with producing S-y0. That is , if y0<S, it

is optimal to produce S-y0 only if z(y0) ≥ K +z(S).

Assume s to be a number such that

z(s)=K+ z(S)

This implies that the optimal policy is as follows:

Order S-y0 if the initial inventory level y0 is at or below s, otherwise do not order.

This type of policy is called an (s,S) policy, with S the order-up-to level and s the reorder

point.

2.3.2 Finite horizon models Here we extend the previous model to a multi-period (T periods) inventory problem. At the

beginning of each period (e.g. month or week), the inventory level of a product is noted.

12

Consequently, it is possible to place on order to increase the inventory level to a certain level.

In this paragraph we make the following assumptions:

No lead-time

Demands for successive periods are independent and follow the same distribution

If the demand exceeds the inventory on hand, the ‘unsupplied’ demand is backlogged

and filled when more inventories becomes available. Backlogged units are thus

viewed as negative inventory.

Unfulfilled demand at the end of the final period can be backlogged at cost c

All remaining inventory left at the final period is salvaged, at the same price c

These assumptions make sure that the expected revenue per period is a constant. Therefore we

will not include this term in the following formulations any more. The objective is to find an

inventory policy that minimizes the cost over T periods. This policy (st,St) is similar to the

(s,S) policy described earlier, except that the two determining parameters may vary from

period to period. The costs consist of:

• Holding cost: there is a holding cost h+ for each unit of product in stock at the end of a

period

• Shortage cost of h- per unit when demand exceeds the inventory

• Ordering cost: a set-up cost K, paid every time the company places an order , and a

proportional purchase cost c

Let yt be the inventory level at the start of period t (before possible ordering). If the inventory

level after ordering is y, then the expected backlog penalty and holding cost for that period is

)()0,max()()0,max()( DdFyDhDdFDyhyGDD∫∫ −+−= −+ , (1)

which is called a one-period loss-function: this function is convex. Considering a policy

Y=(y1,y2,…,yT), where yt are the order-up-to levels (random variables) of period t, the sum of

the total expected proportional purchasing cost and salvage value ∑P becomes

−−−=∑ ∑

=

T

tT

Tt

t DycyycEP1

)()( ,

where Dt is the occurring demand in period t. Because tt

t Dyy −=+1 , the former equation

transforms to

[ ]DcTEP =∑

13

The latter expression shows that P∑ is independent of the used ordering policy, so we can

leave the linear ordering cost component out of the formulation. This observation is quite

intuitive, because all backlogged demand is filled at the end of the last period, while all

remaining inventory left at this period is salvaged, both at the same price c.

To obtain the optimal policy for this model, a dynamic programming formulation has been

developed. Therefore, we define the following two expected cost functions. Let Gt(yt) be the

expected cost for the remaining T-t+1 periods if we do not order in period t and act optimally

in the remaining T-t periods. Let zt(yt) be the minimal expected cost through the remaining T-

t+1 periods if we act optimally in period t and all the remaining T-t periods.

Then it follows that for t=1,…,T,

)()(z)G(y )(yGD

1ttt

t DdFDyt −+= ∫ + ,

tyytt Minyz ≥=)( {K )()( yGyy t

t +−δ }

Where )(xδ is 1 if x>0 and it is 0 otherwise. Note that if we order up to the level y>yt in

period t, the cost for the final T-t+1 periods is K+Gt(y). These two functions are not convex

and may even have many local minima. Veinott (1966) offered a proof for the optimality of

(st,St)-policies under the assumption that –G(y) is unimodal or G(y) is quasiconvex.

So far, we made the assumption that demands are identically distributed and the cost

parameters, c, h- and h+ are time-independent. These can be easily relaxed, and an (s,S) policy

will still be optimal.

2.3.3 Infinite horizon models

2.3.3.1 Description

We consider a discrete time inventory system in which an order can be placed with an outside

supplier at the beginning of each period. We make the assumption that one-period demands

are integer valued and as a consequence, the inventory levels are discrete. Let this discrete

distribution pj=Pr{D=j} (for j=0,1,2,…) be the one-period demand. The objective is to

minimize the long-run expected costs over an infinite horizon. All other assumptions and

notations are identical to those of the previous paragraphs.

Veinott has shown that if

-G(⋅) is unimodal and

Lim y−>∞ G(y) > min y G(y) + K

14

an (s,S) policy is optimal when minimizing long-run average costs over an infinite horizon.

From now on we assume these latter conditions. In the following paragraph we discuss an

algorithm for computing these (s,S) levels. First, however, we define the notation and give a

description of the model.

The following expression for the long-run average cost c(s,S) of a given (s,S)-policy is well

known and given by

c(s,S)=M(S-s)K + )()(1

0jSGjm

sS

j−∑

−−

=

(2)

where

m(0)=(1-p0)-1, M(0)=0 (3a)

m(j) = ( )ljmpj

ll −∑

=0, j = 1,2,… (3b)

and

M(j) = M(j-1) + m(j-1), j = 1,2,… (3c)

Let M(j) be the expected total time until the next order is placed when starting with an

inventory position of respectively s+j units. That is, M(j) is the expected number of periods

until total demand is no less than j units. Then m(j), equal to M(j+1)-M(j) corresponds to the

expected number of periods, prior to placing the next order, for which the inventory level is

exactly S-j. It is obvious that for all j we have

M(j)= [ ] ∑∑∞

+==

+−+10

)(1jk

k

j

kk pkjMp

= ( ) 10

+−∑∞

=

kjMpk

k ,

with M(j)=0 for j .0≤ Consequently,

M(j)= ( ) 11

0+−∑

−

=

kjMpj

kk . (4)

Let ),( ysK be the expected total cost in all periods until placing the next order, when we start

with y units of inventory. Every time we order, it is ‘delivered’ instantly, and the inventory

15

level increases to S. Thus, replenishment times can be viewed as regeneration points. The

theory of regenerative processes tells us that

)(

),(),(sSM

SskSsc−

= (5)

With c(s,S) the long run average cost, that equals the ratio of the expected cost between

successive regeneration points and the expected time between successive regeneration points.

Fix s. To find M(S-s) we can calculate the recursive equation (3). It has been shown that

k(s, ⋅) satisfies the follow discrete equation

k(s,y) = G(y) + K ∑∞

−= syjjp + ( )jyskp

sy

jj −∑

−−

=

,1

0 y>s

and with (1)

k(s,y) = K + ∑−−

=

−1

0)()(

sy

jjygjm y>s

Following from the formulas above, we can redefine k(s,S) as

),(),( SsHKSsk += ,

with H(s,y) the expected holding and shortage cost until placing the next order, when starting

with y units of inventory. The algorithm for computing the optimal (s,S)-levels is shown in

the next paragraph.

2.3.3.2 Algorithm

In this paragraph, we present an algorithm for computing the optimal (s,S)-levels, developed

by Federgruen and Zheng (1990). Earlier, these types of algorithms were considered to be

expensive. This came from the general perception that the policy cost function is, in general,

ill behaved, implying that all possible combinations {(s,S) with s<S} needed to be evaluated.

The first to eradicate this myth were Federgruen and Zipkin (1984): they used a method based

upon an adaptation of the general policy-iteration method for solving Markov-decision

problems, where the special structure of (s,S) policies was exploited in several ways. The

algorithm we are going to use, provides for an efficient search of the optimal reorder levels

and order-up-to levels s* and S* in the (s,S) plane itself. This search is based on a number of

properties of the cost function c(s,S) and tight lower and upper bounds for s* and S*. First we

explain the algorithm theoretically, later we show how this was implemented practically.

16

The lemmas and corollaries of Appendix A suggest the algorithm shown in Fig. 4. Let y* be

the minimum of the G(⋅) function.

Fig.4. Pseudo-code for the algorithm.

We will discuss the algorithm shortly. In Step 0, we start with an initial order-up-to level S0=

y*, with y* an arbitrary minimum of the G(⋅) function. Further on, we obtain a corresponding

optimal reorder level s0 by decreasing s from y* with step size 1 until c(s, S0) ≤ G(s). This

optimality follows from Corollary 1 in Appendix A. In step 1, we search for the smallest

value of S that is larger than S0, which improves on S0. S is incremented with step size 1. To

verify whether a certain value for S improves on S0, we make a single comparison of c(s0, S)

and c(s0, S0). If it does, S0 is updated to S and the new corresponding optimal reorder level s0

is acquired by incrementing the old value of s0 by units of one, as long as c(s, S0) ≤ G(s+1).

The existence of such a reorder level s0, its optimality (for S0) and the fact that s0< y*, are all

guaranteed by Lemma3b in Appendix B. Finally, note that whenever Step 1 is initiated, c0 is

the best available upperbound for c*. Following from Lemma 2c (Appendix B), the search for

an improving value of S may be terminated as soon as G(S)> c0. At the last iteration of the

algorithm, when S0= S* and s0= s* for some optimal policy (s*, S*), we have c0= c* and S0 ≤ −

S *. It follows that the test in the outer while-do loop of Step 1 fails when S:= −

S * + 1.

Consider a two-dimensional integer grid with s on the horizontal axis and S on the vertical

one. Then, every point (s,S) on the grid represents a policy. Denote s0 as the value of s

17

obtained at the end of Step 0. Then we can view Step 1 as going through a vertically-up and

horizontally right path starting from (s0, y*) and ending at (s*, −

S *); see Figure 5 and the

previous observations.

Fig.5. Path for searching the optimal (s,S)-strategy in the grid.

2.3.3.3 Implementation of the algorithm

In order to execute the algorithm, we make use of a tool developed in Excel (Fig.6). In the

following chapters we consider a (continuous) demand that is generated by a stochastic

process. The algorithm only works with discrete distributions; this is solved by generating the

demand by means of Monte Carlo simulation and using those results in a histogram. Recall

that the algorithm uses the G(⋅) and the c(s,S) functions, defined in equation (1) and (5). We

obtain the values of G(⋅) by executing the VBA-code exhibited in Appendix B: this function

describes the expected holding and shorting cost for one period. Furthermore, we have to

calculate the functions m(j) and M(j) as well: these are functions of the discrete demand

probabilities that we generated earlier (Appendix B).

18

Fig.6. Tool in excel that calculates the optimal (s,S)-level. Having evaluated the previous functions, it is easy to calculate the k(s,⋅) function.

Consequently, we can calculate the long run average cost function c as well. The optimal

combination of reorder-point s and order-level S can then be determined by means of the

algorithm: the implementation in VBA is exhibited in Table 1. Finally, this algorithm yields

the combination of s and S with the minimum average cost c(s,S).

Generating demand

Histogram

G(y),m(j) and M(j)

s, S and c(s,S)

Fixed ordering cost

19

Table 1. Implementation of the algorithm to search the optimal (s,S)-values in VBA

Sub orderlevels() orderpoint = Range("yster").Value orderlevel = Range("yster").Value Do orderpoint = orderpoint - 1 bestcost = cost(orderpoint, orderlevel) Loop Until bestcost <= Range("Loss_function")(orderpoint + 1, 1).Value S = orderlevel + 1 Do While Range("Loss_function")(S + 1, 1).Value <= bestcost If cost(orderpoint, S) < bestcost Then orderlevel = S newcost = cost(orderpoint, S) Do While newcost <= Range("Loss_function")(orderpoint + 2, 1).Value orderpoint = orderpoint + 1 newcost = cost(orderpoint, S) Loop bestcost = newcost End If S = S + 1 Loop Range("orderpoint") = orderpoint Range("orderlevel") = orderlevel End Sub

20

3 IRP Model 3.1 Introduction

The Inventory Routing Problem is concerned with the distribution of a single product from

one facility (e.g., a warehouse) to a set of n customers over a certain planning horizon. These

retailers consume the product at a given rate and are served by a fleet of homogeneous

vehicles with restricted capacity. The objective is to determine a distribution plan that

minimizes fleet operating and average total distribution and inventory holding cost with

reducing the chance of a stock-out at any of the sales-points, during the planning horizon. It

implies coordinating inventory replenishment policies and distribution plans in a cost

effective manner. More precisely, it integrates inventory and distribution aspects in the same

planning process.

Aghezzaf et al. (2005) formulated a new model for the long-term IRP when demand rates are

stable and EOQ-like policies are used by the retailers. In this model, multi-tours were used:

this means that a vehicle’s travel plan can contain more than one tour. To solve this model, a

column generation based approximation method was suggested. The resulting sub-problems

were solved using a savings-based approximation method. In Chapter 5 we investigate how

the model performs if we include retailers with stochastic demand. Later on, suggestions will

be made to be able to cope better with this type of retailers in this model.

3.2 A brief survey of solution approaches for the IRP This sub-section is dedicated to a short survey of relevant models and solution approaches

that have been used in the research for IRP problems. We can distinguish IRP models and

their approaches by virtue of the inventory policies used at the sales-points, the service level

restrictions, and the time horizon considered. A popular classification is the following:

Single-period models with stochastic demand

Multi-period models with deterministic demand

Infinite-horizon models with deterministic demand

An example of the first type can be found in Federgruen and Zipkin (1984). They were among

the first to do research on integration of inventory management and routing problems.

Consider a plant with a limited amount of available inventory serving a set of retailers with

21

stochastic demand. The objective was to allocate the available inventory at the warehouse in

such a way that total transportation, inventory and shortage costs at the end of that period

were minimized. They solved this problem, which they modelled as a nonlinear mixed integer

program, with an approximation method. Their approach starts with an initial assignment of

sales-points to routes. From this moment, the problem can be split into an inventory allocation

problem (that determines inventory and shortage costs) and into a Travelling Salesman

Problem (TSP) for each vehicle (that determines transportation costs). This feasible solution

is then iteratively improved by exchanging customers between routes.

The models corresponding to the second type are the closest to the one we describe in the

following paragraphs (Aghezzaf et al. 2005) and that will be tested in the following chapters,

in terms of planning horizon and assumption on demand rates. For instance, Larson (1988)

distinguishes two Inventory/Routing problems: the strategic and tactical IRP. The strategic

IRP estimates the long-term minimum size of the vehicle fleet required to serve the sales-

points. The tactical IRP deals with routing an existing vehicle fleet to serve sales-points

whose actual demands for replenishment could be estimated. The model assigns sales-points

to specific clusters and assumes that all replenishments are made on a single route visiting all

sales-points in a cluster. Consequently, some sales-points are visited more frequently then

required. This inefficiency was corrected by Webb and Larson (1995), using a period/phase

approach.

Finally, in the infinite-horizon models with deterministic demand the product is absorbed by

each retailer at a given constant rate and the problem is to determine an infinite-horizon

shipping policy that minimizes the sum of inventory and vehicle-routing costs. An example of

such type of model can be found in Anily and Federgruen (1990). The routing patterns are

determined using a partitioning scheme of the retailers. The objective is to divide the sales-

points within a partition into regions so as to make demand of each region to match the

vehicle capacity. In the same line of thought, Gallego and Simchi-Levi analyzed a direct

shipping warehouse/ multi-sales-point distribution system with constant and deterministic

demand: they concluded that direct shipping becomes a bad policy when many customers

require significantly less than a truckload. This is quite intuitive.

The model proposed in Aghezzaf et al. (2005) and described more detailed in the next

paragraphs, is a more effective model for the problem in which the concept of ‘multi-tours’ is

introduced. This leads to drastic diminution of fleet fixed costs and size .

22

3.3 Model for long-term IRP In this paragraph we describe the model formulated by Aghezzaf et al. (2005). First we

explain the concept of multi-tours, then we introduce the mixed integer formulation and the

solution approach and finally we discuss the performance analysis.

The model creates a cyclical distribution plan of a single product distributed from a single

distribution center r to a set of retailers (sale-points) S. These sales-points are served by an

assigned vehicle and in such a way stock-outs are being avoided. Each sales-point i∈S

consumes the product at a known rate of di (in units/hour) and utilizes an EOQ-like policy. It

is assumed that these retailers can maintain a local inventory. For the distribution of the

product to the different clients, a fleet of homogeneous vehicles V, with capacityκ , is being

used. The objective is minimizing the expected distribution and inventory costs during the

planning horizon, avoiding stock-outs at any of the retailers.

In a later stage of the thesis, we will incorporate sales-points with stochastic demand in the

model; i.e. retailers for which the order-up-to strategy would be the optimal inventory

strategy. To be able to execute the model when using these types of retailers, we use the

average of their demand distributions. Because the sales-points with order-up-to policies do

not have stable demand rates, stock-outs can occur. The purpose of the following chapters is

to look how this affects inventory and distribution costs, if we leave the model unmodified.

This will be monitored and evaluated by means of simulation. The final step then is to

propose a new strategy, that can cope better with retailers with different inventory policies,

i.e. EOQ and Order-up-to.

3.3.1 Multi-tours The concept of vehicle multi-tours implies that a vehicle’s travel plan can contain more than

one tour; the vehicle leaves the distribution centre to replenish a set of sales-points in a first

tour, returns to the distribution centre to be reloaded and then leaves again to replenish a

different set of sales-points in a different tour.

In this model vehicles are considered to have a limited capacity К and to be travelling at an

average speed v. Each of them replenishes a set of sales-points C in a cyclical manner. If the

vehicle only makes one tour through the distribution centre and his set of sales-points, then

the most cost-effective way to supply these sales-points, in terms of transportation costs, is to

travel along the TSP-tour (Travelling Salesman Problem). The cycle time T(C), the time

23

between two consecutive iterations of the tour, is bounded from below by the travel time of

the TSP-tour (Tmin) and bounded upwards by the limited vehicle capacity (Tmax). To avoid

stock-outs at the retailers, the vehicle should deliver qi=di⋅t* if T(C) equals ∗t . However,

because of the limited capacity of each vehicle, the load ∑∈Ci

iq can be at most the capacityκ .

Hence, the maximal cycle time equals∑∈

=

Ciid

T κmax .

The condition for a tour to be feasible is that )()( maxmin CTCT ≤ . Another possibility would be

replenishing the set of sales-points C by making multiple tour through disjoint subsets C1,

C2,…, Cm. If so, )(...1

minmin imi

CTT ∑=

= and { })(min)( max....1max imi cTCT == .

Fig.7. Example with seven sales-points.

The following example (see Fig.7) shows how savings, in term of fleet utilization, can be

achieved when working with multi-tours instead of single tours. The logistic system consists

of seven retailers, with their demand rate shown in the figure, served by vehicles with a

capacity of 100 ton. The total demand rate equals 10 ton/hour. The TSP-tour through the

seven sales-points and the DC (Fig.8(a)) gives an infeasible solution: the minimal cycle time

is 11.5 hours, while the maximal cycle time is smaller, namely 10 hours (100/10). A feasible

solution can be found if we allow a vehicle to make multiple tours. The multi-tour shown in

Fig.8 (b) has two tours, where the left tour has Tmin= 8 hours and Tmax=16.67 hours while the

right tour has Tmin= 4 hours and Tmax=25 hours. Consequently the minimal cycle time of the

multi-tour is 4 + 8 = 12hrs and the maximal cycle time is min(16.67,25)=16.67hrs. This

means that this multi-tour is feasible and only one vehicle is needed.

24

.

Fig.8. An infeasible tour and a feasible multi-tour.

3.3.2 Mixed integer formulation of the problem and solution approach In order to define the objective function to be minimized, we consider four cost components

for each multi-tour. The first cost component is the fixed operation cost of the vehicle ψ (in

€/hr). A second cost component is the transportation cost. Let δ be the travel cost per km.

Then, the transportation cost can be expressed as δ ⋅ Tmin⋅ v per iteration of the multi-tour,

because the vehicle travels Tmin(C) hours per cycle at a velocity of v km/hr. The third cost

component is the delivery handling cost at the retailers, defined by ∑∈Ci

iϕ per cycle, with iϕ

the cost per delivery at sales-point i. The last component consists of the stock holding cost:

the quantity delivered at every sales-point i is given by qi=di⋅T(C), and is just enough to cover

demand until the next delivery. Thus, stock holding cost per cycle at sales-point i is given by

ηi ⋅ qi ⋅T(C)/2, with ηi the holding cost per ton per hour. For the total stock holding cost per

multi-tour we have to add the costs of the different retailers. In order to calculate cost rates

per hour we have to divide the cost rates per multi-tour by T(C).

The cost rate of a multi-tour is then

Because the multi-tour cost rate varies with its cycle time, there exists a theoretical optimal

cycle time for which cost rate is minimized. This is an extension of the EOQ-model, as

described in Chapter 2. Just as in this model we can obtain the optimal policy (cycle time) by

balancing the holding costs with the sum of delivery handling costs at the sales-points and

transportation costs. This optimal cycle time looks like this:

25

If this theoretical optimal cycle time turns out to be greater than the maximal cycle time or

smaller than the minimal cycle time, it is not actually feasible. In this case, the effective

optimal cycle time has to be chosen as close as possible to the theoretical one, respecting the

boundaries set by the minimal an maximal cycle time. Otherwise, if the theoretical optimal

value falls between these boundaries, it is chosen as the actual optimal cycle time.

We present the assumptions made when developing the model:

The supplier has an infinite stock

Time for loading and unloading vehicles relatively small and is not taken into account

(no cost attached)

Inventory capacity at the sales-point is large enough

Transportation costs are proportional to travel times

The length of a trip from sales point i ∈S+ }{rS ∪= to sales-point J∈S+ will be denoted by tij

(in kilometres). The model variables are shown in Fig.9 and the nonlinear mixed formulation

in Fig.10.

Fig.9. The model variables.

For a more detailed description of the constraints, see Appendix C. To solve the problem in

Fig.10, an approximation algorithm, based on a column generation process was proposed. The

multi-tour generater used in this method is an extension of the savings-based heuristic that is

used to solve the VRP.

3.3.3 Performance analysis They observed that their model leads to drastic savings in terms of fleet operating costs (up to

50%), since the model reduces the size of required fleet to serve all retailers. In delivery costs

rates, important average savings were realized as well. In transportation costs, however, only

26

small average savings could be noted. Surprisingly, the number of sales-points and the vehicle

capacity did not have an important impact on average savings. The geographical distribution

however, did: if the retailers were clustered, the savings increased.

Fig.10. The initial nonlinear mixed integer formulation of the IRP.

27

4 Simulation approach

4.1 Introduction As in the previous chapter, we consider a distribution problem in which a warehouse (with

infinite stock) distributes a single product to several retailers: the deliveries are performed by

a fleet of homogeneous vehicles with limited capacity. The model developed by Aghezzaf et

al. (2005) assumed known and deterministic demand rates for every sales-point, using EOQ-

like policies. This time however we will also encounter sales-points, for which the optimal

inventory lies within the class of the (s,S) policies.

One of the objectives of this thesis is to analyze the performance of the IRP model when

implementing these types of sales-points. We use two critical factors to test the effectiveness

of the model. The first one reflects the impact of the percentage of sales-points experiencing

stochastic demand on the different types of cost rates. The percentage of order-up-to retailers

examined varies from 0% to 50% : it is quite intuitive to see that if more then half of the

retailers implement order-up-to strategies, the model, designed for EOQ-policies, would not

be appropriate at all. The other factor reflects the influence of the number of sales-points on

the cost rate. We examined nine problems, with size varying from small (30) to large (120).

The simulations and their analysis are elaborated in Chapter 5.

The following is an outline of the remainder of this chapter. In the second paragraph, we

describe how to define the input data for the IRP model. Paragraph 3 discusses the output of

the model. Paragraph 4 gives a description of the different performance measurements to be

evaluated during simulation. Finally, in paragraph 5, we give more insight into the tools used

during the execution of the simulations.

4.2 Input of the IRP model When running the IRP model in order to obtain the optimal distribution plan, the data of the

problem considered is inserted in the model by means of a dat-file: Table 2 shows an extract

of such a file. The data of all the problems examined can be found in Appendix D.

28

Table 2. Extract of a dat file Capacity Speed FixedCost VarCost Vehicle

50 50 30 1

Depot X Y delTime delCost

DC 150 150 0.5 20

Name X Y demRate delTime delCost holdCost

1 131 127 0.375 0.25 10 0.1

2 53 105 0.325 0.25 10 0.1

3 209 234 1 0.25 10 0.1

4 214 107 1.175 0.25 10 0.1

5 227 136 1.125 0.25 10 0.1

6 125 213 0.9 0.25 10 0.1

7 136 63 0.075 0.25 10 0.1

8 183 7 0.65 0.25 10 0.1

9 113 139 1.125 0.25 10 0.1

In this table we define the following parameters:

Capacity: the limited capacity of the vehicles serving the sales-points (in tons)

Speed: the average value of the vehicle speed (in km/h)

FixedCost: the fixed operational cost of a vehicle (in €/hr)

Varcost: the travel cost per kilometre of a vehicle (in €/km)

DC: this represents the warehouse

X,Y: these are the coordinates of the facility; distances between facilities are

calculated with the Euclidean distance formula

delTime: for DC and retailers this is respectively the loading and unloading time of a

truck

delCost: the cost associated with loading or unloading a vehicle ((in €/hr)

holdCost: the cost of holding one ton of the product in inventory for one hr, expressed

in €/hr⋅ton

demRate: the demand rate at each retailer (in tons/hr)

29

As we see, the demand occurring at each retailer is quantified by an average demand rate. The

following subsections will describe how this rate is defined for the different types of retailers.

4.2.1 Retailers with deterministic demand (EOQ) The sales-points using an EOQ-like inventory policy have a demand that resembles a normal

distribution (µ ,σ) with a small variance. Because the variability is very low, we consider the

demand to be stable (in time). Consequently, the average value of is inserted into the model.

4.2.2 Retailers with stochastic demand (Order-Up-To) Sales-points implementing order-up-to policies have a more unstable demand. In this thesis

we will model the weekly demand as a stochastic process. This process is depicted below

(Fig.11). It starts with generating a random number R between 0 and 1. If this number is

below a threshold α, the demand is generated from a first normal distribution with small

variance. If R exceeds the threshold, the demand follows a second normal distribution with

small variance.

Fig.11. The stochastic process generating the weekly demand.

When we consider a retailer to be experiencing a demand as described above, we assume its

average demand rate per hour to be as listed in the dat-file. Based on this average demand

rate, we develop a stochastic process that generates the weekly demand.

This type of process could be a reflection of the impact of certain (weekly-dependent) factors

on the demand. For instance, the effect of the weather forecast on the bookings for a beach-

sided hotel, and consequently on the client-related demands of the following week. Starting

from the average demand rate per hour, we can compute the average demand rate per week by

multiplying it by forty: we consider a working week of five days (eight hours each). In order

Random number R ∈ [0,1]

R<α: N(µ1 ,σ1)

R> α: N(µ2 ,σ2)

30

to obtain µ1, we multiply the average week demand for instance by 0.7 and assign the integer

of the result to µ1. Let Z be the integer value of the double of the average weekly demand.

The average of the second distribution, is calculated by subtracting the µ1−value from Z.

Considering the variances of both normal distributions N1 and N2 to be one and the threshold

to be 0.5, it is easy to see that the average demand (per hour) generated by this stochastic

process is approximately the same as the demand rate listed in the data-file. This is shown by

the following example. Check the demand rate (per hour) of the first retailer in Table 2: 0.375

ton/hour. The average demand per week becomes 40⋅0.375 = 15 tons/week. The average µ1 of

the first normal distribution is 11 (15⋅0.7 = 10.5) and µ2 equals 25 ( 2⋅15-11=19). The

stochastic process above, with N1(11,1) and N2(19,1), yields a demand probability as in

Fig.12.

Distribution of the stochastic process

0

0,05

0,1

0,15

0,2

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

Demand

Probability

Fig.12. Demand probability generated by the stochastic process.

This figure was obtained by generating weekly demands using the Monte-Carlo method in

Excel. Notice that is very important to determine whether the optimal inventory policy for

such demand is indeed a (s,S)-policy, so that the algorithm covered in subsection 1.3.3.2. can

be used. Veinott (1966) showed that an order-up-to policy is optimal when minimizing long-

run average costs if

-G(⋅) is unimodal

Lim y−>∞ G(y) > min y G(y) + K

with K the fixed ordering cost. Recall that G(y) represents the one-period loss function, the

expected shortage penalty cost and holding cost for one period, when starting with an

inventory y:

31

)()0,max()()0,max()( DdFyDhDdFDyhyGDD∫∫ −+−= −+ ,

with +h holding cost and −h the shortage penalty cost, as we do not consider backlogs.

Denote that y is the inventory after ordering, which happens at the beginning of every period

(every week). We assume that there is no lead-time. This is acceptable: when this retailer

would be directly served by the warehouse, using an order-up-to policy, the travelling time

between warehouse and sales-point is negligible in comparison to the length of the period. In

the IRP model from the previous chapter there is no such thing as a fixed ordering cost.

However, we could define the fixed ordering cost as the cost (vehicle costs, loading costs and

delivery costs) accrued when going from the DC to that retailer and back. Now we can check

the example from above on the optimality of (s,S) strategies.

The holding cost can easily be found in Table 2: €0.1/hr⋅ton. We assume a shortage penalty

cost of 10 euro per unit of demand lost because of insufficient stock. The formula for

calculating G(⋅), starting from the probability function generated by the stochastic process, is

shown in Appendix B. The loss function for this sales-point is shown in Fig.13.

Loss function

020406080

100120140160

0 20 40 60 80 100

120

140

160

180

200

220

240

260

280

300

320

340

360

380

inventory position

Loss function

Fig.13. The loss function for a retailer with demand generated by a stochastic process.

The definition of a unimodal function states that there is only a single local maximum: we can

see on Fig.13 that –G(⋅) is indeed such a function. So hereby we fulfil the first condition

drafted by Veinott. Now we try to evaluate the second condition, that is Lim y−>∞ G(y) >

min y G(y) + K. The minimum of the loss function is obtained from the function above and

yields 0.6533€ for y=21. In order to get the fixed order cost, as defined above, we need to

calculate the Euclidean distance between the DC and the retailer. Using the coordinates of

both facilities (see Table 2), we get

32

833.29)127150()131150( 22 =−+−=Dist km

The vehicle drives at 50km/hr, so with a variable transportation cost of €1/km, the total

travelling cost becomes €59.7 (=2 ⋅ 29.833 km ⋅ €1/km). The loading cost at the DC equals

€10 (DeltimeDC ⋅ DelcostDC = 0.5⋅20) and the delivery (unloading) cost at the retailer equals

€2.5 (Deltimeretailer ⋅ Delcostretailer = 0.25⋅10). So, the fixed order cost is the sum of the three

costs calculated above: K equals €90.1 (59.7 + 10 + 2.5 = 72.2). The loss function generated

in excel shows us that for y=1995, G(y) equals 198€ . This is already much more then 72.85€

(=0.65 + 72.2). Because from y=27 (minimum) onwards, the slope of the loss function

remains positive (unimodal function), we can safely conclude that we have fulfilled the

second condition as well. So this type of demand, generated by our stochastic process, has an

order-up-to strategy as an optimal policy.

4.3 Output of the IRP model The IRP model returns the optimal routing by means of a txt-file. For each vehicle, the

different sub-tours are exhibited, as well as the different types of costs defined in Chapter 3.

However, one extra type of costs is considered: the loading cost at the warehouse. We can

define this cost as a handling cost. It is important to inform you that the model used to

calculate the optimal distribution plan is a slightly more extensive (and more advanced)

version of the one published by Aghezzaf et al.: besides considering this extra cost type, only

cycle-times (of the multi-tours) multiples of eight hours are calculated. Just as we did in the

previous paragraph, we assume working days of eight hours. This implies that every multi-

tour executed by the vehicle is completed after a round number of days. Table 3 shows the

data that is returned for every vehicle used in the optimal routing scheme.

Each of the first eight lines represents a sub-tour: the travel time shown at the right side

consists of the driving time of the vehicle plus the time occupied by handling loads (loading

and unloading the vehicle). Consequently, the total travel time is the sum of the travel time of

the different sub-tours. This is the output of the model for the data-file of which Table 2 was

an extract. The maximum cycle time denotes the upper bound of the cycle time, imposed by

the limited capacity of the vehicle: ∑∈

=

Ciid

T κmax ,

33

with κ the capacity of the vehicle, di the average demand rate of retailer i and C the subset of

retailers served by this vehicle. The EOQ-time equals the variable Tν defined in Fig.9 in the

previous chapter. As we mentioned before, the cycle is always a multiple of eight hours. The

idle time is the percentage of time in one cycle, in which the vehicle ‘waits’ at the DC,

without loading any products. The capacity utilization reflects the average occupation of the

vehicle capacity, for the different sub-tours. The transportation cost defines the cost rate

incurred by the variable transportation cost (without considering handling costs).

Table 3. Data returned from IRP model for every vehicle

The loading and drop cost rates exhibit the cost rate induced by respectively loading the truck

at the warehouse and unloading it at the sales-point. Finally, the cost rate per vehicle is the

sum of the cost rates described above and the fixed vehicle operation cost rate. Table 4 depicts

the total cost rates, considering all vehicles: these are the cost rates for the whole distribution

plan.

DC -> 21 -> 20 -> DC. (Time = 3.82643) DC -> 29 -> 9 -> DC. (Time = 3.10819) DC -> 2 -> 22 -> 30 -> DC. (Time = 1.82054) DC -> 8 -> 25 -> DC. (Time = 3.47344) DC -> 17 -> 24 -> DC. (Time = 4.12749) DC -> 14 -> DC. (Time = 1.82835) DC -> 1 -> 16 -> DC. (Time = 1.12555) DC -> 27 -> 5 -> DC. (Time = 0.54) Travel time: 23.85 Minimal cycle time: 0 Maximal cycle time: 32.7869 EOQ time: 45.1806 Cycle time: 32 Idle time: 25.4688% Capacity utilization: 87.2% Transportation cost: 24.7656 Fixed operating cost: 30 Loading cost: 5 Drop cost: 5 Hold cost: 17.44 Total cost rate per vehicle: 82.2056

34

Table 4. Total cost rates

The model returns as well, for every vehicle, a list with the order of the facilities that the

vehicle visits on his multi-tour, accompanied by the facilities’ coordinates. Furthermore,

every sub-tour is dedicated to a particular day of the multi-tour: these sub-tours are calculated

with the requirement that they should be completed in less than eight hours. This is another

constraint that was not implemented in the model by Aghezzaf et al. An example of those data

is shown in Table 4.

We can easily visualize the track covered by a vehicle by using the first part of the data ( in

Table 5 ) and inserting it into a scatterplot in Excel (Fig.14). Notice that the diamond-shaped

points represent the different retailers served by the vehicle, and the centre point being the

warehouse.

Track of a vehicle covering its multi-tour

020406080

100120140160180

0 50 100 150 200

X coordinate

Y c

oord

inat

Track of a vehicle

Fig.14. Track of a vehicle covering its multi-tour.

Total cost: 151.771 (2) Fixed Vehicle costs: 60 Transportation costs: 43.7377 Loading costs: 7.91667 Delivery costs: 7.91667 Holding costs: 32.2

35

Table 5 .Order in which the facilities are visited by a single vehicle and the assignment of the various sub-tours to different days

DC 86 86 21 148 123 20 160 114 DC 86 86 29 121 31 9 119 35 DC 86 86 2 80 81 22 63 78 30 62 82 DC 86 86 8 85 160 25 89 94 DC 86 86 17 17 99 24 22 58 DC 86 86 14 47 80 DC 86 86 1 96 74 16 90 82 DC 86 86 27 86 87 5 86 86 DC 86 86 Tour 1 assigned to day 2 Tour 2 assigned to day 2 Tour 3 assigned to day 3 Tour 4 assigned to day 3 Tour 5 assigned to day 1 Tour 6 assigned to day 1

36

4.4 Performance measurements In this paragraph, we describe the different performance measurements that need to be

evaluated during simulation.

4.4.1 Cost rates It is obvious that we need to monitor the different types of cost rates in order to evaluate the

performance of the IRP model: these cost rates are the ones described in the previous

paragraph and the shortage penalty cost. Notice that when changing the optimal order policy

of an amount of retailers from EOQ to order-up-to policies, which implies changing the

constant demand rate per hour to a stochastic process that generates the weekly demand, that

all handling costs (loading and deliveries), transportation costs and fixed vehicle operation

costs per multi-tour stay the same: all vehicles still execute their same multi-tour. For these

values we can rely on the output of the IRP model. Nevertheless, the holding cost and the

shortage penalty cost will change. Every sales-point with an EOQ-like policy has a shortage

penalty cost of zero because no stockouts can occur as a consequence of the constant and

deterministic demand; when implementing an order-up-to strategy, demand is stochastic, so

stockouts become possible. Thus, the only cost rates that need monitoring during simulation

are the shortage penalty cost and the holding cost.

4.4.2 Service levels The service level can be defined as a measure for the possiblity to satisfy the customers’

demand. Many client-orientated companies try to achieve a 100% service level. For a retailer

using an EOQ-like strategy, this will be obtained. For the sales-points with order-up-to

strategy however, where stockouts can occur, the service level is very likely to drop below

100%.

We can define different types of service levels, depending on the nature of the criterium on

which it is based:

The stockout probability

The percentage of fulfilled demand (fill rate)

37

4.4.2.1 Service level based on stockout probability in a cycle (P1)

This service level is the chance of not having a stockout during a cycle (between two

replenishments). It is the complement of the probability of a stockout occuring: P1= 1-

P(Demand during cycletime > Cycletime ⋅ Average demand rate per hour). Because of the

complexity of the stochastic process and the fact that this process is executed only once a

week, it is to difficult to calculate the expected value of P1 theoretically. Therefore, we will

examine this value by means of simulation.

4.4.2.2 Service level based on stockout probability in a cycle (P2)

In this case we consider the service level based on the fill rate. This service level is the

percentage of the total demand that could be fulfilled. Because we don’t consider backorders

this is the same as the total demand served immediately. It equals the number of tons supplied

at the customers divided by the total demand during a certain time periode. The same

observation for the expected value counts for P2.

4.4.3 Average time between stockout We have defined breach as the average time between stock-outs.

4.5 Simulation method In this paragraph we give an overview of the tools used for the simulation of the various

problems considered in the next chapter. Figure 15 shows the interface of the Excel file that

was constructed in order to be able to execute the simulations as fast and effective as possible.

The following subsections are dedicated to the description of the various parts of the Excel

file.

4.5.1 Parameters In this part we insert the non-retailer-dependent data, concerning cost rates and

delivery/loading times, obtained from the dat-file (see Table 1). Specific parameters for the

simulation are listed as well: the percentage of order-up-to retailers and the number of

simulation hours. (Table 6)

38

Fig.15. Simulation sheet.

4.5.2 Optimal distribution plan generated by the IRP model Based on the output of the IRP model (Table 3), we construct a tabel containing the data of

the multi-tour of every vehicle. This is done manually. See Table 7.

4.5.3 Retailer-dependent properties In this table we show the salespoint-related properties (Table 8). The second column shows us

the average demand rate per hour: In order to acquire this data from the dat-file we had to use

a converting function, due to the fact that the integer and decimal values of the demand rate in

the dat-file are separated by a comma. So the objective of this converting function is to

remove the comma, replacing it by a point, in order to make it possible for Excel to make

calculations with these values. The formulation of the function in VBA can be found in

Appendix E. In the third column we show the number of the vehicle that serves this retailer:

this is assigned by a function that, if the number of the retailer is given, runs through the

different sub-tours of every vehicle (Table 7) and if it encounters this number in a certain sub-

tour, it will assign the retailer to the corresponding vehicle. Again, this function can be found

in Appendix E. The fourth column, which states µ1, decides which type of demand we assume

Parameters Distribution plan according to IRP model

Results of the simulation for the whole logistic system

Properties of the retailers

39

that retailer to experience. If the value is zero, we assume that the optimal inventory policy

for that retailer is EOQ and that the demand per hour is constant and deterministic, equalling

the demand rate from column 2. However, is this value positive, then we presume the optimal

strategy for that retailer to be an (s,S)-policy, with a weekly demand generated by a stochastic

process as in 4.2.2. . The average and variance of the two normal distributions in the process

are calculated by the function in Table 9, resulting in an array consisting of respectively

µ1 , σ1, µ2 and σ2.

Table 6. Parameters for the simulation

Table 7. Data of the multi-tour of every vehicle

40

In this table we show the salespoint-related properties (Table 8). The second column shows us

the average demand rate per hour: In order to acquire this data from the dat-file we had to use

a converting function, due to the fact that the integer and decimal values of the demand rate in

the dat-file are separated by a comma. So the objective of this converting function is to

remove the comma, replacing it by a point, in order to make it possible for Excel to make

calculations with these values. The formulation of the function in VBA can be found in

Appendix E.

In the third column we show the number of the vehicle that serves this retailer: this is

assigned by a function that, if the number of the retailer is given, runs through the different

sub-tours of every vehicle (Table 7) and if it encounters this number in a certain sub-tour, it

will assign the retailer to the corresponding vehicle. Again, this function can be found in

Appendix E. The fourth column, which states µ1, decides which type of demand we assume

that retailer to experience. If the value is zero, we assume that the optimal inventory policy for

that retailer is EOQ and that the demand per hour is constant and deterministic, equalling the

demand rate from column 2. However, is this value positive, then we presume the optimal

strategy for that retailer to be an (s,S)-policy, with a weekly demand generated by a stochastic

process as in 4.2.2. . The average and variance of the two normal distributions in the process

are calculated by the function in Table 9, resulting in an array consisting of respectively

µ1 , σ1, µ2 and σ2.

Table 8. Retailer-dependent properties

41

Table 9. The VBA formulation to determine the parameters of the two normal distributions in the stochastic process

Consequently, we can assume a retailer to be experiencing stochastic demand, just by

calculating the array from Table 8 and inserting its values into the corresponding columns. If

we do not do this, the value in the column of µ1 stays zero, implying that we consider an

EOQ-strategy.

Then for every retailer utilizing an EOQ-policy, we calculate the holding cost rate (column

holdingeoq in Table 8). The holding cost rate (per hour) of retailer i is easily obtained by the

welknown formula:

Holding cost rate= ηi ⋅ Di ⋅T(C)/2

with ηi the holding cost (see Table 6), Di the average demand rate of the retailer and T(C)

being the cycle time (Table 7) of the multi-tour covered by the vehicle serving this sales-

point. In the column holdingoup of Table 8 we calculate the holding cost for the sales-points

that utilize order-up-to strategies. This is computed by the function depicted in Table 10.

Looking at this VBA expression, we can see that every week there is a weekly demand

generated by a stochastic process demandoup: the VBA expression is listed in Table 11. For

the following week, this demand is then equally spread over 40 hrs (we assumed a working

week of five days of eight hours). In the function exhibited in Table 11, we assume that the

parameters for the two distributions of the stochastic processes are already calculated. The

penalty shortage cost is the cost incurred when demand can not be satisfied because of

demrate = Demand_rate(num) # this is the average demand rate/hour demrateoupweek = 40 * demrate # the average demand per week demrateoupweekint = CInt(demrateoupweek) # making the value integer rate1 = demrateoupweekint * 70 / 100 # multiplying by 70 rate1int = CInt(rate1) # making the value integer, obtaining µ2 rate2int = 2 * demrateoupweekint - rate1 # calculating µ2 in order to make the average demand generated approxima- demarray(0) = rate1int tely equal to the average demand from demarray(1) = 1 the dat-file demarray(2) = rate2int demarray(3) = 1

42

insufficient stock. This can only occur for facilities using the order-up-to policy. The VBA

expression to calculate this cost rate is very similar to the one for calculating the holding cost

rate. In fact, just by adding two equations, we can use the expression in Table 10 for

calculating penalty shortage costs too. The variables concerning the penalty cost are written in

bold letters. For retailers with the EOQ-policy, the both types of service levels P1 and P2 are

equal to 100%. We can already expect that the sales-points with stochastic demand (order-up-

to) will not be performing that well. Again, by adding some equations to Table 10, we can

easily calculate these service levels: this time we use underlined letters for the variables

concerning P1 and P2. Finally we calculate the average number of time between stock-outs;

this function is listed in Appendix E.

4.5.4 Results for the whole logistic system Summarizing the results obtained from simulation we construct a table (Table 12) consisting

of the different cost rates and service levels for the whole logistic system. Recall that only the

penalty and holding costs are a result of the simulation: the other costs are obtained from the

output of the IRP model. P2total denotes the total fill rate for all retailers and P1oup shows

the average servicelevel P1 for the retailers with stochastic demand. Finally, breaches

represents the average time between stock-outs for retailers with stochastic demand.

43

Table 10. The VBA function calculating the cost rates and service levels for retailers with an

(s,S)policy

delivery = cycleT * demrate # delivery at retailer every cycle voorraad = delivery # defining the starting inventory level demandweek = demandoup(retailer) # generating weekly demand by stochastic process demand = demandweek / 40 # calculating demands per hour from weekly demand For i = 1 To numberhoursim rest = i Mod cycleT If rest = 0 Then # after each cycle, replenishment by the stock = stock + delivery vehicle numbreaches= numbreaches + breach breach=0 numbercycle=numbercycle +1 End if stock = stock – demand #adapting the stock (after demand) Totaldemand= Totaldemand + demand If stock >= 0 Then stockage = stockage + stock Else voorraad = 0 # there is no backlogging shortage=shortage - stock breach =1 End If rest = i Mod cycleT If rest = 0 Then # after each cycle, replenishment by the stock = stock + delivery vehicle numbreaches= numbreaches + breach breach=0 numbercycle=numbercycle End If rest2 = i Mod 40 # every week new weekly demand If rest2 = 0 Then by stochastic process demandweek = demandoup(retailer) demand = demandweek / 40 # that week, the weekly demand End If generated is equally spread over Next I the 40 hrs totalstockagecost = stockage * Hcost holdingrate = totalstockagecost / numberhoursim holdingoup = holdingrate penalty=shortage* penalty cost/ numberhoursim P1= numbreaches *cycleT / numbercycles P2= 1 - shortage / Totaldemand

44

Table 11. Executing the stochastic process

Table 11. Executing the stochastic process

Table 12. Results for the whole logistic system

Transportation 63,2938Vehicle operating 60Penalty 0Loading 14,5Delivery 12,5Holding 65TOTAL COST 215,2938P2total 1P1oup 1betweenshortages 5000

randomvariable = Rnd() randomvariable2 = Rnd() If randomvariable <= 0.5 Then standnorm = WorksheetFunction.NormSInv(randomvariable2) avg = mu1 dev = sigma1 dema = standnorm * Sqr(dev) + avg Else standnorm = WorksheetFunction.NormSInv(randomvariable2) avg = mu2 dev = sigma2 dema = standnorm * Sqr(dev) + avg End If If dema > 0 Then demandoup = dema Else demandoup = 0

45

5 Simulation results and analysis In order to examine the performance of the IRP model when including order-up-to retailers,

we consider multiple logistic problems, with size varying from 30 to 120 sales-points. Let n

denote the number of sales-points. We determine the performance of the model by evaluating

different measurement gauges (cost rates, service levels,...) for each problem. These are

obtained by means of simulation: while executing these simulations we also vary the

percentage of order-up-to retailers β from 0% to 50%. The first paragraph will display the

results from these simulations: we classify the various problems according to their size( small,

medium and large). The second paragraph will investigate the impact of the percentage of

order-up-to retailers and the size of the logistic problem on these gauges. The input data files

for the IRP model of the different logistic problems can be found in Appendix D. We used a

shortage cost of 20€/(ton of sales lost). All simulations were executed a numerous times

considering a time period of approximately a year.

5.1 Results 5.1.1 Small problems 5.1.1.1 n=30

First, we look at the fluctuations of the inventory level at a single retailer with stochastic

demand (order-up), when being replenished in accordance with the distribution plan generated

by the IRP model: this implies one replenishment every cycle time of the vehicle. Take for

example the first retailer in this problem, retailer 1, with an average demand rate of 1.25

ton/hour. If we assume that this sales-point experiences a stochastic demand and its optimal

policy would be order-up-to, we can calculate (as explained before) the parameters of the

distributions determining the stochastic process. This yields N1(35,1) and N2(65,1), and a

probability distribution for the weekly demand similar to Fig. 12. By recording the inventory

positions during simulation we can generate the dynamics of the inventory level as depicted in

Fig. 16. Every upward movement of the inventory level is due to replenishment by the

appropriate vehicle: the cycle time of its corresponding vehicle is 32. As mentioned in the

previous chapter, we assume the weekly demand to be generated by a stochastic process: this

demand is then equally spread over the whole week (40hrs). This is why we see the weekly

demand shifting from approximately 35 to approximately 65. This change of weekly demand

46

(and thereby demand per hour for that week) can be noticed from the change of the slope of

the inventory level as well. The simulation yields a P1 of 0.886, meaning that there is a 88.6%

chance that no shortage will occur between two replenishments. The average fill rate becomes

0.976, indicating that 97.6 % of the demand at this retailer can be fulfilled.

Fluctuations of the inventory level

01020304050607080

inhours

Period

in to

n Weekly demandInventory level

Fig.16. Fluctuations of the inventory level at a retailer with stochastic demand

This might look surprisingly high but as you see on Figure 16, the cycle time of its

replenishing vehicle is smaller than a week: even if the ‘bigger’ weekly demand is generated,

the retailer will always be replenished before the end of that week. Consequently, not a lot of

sales will be lost, but this has high holding costs as a consequence. However, if the cycle time

of the replenishments is equal to a week, consequent generations of the big weekly demand

could cause major stock-outs. If the cycle time of the replenishments is more than a week, the

chance of stock-outs drops again, because the retailer experiences a mix of weekly demands.

Table 13 shows the results of the total logistic system, acquired by simulation, when

increasing β up to 50 percent: the retailers for which the inventory policy is changed are

picked randomly.

All costs and service levels are expressed in respectively €/hr and percentage. Notice that only

the penalty and holding cost rates change when increasing the number of order-up-to retailers

(Fig.17); the other cost rates stay the same because we utilize the same multi-tours as

generated in the IRP-model (where only EOQ-policies are considered). Fig.17 shows us how

the cost rates change along with the OUP percentage; notice that the holding cost rate doubles

when β=50%, while the shortage cost stays small.

Replenishment Shift of weekly demand

47

Table 13. Simulation results for logistic system with 30 retailers 0% 10% 20% 30% 40% 50%

Transportation 47,7377 43,7377 43,7377 43,7377 43,7377 43,7377

Vehicle operating 60 60 60 60 60 60

Penalty 0 1,043739053 1,475365881 2,56130425 2,771936124 4,096686624

Loading 7,91667 7,91667 7,91667 7,91667 7,91667 7,91667

Delivery 7,91667 7,91667 7,91667 7,91667 7,91667 7,91667

Holding 32,2 41,64987823 47,13424482 55,45348104 58,87229032 69,45860935

TOTAL COST 108,282137 162,2646573 168,1806507 177,5858253 181,2152664 193,126336

P2total 1 0,996939182 0,995673414 0,992488844 0,991871155 0,987986256

P1oup 1 0,87688172 0,878524492 0,872023098 0,874148746 0,877287933

betweenshortages 5000 20,90853402 22,83466631 22,07749389 22,94600754 20,63509761

Changing cost rate in function of percentage OUP

0

50

100

150

200

250

0% 10% 20% 30% 40% 50%

percentage

in e

uro/

h Penalty cost rateHolding cost rateTotal cost

Fig.17. Changing cost rates depending of OUP percentage(n=30).

Table 13 shows us a few types of service-levels as well: P1oup denotes the average

probability not to stock-out between replenishments for the OUP-retailers. Furthermore,

betweenshortages reflects the average time period (in hours) between consequent shortages

( for OUP retailers). The value of betweenshortages for all sales-points using EOQ-like

strategies (0%) is 5000: in fact this is infinite because stockouts can not occur under these

assumptions but for the simplicity of the simulation model we used this big number.

The total fill rate, the percentage of total demand fulfilled, drops slightly but still amounts to

98.7% if fifty percent of the logistic system has a stochastic demand. This could be surprising,

but if we look for example at the retailer discussed above, we notice that its single fill rate

values 0.976 too. This is why we have such small penalty costs as well.

48

5.1.1.2 n=39

The result of the simulation for this logistic system is shown in Table 14: just as in the

problem discussed above we notice high fill rates. In paragraph 2 we examine the impact of

the size of the logic problem on these service levels.

Table 14. Result of the simulation for n=39 0% 10% 20% 30% 40% 50%

Transportation 105,038 105,038 105,038 105,038 105,038 105,038


Penalty 0 0,623976719 2,688866758 3,945719048 5,14271234 6,757305762

Loading 11 11 11 11 11 11

Delivery 9,75 9,75 9,75 9,75 9,75 9,75

Holding 49,65 59,88515085 68,77551367 82,73206224 93,98941822 109,051748

TOTAL COST 265,438 276,2971276 287,2523804 302,4657813 314,9201306 331,5970538

P2total 1 0,998743249 0,994584357 0,992052932 0,98964207 0,986390119

P1oup 1 0,980448381 0,969533378 0,960552261 0,968840213 0,965897577

breach 5000 19,35622741 31,89506925 24,98691417 29,08008272 25,67068317

Cost rates for n=39

050

100150

200250

300350

0% 10% 20% 30% 40% 50%

Beta

Penalty cost rateHolding cost rateTotal cost

Fig.18. Changing cost rates depending of OUP percentage(n=39).

Again, we see the total cost rate increasing in a quasi linear trend (Fig.18); one of the

objectives in paragraph 2 is to examine whether the magnitude of the increase of this cost

rate is dependent of the number of retailers in the logistic system.

Considering the results from the two previous problems, we can already conclude, as

expected, that the total cost rate increases with β . Because this trend has already become

49

clear, we will only show the data tables obtained from simulation for the remaining problems

in the following subsections.

5.1.1.3 n=50

Table 15. Simulation results for n=50 0% 10% 20% 30% 40% 50%

Transportation 63,2938 63,2938 63,2938 63,2938 63,2938 63,2938


Penalty 0 3,10009503 2,453089297 6,21531812 4,059796599 8,561043356

Loading 14,5 14,5 14,5 14,5 14,5 14,5

Delivery 12,5 12,5 12,5 12,5 12,5 12,5

Holding 65 78,03677897 105,0281366 126,1256908 132,155795 138,25862

TOTAL COST 215,2938 231,430674 257,7750259 282,634809 286,5093916 297,1134633

P2total 1 0,995230623 0,996226016 0,990437972 0,993754159 0,986829164

P1oup 1 0,980657278 0,978141746 0,979728618 0,977930078 0,968589186

breach 5000 27,60841338 26,91091259 27,97187681 26,85844382 40,75720419

5.1.2 Medium-sized problems We define medium-sized problems as logistic configurations with number of retailers ranging

from 60 to 90; just as for the small problems, we have examined three of them.

5.1.2.1 n=67


Transportation 106,655 106,655 106,655 106,655 106,655 106,655


Penalty 0 2,621830163 3,865868001 6,724244976 8,019805895 11,13998269

Loading 20,5417 20,5417 20,5417 20,5417 20,5417 20,5417

Delivery 20,9167 20,9167 20,9167 20,9167 20,9167 20,9167

Holding 77,43 106,3478836 123,5823724 146,3630667 163,2178876 182,2327615

TOTAL COST 345,5434 377,0831138 395,5616404 421,2007117 439,3510935 461,4861442

P2total 1 0,997200395 0,995872004 0,992819813 0,991436406 0,988104663

P1oup 1 0,965337108 0,969847866 0,961069294 0,975620975 0,966064891

breach 5000 39,25542079 27,66967735 23,68136462 29,14453274 25,37668849

50

5.1.2.2 n= 81


Transportation 98,2712 98,2712 98,2712 98,2712 98,2712 98,2712


Penalty 0 4,71613251 6,147005645 8,367981365 10,36073196 12,59810546

Loading 22 22 22 22 22 22

Delivery 25,75 25,75 25,75 25,75 25,75 25,75

Holding 78,11 106,1034102 131,2537176 155,2302048 177,2250257 210,0837824

TOTAL COST 344,1312 376,8407427 403,4219232 429,6193862 453,6069576 488,7030879

P2total 1 0,995300316 0,993874434 0,991661204 0,989675404 0,987445834

P1oup 1 0,973218543 0,960689991 0,963162125 0,971510203 0,964623864

breach 5000 29,63685677 35,55152717 20,50025743 33,00521284 38,49307244

5.1.2.3 n= 93


Transportation 143,157 143,157 143,157 143,157 143,157 143,157


Penalty 0 2,738723916 6,156066826 9,907328774 11,80699966 18,28610876

Loading 25,0417 25,0417 25,0417 25,0417 25,0417 25,0417

Delivery 37,0208 37,0208 37,0208 37,0208 37,0208 37,0208

Holding 78,04 108,4639871 128,105328 159,9709807 177,7395713 216,864901

TOTAL COST 463,2595 496,422211 519,4808948 555,0978095 574,7660709 620,3705098

P2total 1 0,997695647 0,994820306 0,991664006 0,990065629 0,984614128

P1oup 1 0,943468666 0,966056314 0,962716375 0,967989835 0,970096267

breach 5000 70,38056986 33,13811371 33,82457161 38,98511426 23,11993685

5.1.3 Large problems For this class of problems we examine three problems with respectively 98, 110 and 120

sales-points.

51

5.1.3.1 n=98


Transportation 134,274 134,274 134,274 134,274 134,274 134,274


Penalty 0 5,778597356 7,29958684 9,89195189 15,83595784 16,21571332

Loading 27,4167 27,4167 27,4167 27,4167 27,4167 27,4167

Delivery 29,2708 29,2708 29,2708 29,2708 29,2708 29,2708

Holding 107,95 139,308301 182,0110368 250,6824116 242,4774765 271,8129045

TOTAL COST 448,9115 486,0483983 530,2721237 601,5358635 599,2749343 628,9901178

P2total 1 0,995417449 0,994211271 0,99215547 0,987441746 0,987140592

P1oup 1 0,97078339 0,972940677 0,973757202 0,975529794 0,978316347

breach 5000 28,19484075 16,47364349 50,87583562 64,88376831 36,69428272

5.1.3.2 n=110


Transportation 167,255 167,255 167,255 167,255 167,255 167,255


Penalty 0 2,849675071 6,811114467 8,251993647 17,36879251 20,9688484

Loading 31,25 31,25 31,25 31,25 31,25 31,25

Delivery 30,4375 30,4375 30,4375 30,4375 30,4375 30,4375

Holding 126,17 157,7096782 195,5329423 283,1594779 256,6228985 304,627

TOTAL COST 505,1125 539,5018532 581,2865568 670,3539715 652,934191 704,5383484

P2total 1 0,99793427 0,995062621 0,994018127 0,987409357 0,984799675

P1oup 1 0,807233627 0,889423265 0,887820137 0,846940371 0,913040078

breach 5000 27,21837854 33,84379464 30,59874765 28 32,97915363

52

5.1.3.3 n=120


Transportation 300,784 300,784 300,784 300,784 300,784 300,784


Penalty 0 5,57736045 6,210133276 9,085515093 16,20105419 20,43098151

Loading 34,5 34,5 34,5 34,5 34,5 34,5

Delivery 31,8542 31,8542 31,8542 31,8542 31,8542 31,8542

Holding cost 151,15 180,2810508 240,9990681 285,8758742 313,9660057 305,6758075

TOTAL COST 788,2882 822,9966112 884,3474014 932,0995892 967,3052598 963,244989

P2total 1 0,996414426 0,996007629 0,994159103 0,989584665 0,986865329

P1oup 1 0,887225806 0,849625448 0,888804062 0,861842294 0,885999283

breach 5000 15,80461814 41,35198083 29,98764555 23,68266239 48,37278553

5.2 Analysis This is an outline of the remaining paragraph. First, we investigate the impact of the number

of sales-points on the different types of cost rates when changing β. Secondly, we look for its

effect on the various service levels.

5.2.1 Cost rates The loading/unloading and transportation cost rates are not considered because they don’t

change when varying the number of OUP retailers.

5.2.1.1 Shortage costs

Because stock-outs can not occur when all retailers experience a deterministic demand, the

shortage cost rate is zero when β =0. Table 22 shows the percentage of the total cost rate

caused by shortage costs, for the different problems and β-levels. Analysing the table, we see

that this percentage ranges between 2 and 3 percent for all logistic problems when β equals

50. We can’t detect a trend, that indicates the impact of the size of the problem on this

percentage: therefore we conclude that the number of sales-points has no impact on the

fraction of the total cost rate caused by shortages.

53

Table 22. Percentage of total cost rate incurred by shortages No. Of sales-

points Percentage of OUP retailers

0% 10% 20% 30% 40% 50%

30 0 0,643233 0,877251 1,442291 1,529637 2,121247

39 0 0,225835 0,936064 1,304517 1,633021 2,037806

50 0 1,339535 0,95164 2,199063 1,416986 2,881405

67 0 0,695292 0,977311 1,596447 1,825375 2,413937

81 0 1,251492 1,523716 1,947766 2,284077 2,577865

93 0 0,551692 1,185042 1,78479 2,054227 2,947611

98 0 1,188893 1,376574 1,644449 2,64252 2,578055

110 0 0,528205 1,171731 1,23099 2,660114 2,976254

120 0 0,677689 0,702228 0,974737 1,674865 2,121058

5.2.1.2 Holding cost

The holding cost for β=0 is not equal to zero, in contrary to the shortage cost; Table 23

exhibits the shortage cost relative to the one for β=0, showing the relative increase when

varying the number of sales-points and the β-level. For instance, the value for n =50 and β=20

is calculated by:

1,61581765

6105,028136= ( see Table 15)

Table 23 .Relative increase of holding cost No. Of sales-

points n Percentage of OUP retailers β

0% 10% 20% 30% 40% 50%

30 1 1,293474 1,463796 1,722158 1,828332 2,1571

39 1 1,206146 1,385207 1,666305 1,89304 2,19641

50 1 1,200566 1,615817 1,940395 2,033166 2,127056

67 1 1,373471 1,596053 1,890263 2,107941 2,353516

81 1 1,358384 1,68037 1,987328 2,268916 2,689589

93 1 1,389851 1,641534 2,049859 2,277544 2,778894

98 1 1,290489 1,686068 2,322209 2,246202 2,517952

110 1 1,249978 1,549758 2,244269 2,033945 2,414417

120 1 1,192729 1,594436 1,891339 2,077182 2,022334

54

The table above reveals that for every problem considered, if we move β up to 50%, the

holding cost rate will be more than doubled; this explains why this cost rate influences the

total cost rate so much. The table gives no indication of the relative increase of the holding

cost rate being dependent of the number of retailers. Recalling that the penalty and shortage

costs are the only cost rates changing and keeping in mind the conclusion of the previous

subsection, we can conclude as well that the percentage of the total cost rate incurred by

holding inventory is independent of the size of the problem.

5.2.1.3 Total cost rate

Table 24 shows the value of the total cost rate for every problem, varying the β-level from 0

to 50%, relative to the total cost rate implementing β equal to 0.

Table 24. Relative increase of total cost rate No. Of sales-

points Percentage of OUP retailers (in%)

0 10 20 30 40 50

30 1 1,498536 1,553171 1,640029 1,673547 1,783548

39 1 1,04091 1,082183 1,139497 1,186417 1,249245

50 1 1,074953 1,197317 1,312787 1,330783 1,380037

67 1 1,091276 1,144752 1,218952 1,271479 1,335537

81 1 1,09505 1,172291 1,248417 1,318122 1,420107

93 1 1,071586 1,12136 1,198244 1,2407 1,339143

98 1 1,082727 1,18124 1,339988 1,334951 1,401145

110 1 1,068083 1,150806 1,327138 1,292651 1,394815

120 1 1,04403 1,121858 1,182435 1,227096 1,221945

Notice that the total cost rate, when β equals 50%, has values varying from 1.22 to 1.78 times

the total cost, when only implementing EOQ retailers. Just as for the other cost rates, we can

conclude that the size of the logistic problem doesn’t influence the values exhibited in the

table above. In the previous chapter, we already remarked that the total cost rates for a certain

problem changes in function of the β-levels quasi linear. Therefore we calculate the average

slope of the total cost rate for all the problems examined.(Table25)

55

Table 25. Average slope of the total cost rate for every logistic problem Nr of sales-points Average slope (in euro/hr*hr)

30 1,135417

39 1,045531

50 1,067189

67 1,059714

81 1,072746

93 1,060275

98 1,070786

110 1,070361

120 1,041229

Notice that this slope is almost equal for the different logistics systems; again, the number of

salespoint does not have an impact.

5.2.2 Servicelevel P1 Table 26 gives an overview of the total fill rate: remarkably, this fill rate exceeds 98% most of

the time. For every problem , we see that the fill rate decreases only slightly when increasing

the β-level.

Table 26 . The total fill rate P2total No. Of sales-

points Percentage of OUP retailers

0% 10% 20% 30% 40% 50%

30 1 0,996939 0,995673 0,992489 0,991871 0,987986

39 1 0,998743 0,994584 0,992053 0,989642 0,98639

50 1 0,995231 0,996226 0,990438 0,993754 0,986829

67 1 0,9972 0,995872 0,99282 0,991436 0,988105

81 1 0,9953 0,993874 0,991661 0,989675 0,987446

93 1 0,997696 0,99482 0,991664 0,990066 0,984614

98 1 0,995417 0,994211 0,992155 0,987442 0,987141

110 1 0,997934 0,995063 0,994018 0,987409 0,9848

120 1 0,996414 0,996008 0,994159 0,989585 0,986865

56

5.3 Conclusion Analysis of the simulation results yielded that the number of retailers in the logistic system

does not influence the performance of the IRP model when working with retailers with

stochastic demand. However, the percentage of order-up-to retailers, does, as expected.

Another result from the analysis was the following: when α reaches 50%, the holding cost rate

is approximately doubled, independent of the number of retailers. The increase of the shortage

cost is almost insignificant compared to that of the holding cost. Consequently, the increase in

the total cost rate is mostly due to the holding cost. This is reflected in a high fill-rate: when

increasing α from 0 to 50%, this service level only decreases slighty, ending with a P2 above

0.98. We can conclude that the IRP model performs very well in terms of shortages and fill-

rates, but that the total costs increase significantly when increasing β.

57

6 Implementing a different replenishment strategy

6.1 Introduction The results from the previous chapter demonstrate that even when some of its retailers

experience stochastic demand, the IRP model still produces a distribution plan with a high fill

rate. However, the total cost rate (approximately) doubles when β becomes 50%. Therefore

we will examine the impact on the total cost if we start implementing a different strategy. In

this line of thought, we will serve the retailers with deterministic demand according to a

distribution plan generated by the IRP model; the retailers with stochastic demand are served

using a “direct shipping“ strategy. The latter means that for every retailer, a single vehicle is

assigned to replenish its inventory, as soon as an order is made. Remember that for the type of

stochastic demands defined earlier, the order-up-to strategy was proved to be the optimal in

the long run. This implies that we only order at the beginning of a period (a week) if the

inventory drops below the reorder point s. We will test this new strategy on the logistic

problem with n= 39 and β= 20%. We will assume that the first eight retailers on the dat-file

experience stochastic demand. For the other retailers, encountering deterministic demand, the

distribution plan is determined by the IRP model.

This is an outline of the remainder of this chapter. The second paragraph is dedicated to

computing the average costs involved with using this strategy. Later, in paragraph 3, we make

a suggestion in order to reduce the total cost associated with the strategy.

6.2 Costs associated with the new strategy Calculating the total cost rate, we make a division between the costs caused by retailers with

deterministic and stochastic demand; the first subsection discusses the cost due to EOQ-

retailers and the second is dedicated tot the costs corresponding with the OUP-retailers.

6.2.1 Deterministic demand The cost rates as a consequence of sales-points with deterministic demand are easily obtained

from the results of the IRP model:

58

Table 27. Cost rates for the retailers with deterministic demand (in euro/h)

Type of cost rate

in

Euro/hr

Vehicle costs 90

Transportation costs 875.531

Dispatching costs 8.875

Delivery costs 814.583

Holding costs 39.61

Total cost 234.184

6.2.2 Stochastic demand The total cost rate caused by the retailers with this type of demand consists of two parts: the

fixed operational cost of the used vehicles and the long run average costs of the different

sales-points. Recall (from section 1.1.3.) that this long run average cost is defined as follows:

)(),(),(sSM

SskSsc−

=

which equals the ratio of the expected cost between successive regeneration points and the

expected time between successive regeneration points. The different functions and variables

are defined in equations (1) to (5) in Chapter 1. In order to calculate this cost, we need to

know the optimal reorder point s and order-up-to point S. This implies executing the

algorithm exhibited in section 1.1.3.

To be able to do so, we need to evaluate the parameters of the two normal distributions used

in the stochastic process that generates the weekly demand. Table 28 shows the average

hourly demand and the parameters for the stochastic process of the OUP-sales-points. These

were calculated in the same way as in Chapter 4.

Table 28. Parameters of the stochastic process for the different retailers Sales-point Demand rate (ton/hr) mu1 sigma1 mu2 sigma2

1 0,5 14 1 26 1

2 0,5 14 1 26 1

3 0,6 17 1 31 1

4 0,225 6 1 12 1

5 0,6 17 1 31 1

6 0,1 3 1 5 1

7 0,7 20 1 36 1

8 0,95 27 1 49 1

59

Recall that this algorithm works with a discrete distribution pj=Pr{D=j} (for j=0,1,2,…)

representing the one-period demand; in our case we consider the weekly demand generated by

the stochastic process. Because the latter is difficult to describe theoretically, we simulated its

distribution by means of Monte-Carlo. This generated distribution was made discrete by

means of data-analysis: e.g. every value in the range [γ-0.5, γ+0.5] was assigned to the integer

value γ. When calculating the optimal (s,S)-values and their corresponding long run average

costs, we need to determine the fixed ordering cost K. This cost consists of three parts:

Travelling cost: this equals the distance covered by a vehicle going from the

warehouse to the retailer and back, times the variable vehicle cost

Loading cost: the cost for loading the truck at de DC

Delivery cost: the cost for unloading the goods at the retailer

For the first type of cost, we need to calculate the Euclidean distance between the DC and the

different retailers:

Distance 22 )()( retailerDCretailerDC YYXX −+−=

Then, the travelling cost for every retailer is calculated as follows:

Travelling cost= 2⋅Distance. Variable vehicle cost

The two following cost rates are independent of the retailer. We can calculate the loading cost

by multiplying the loading time by the loading cost: 0.5hr⋅20€/hr=10€. Finally, the delivery

cost is obtained by multiplying the delivery time by the delivery cost : 0.25hr⋅10€/hr=2.5€.

The fixed cost and its components are shown in Table 29. We are now able to calculate the

(s,S) level for each retailer and its corresponding long run average cost, using the algorithm in

Chapter 2. The results are shown in Table 30.

Table 29. Constructing the fixed order cost

Retailer Travelling cost (in euro) Loading cost(in euro)

Delivery cost (in

euro) K (in euro)

1 297,8657416 10 2,5 310,3657416

2 211,0260647 10 2,5 223,5260647

3 97,67292358 10 2,5 110,1729236

4 186,6976165 10 2,5 199,1976165

5 250,3197955 10 2,5 262,8197955

6 91,23595782 10 2,5 103,7359578

7 32,24903099 10 2,5 44,74903099

8 43,8634244 10 2,5 56,3634244

60

Table 30. The results generated by the (s,S)-algorithm

Retailer s S

Long run average

cost(€/week)

Average cost

rate(€/hr)

1 23 269 24,9173 0,6229325

2 23 230 21,09830781 0,527457695

3 28 182 15,96323388 0,399080847

4 10 145 13,71758947 0,342939737

5 28 273 24,98427728 0,624606932

6 3 68 6,456806525 0,161420163

7 34 135 10,76616542 0,269154136

8 47 179 13,99395001 0,34984875

Consequently, these retailers yield a total cost per hour of 3,297€/hr. Because we assumed

that every retailer is served by a different vehicle, we have a fixed vehicle cost of 240 €/hr

(=30€/(hr⋅vehicle)*8 vehicles), producing a total cost for the OUP-retailers of 243.297€/hr.

Adding up the cost rate for the retailers with constant demand (Table27), we become the total

cost rate for the whole logistic system: 477€/hr (= 234.184+ 243.297). This is much more then

the total cost rate when all retailers were implemented in the IRP algorithm (287€/hr). It is

clear that a huge amount of this cost rate is caused by the vehicles that replenish the retailers

with stochastic demand. A next step is trying to determine how to diminish the number of

fleet; this is discussed in the next paragraph.

6.3 Diminishing the used fleet In the previous paragraph we assumed that every retailer with stochastic demand had his own

vehicle to replenish. This yielded enormous fixed vehicle costs. The objective of this

paragraph is to investigate how we can let the number of vehicles drop: we can do this by

letting a vehicle serve multiple retailers. If we could serve all retailers just by one vehicle, the

new strategy could yield a lower cost rate than the one obtained with the IRP-model; 243.297

+30=273.297€/hr.

M(j) is defined as the expected total time until the next order is placed when starting with an

inventory position of respectively s+j units. Consequently, M(S-s) corresponds to the average

time between replenishments. We have computed this values for each retailer using the same

algorithm as above:

61

Table 31. Expected number of time between orders Retailer Expected number of weeks between replenishment

1 12,51438

2 10,64467

3 6,921391

4 14,62517

5 10,57312

6 16,06614

7 4,176773

8 4,074432

Notice that the minimum average number of weeks between replenishment is 4.074432 for

retailer 8; this is not startling, because after every replenishment, its inventory mounts up to

179 with an approximate week demand of 40. The optimal order-up-to level is so high

because of the high fixed ordering cost. The structure of these demands makes it difficult to

determine the probability that in a week the total demand could be replenished by one vehicle,

recalling the limited capacity of the vehicle. However, it is easy to see that the chance that

every retailer needs to be replenished in the same week is very small, just as the chance that

every ordering retailer needs his total order-up-to level.

However the OUP-retailers experience lower holding and shortage costs when utilizing such a

direct shipment strategy, these cost savings can hardly match the savings that the IRP-model

makes in terms of fleet operating costs, which becomes an acceptable solution when

implementing demands with stochastic demand ( of the form we defined earlier).

62

7 Conclusion The objective of this thesis was to find out how the IRP model, that usually works with sales-

points with constant and deterministic demand, would response if we would implement

retailers with a stochastic demand. For the type of stochastic demand considered , i.e. a

stochastic process that generates the weekly demand, good results were obtained in terms of

fill rate and shortage costs. However, this yielded a big increase of the holding cost,

increasing the total cost rate. Therefore, a new distribution strategy was proposed, using the

„optimal“ replenishment strategy for both inventory policies: a distribution plan generated by

the IRP-model for the EOQ-retailers and a direct shipping method for the OUP-retailers.

However, the latter doesn’t take into account the fixed vehicle operating cost; once you use a

vehicle, you keep paying this cost per hour. Therefore, this new strategy did not perform as

well as expected. The savings achieved in terms of holding and shortage cost are almost

neutralized by the big increase of fleet operating costs. Because of the drastic savings in terms

of fleet operating costs that are made with the IRP model, this model does not perform that

bad on logistic systems with EOQ-retailers and OUP-retailers, for which the demand is

generated as mentioned before.

63

Appendix A. Properties of c(⋅,⋅) and bounds for(s*,S*)

66

Appendix B. Calculating the optimal (s,S)-values Loss Function G Function Verliesfunctie(num) Dim N As Double Dim J As Double Dim Demand As Range Dim Probability As Range Set Demand = Range("Demand") Set Probability = Range("Probability") Total = 0 For i = 1 To 200 N = 0.1 * WorksheetFunction.Max(num - Demand(i), 0) * Probability(i) J = 20 * WorksheetFunction.Max(Demand(i) - num, 0) * Probability(i) Total = Total + N + J Next i Verliesfunctie = Total End Function Sub loss() Dim Loss_function As Range Set Loss_function = Range("Loss_function")

For i = 1 To 1999 Range("Loss_function").Cells(i, 1) = Verliesfunctie(i)

Next i End Sub Function m(j) Sub mvalues() Dim Q As Double Dim Total As Double Dim aantal As Integer Dim Tussenoplossing(400) As Double Dim Probability As Range Set Probability = Range("Probability") aantal = num step = (1 - Probability(1)) Tussenoplossing(0) = step Total = 0 For J = 1 To 389 For i = 1 To J

67

Q = Probability(i + 1) * Tussenoplossing(J - i) Total = Total + Q Next i Tussenoplossing(J) = (Total) / (step) Total = 0 Next J For i = 1 To 390

Range("m").Cells(i, 1) = Tussenoplossing(i - 1) Next i End Sub Function M(j) Sub valueM() Dim Q As Double Dim Total As Double Dim aantal As Integer Dim Tussenoplossing(400) As Double Dim Probability As Range Set Probability = Range("Probability") Dim m As Range Set m = Range("m") Tussenoplossing(0) = 0 For J = 1 To 389 Q = Tussenoplossing(J - 1) + Range("m")(J, 1).Value Tussenoplossing(J) = Q Next J For i = 1 To 390 Range("Mvalue").Cells(i, 1) = Tussenoplossing(i - 1) Next i End Sub Function K(s,⋅) Function kcost(down, up) Vastekost = Range("K").Value For i = 0 To orderlevel - orderpunt - 1

waarde = Range("m")(i + 1, 1).Value * Range("Loss_function")(up - i + 1, 1).Value totaal = totaal + waarde

Next i

68

kcost = totaal + Vastekost End Function Function c(s,S) Function cost(lower, up) cost = kcost(lower, up) / Range("Mvalue")(up - lower + 1).Value End Function Algorithm Sub orderlevels() orderpoint = Range("yster").Value orderlevel = Range("yster").Value Do orderpoint = orderpoint - 1 bestcost = cost(orderpoint, orderlevel) Loop Until bestcost <= Range("Loss_function")(orderpoint + 1, 1).Value S = orderlevel + 1 Do While Range("Loss_function")(S + 1, 1).Value <= bestcost If cost(orderpoint, S) < bestcost Then orderlevel = S newcost = cost(orderpoint, S) Do While newcost <= Range("Loss_function")(orderpoint + 2, 1).Value orderpoint = orderpoint + 1 newcost = cost(orderpoint, S) Loop bestcost = newcost End If S = S + 1 Loop Range("orderpoint") = orderpoint Range("orderlevel") = orderlevel End Sub

69

Appendix C. Discussion constraints IRP-model

70

Appendix D. Data-files for the different logistic problems N=30 Capacity Speed FixedCost VarCost Vehicle

50 50 30 1 Depot X Y delTime delCost DC 86 86 0.5 20 Name X Y demRate cap delTime delCost holdCost

1 96 74 1.25 50 0.25 10 0.1 2 80 81 0.1 50 0.25 10 0.1 3 49 99 0.6 50 0.25 10 0.1 4 7 101 0.9 50 0.25 10 0.1 5 86 86 0.225 50 0.25 10 0.1 6 137 71 0.125 50 0.25 10 0.1 7 152 87 0.325 50 0.25 10 0.1 8 85 160 1.2 50 0.25 10 0.1 9 119 35 0.2 50 0.25 10 0.1

10 83 57 0.3 50 0.25 10 0.1 11 83 41 0.15 50 0.25 10 0.1 12 141 48 0.45 50 0.25 10 0.1 13 56 56 0.075 50 0.25 10 0.1 14 47 80 1.2 50 0.25 10 0.1 15 81 122 0.925 50 0.25 10 0.1 16 90 82 0.225 50 0.25 10 0.1 17 17 99 0.725 50 0.25 10 0.1 18 124 53 0.075 50 0.25 10 0.1 19 80 147 0.9 50 0.25 10 0.1 20 160 114 0.6 50 0.25 10 0.1 21 148 123 0.675 50 0.25 10 0.1 22 63 78 1.125 50 0.25 10 0.1 23 120 74 0.15 50 0.25 10 0.1 24 22 58 0.775 50 0.25 10 0.1 25 89 94 0.325 50 0.25 10 0.1 26 61 127 0.8 50 0.25 10 0.1 27 86 87 0.8 50 0.25 10 0.1 28 128 76 0.375 50 0.25 10 0.1 29 121 31 1.225 50 0.25 10 0.1 30 62 82 0.25 50 0.25 10 0.1

N=39 Capacity Speed FixedCost VarCost Vehicle

50 50 30 1 Depot X Y delTime delCost DC 159 159 0.5 20

71

Name X Y demRate cap delTime delCost holdCost

9 164 193 1.125 50 0.25 10 0.1 10 151 173 0.6 50 0.25 10 0.1 11 176 161 0.475 50 0.25 10 0.1 12 153 185 0.05 50 0.25 10 0.1 13 130 215 0.5 50 0.25 10 0.1 14 9 202 1.05 50 0.25 10 0.1 15 198 128 1.225 50 0.25 10 0.1 16 51 51 0.45 50 0.25 10 0.1 17 146 241 0.75 50 0.25 10 0.1 18 133 250 1.2 50 0.25 10 0.1 19 139 202 1.225 50 0.25 10 0.1 20 239 25 0.6 50 0.25 10 0.1 21 185 244 0.475 50 0.25 10 0.1 22 192 97 1.25 50 0.25 10 0.1 23 158 89 1.05 50 0.25 10 0.1 24 88 116 0.725 50 0.25 10 0.1 25 196 20 0.35 50 0.25 10 0.1 26 129 159 0.175 50 0.25 10 0.1 27 120 232 0.425 50 0.25 10 0.1 28 255 116 0.85 50 0.25 10 0.1 29 217 99 1.175 50 0.25 10 0.1 30 177 109 0.125 50 0.25 10 0.1 31 179 45 0.3 50 0.25 10 0.1 32 152 149 0.65 50 0.25 10 0.1 33 144 222 0.975 50 0.25 10 0.1 34 83 68 0.6 50 0.25 10 0.1 35 26 145 0.275 50 0.25 10 0.1 36 210 33 0.45 50 0.25 10 0.1 37 133 78 0.35 50 0.25 10 0.1 38 254 223 0.7 50 0.25 10 0.1 39 192 142 0.5 50 0.25 10 0.1



1 42 26 1.125 50 0.25 10 0.1 2 37 44 0.875 50 0.25 10 0.1 3 62 38 0.95 50 0.25 10 0.1 4 56 90 0.375 50 0.25 10 0.1 5 84 66 0.55 50 0.25 10 0.1 6 76 57 0.275 50 0.25 10 0.1 7 47 74 0.55 50 0.25 10 0.1 8 79 51 0.65 50 0.25 10 0.1 9 127 76 0.6 50 0.25 10 0.1

72

10 69 7 0.775 50 0.25 10 0.1 11 110 80 0.05 50 0.25 10 0.1 12 102 42 1.1 50 0.25 10 0.1 13 85 64 0.775 50 0.25 10 0.1 14 92 124 0.875 50 0.25 10 0.1 15 70 149 0.35 50 0.25 10 0.1 16 75 98 0.075 50 0.25 10 0.1 17 77 71 0.975 50 0.25 10 0.1 18 34 105 0.8 50 0.25 10 0.1 19 69 10 0.875 50 0.25 10 0.1 20 99 33 0.325 50 0.25 10 0.1 21 75 75 0.375 50 0.25 10 0.1 22 77 57 1.075 50 0.25 10 0.1 23 112 92 0.6 50 0.25 10 0.1 24 99 12 0.45 50 0.25 10 0.1 25 81 122 0.025 50 0.25 10 0.1 26 140 85 0.325 50 0.25 10 0.1 27 78 16 0.85 50 0.25 10 0.1 28 114 103 1.125 50 0.25 10 0.1 29 111 99 0.625 50 0.25 10 0.1 30 75 75 1.125 50 0.25 10 0.1 31 57 54 1.1 50 0.25 10 0.1 32 32 103 1.2 50 0.25 10 0.1 33 75 66 1.15 50 0.25 10 0.1 34 136 76 1.025 50 0.25 10 0.1 35 86 47 0.475 50 0.25 10 0.1 36 109 29 0.125 50 0.25 10 0.1 37 69 74 0.35 50 0.25 10 0.1 38 85 65 1.15 50 0.25 10 0.1 39 79 83 1.025 50 0.25 10 0.1 40 38 78 0.3 50 0.25 10 0.1 41 34 105 1.225 50 0.25 10 0.1 42 119 40 0.525 50 0.25 10 0.1 43 76 74 0.025 50 0.25 10 0.1 44 39 81 0.325 50 0.25 10 0.1 45 77 71 0.35 50 0.25 10 0.1 46 45 65 0.2 50 0.25 10 0.1 47 72 40 0.475 50 0.25 10 0.1 48 112 117 1.225 50 0.25 10 0.1 49 36 72 0.3 50 0.25 10 0.1 50 49 84 0.45 50 0.25 10 0.1



1 99 134 0.725 50 0.25 10 0.1

73

2 107 95 1.175 50 0.25 10 0.1 3 156 42 1.125 50 0.25 10 0.1 4 95 86 0.475 50 0.25 10 0.1 5 54 154 0.925 50 0.25 10 0.1 6 42 77 1 50 0.25 10 0.1 7 12 123 0.775 50 0.25 10 0.1 8 73 64 1.15 50 0.25 10 0.1 9 73 94 1.225 50 0.25 10 0.1

10 81 106 0.325 50 0.25 10 0.1 11 106 51 0.35 50 0.25 10 0.1 12 60 100 0.575 50 0.25 10 0.1 13 150 45 0.475 50 0.25 10 0.1 14 111 89 0.05 50 0.25 10 0.1 15 112 53 0.4 50 0.25 10 0.1 16 56 111 1 50 0.25 10 0.1 17 123 32 0.1 50 0.25 10 0.1 18 68 84 0.275 50 0.25 10 0.1 19 18 48 0.6 50 0.25 10 0.1 20 62 73 0.175 50 0.25 10 0.1 21 60 91 0.825 50 0.25 10 0.1 22 129 17 0.575 50 0.25 10 0.1 23 113 106 1.225 50 0.25 10 0.1 24 8 79 0.65 50 0.25 10 0.1 25 42 96 1.1 50 0.25 10 0.1 26 92 89 0.3 50 0.25 10 0.1 27 97 64 0.675 50 0.25 10 0.1 28 98 60 0.625 50 0.25 10 0.1 29 82 86 1.025 50 0.25 10 0.1 30 82 77 1.2 50 0.25 10 0.1 31 107 100 0.45 50 0.25 10 0.1 32 36 117 0.85 50 0.25 10 0.1 33 107 96 0.375 50 0.25 10 0.1 34 66 160 0.15 50 0.25 10 0.1 35 75 97 0.525 50 0.25 10 0.1 36 145 57 1.175 50 0.25 10 0.1 37 100 43 0.675 50 0.25 10 0.1 38 152 81 0.95 50 0.25 10 0.1 39 62 76 0.075 50 0.25 10 0.1 40 59 12 0.6 50 0.25 10 0.1 41 49 134 0.325 50 0.25 10 0.1 42 79 94 0.7 50 0.25 10 0.1 43 60 97 0.65 50 0.25 10 0.1 44 63 119 0.675 50 0.25 10 0.1 45 65 110 0.775 50 0.25 10 0.1 46 33 38 0.575 50 0.25 10 0.1 47 77 95 0.6 50 0.25 10 0.1 48 87 107 0.2 50 0.25 10 0.1 49 68 111 0.85 50 0.25 10 0.1 50 89 89 0.75 50 0.25 10 0.1 51 85 85 0.1 50 0.25 10 0.1 52 150 73 1.225 50 0.25 10 0.1 53 141 85 0.025 50 0.25 10 0.1 54 92 75 0.7 50 0.25 10 0.1

74

55 46 128 0.65 50 0.25 10 0.1 56 79 48 0.7 50 0.25 10 0.1 57 137 14 0.5 50 0.25 10 0.1 58 35 80 0.725 50 0.25 10 0.1 59 70 6 0.975 50 0.25 10 0.1 60 109 68 0.775 50 0.25 10 0.1 61 113 33 1.15 50 0.25 10 0.1 62 62 145 1.125 50 0.25 10 0.1 63 82 88 1.025 50 0.25 10 0.1 64 93 96 0.725 50 0.25 10 0.1 65 112 151 1.225 50 0.25 10 0.1 66 107 34 1.075 50 0.25 10 0.1 67 31 31 1.1 50 0.25 10 0.1



1 16 50 0.6 50 0.25 10 0.1 2 128 105 0.025 50 0.25 10 0.1 3 76 77 1 50 0.25 10 0.1 4 54 116 1.15 50 0.25 10 0.1 5 126 139 0.8 50 0.25 10 0.1 6 40 131 1.25 50 0.25 10 0.1 7 76 80 0.9 50 0.25 10 0.1 8 21 103 0.3 50 0.25 10 0.1 9 58 28 0.975 50 0.25 10 0.1

10 114 105 1.2 50 0.25 10 0.1 11 104 70 0.35 50 0.25 10 0.1 12 75 81 0.75 50 0.25 10 0.1 13 92 64 0.05 50 0.25 10 0.1 14 130 53 0.275 50 0.25 10 0.1 15 8 95 0.3 50 0.25 10 0.1 16 85 43 1.05 50 0.25 10 0.1 17 89 32 0.625 50 0.25 10 0.1 18 117 13 1.05 50 0.25 10 0.1 19 49 113 0.475 50 0.25 10 0.1 20 5 88 0.025 50 0.25 10 0.1 21 55 20 0.05 50 0.25 10 0.1 22 102 148 0.275 50 0.25 10 0.1 23 115 95 0.525 50 0.25 10 0.1 24 128 98 0.45 50 0.25 10 0.1 25 75 54 0.275 50 0.25 10 0.1 26 73 38 1.075 50 0.25 10 0.1 27 73 82 0.675 50 0.25 10 0.1 28 94 92 0.625 50 0.25 10 0.1 29 131 93 0.4 50 0.25 10 0.1

75

30 114 98 0.825 50 0.25 10 0.1 31 93 119 0.6 50 0.25 10 0.1 32 77 90 0.15 50 0.25 10 0.1 33 53 45 0.625 50 0.25 10 0.1 34 98 17 0.025 50 0.25 10 0.1 35 85 95 0.225 50 0.25 10 0.1 36 60 58 0.1 50 0.25 10 0.1 37 86 52 0.85 50 0.25 10 0.1 38 47 70 0.9 50 0.25 10 0.1 39 76 118 1.2 50 0.25 10 0.1 40 58 113 0.55 50 0.25 10 0.1 41 100 71 1.075 50 0.25 10 0.1 42 141 86 0.65 50 0.25 10 0.1 43 79 81 1.225 50 0.25 10 0.1 44 67 80 0.925 50 0.25 10 0.1 45 69 134 0.1 50 0.25 10 0.1 46 73 81 0.375 50 0.25 10 0.1 47 114 64 0.525 50 0.25 10 0.1 48 70 21 1.25 50 0.25 10 0.1 49 79 77 1.1 50 0.25 10 0.1 50 120 60 0.25 50 0.25 10 0.1 51 131 101 0.375 50 0.25 10 0.1 52 77 53 0.05 50 0.25 10 0.1 53 82 83 0.75 50 0.25 10 0.1 54 115 34 0.8 50 0.25 10 0.1 55 59 70 0.05 50 0.25 10 0.1 56 16 78 0.875 50 0.25 10 0.1 57 71 76 0.05 50 0.25 10 0.1 58 114 66 1.25 50 0.25 10 0.1 59 2 90 0.575 50 0.25 10 0.1 60 81 118 1.2 50 0.25 10 0.1 61 80 78 0.275 50 0.25 10 0.1 62 8 74 0.25 50 0.25 10 0.1 63 94 58 0.775 50 0.25 10 0.1 64 13 117 1.025 50 0.25 10 0.1 65 62 103 0.15 50 0.25 10 0.1 66 76 78 1.25 50 0.25 10 0.1 67 60 55 0.125 50 0.25 10 0.1 68 78 83 0.925 50 0.25 10 0.1 69 72 25 0.675 50 0.25 10 0.1 70 65 88 0.225 50 0.25 10 0.1 71 81 78 0.8 50 0.25 10 0.1 72 113 92 0.7 50 0.25 10 0.1 73 73 66 0.35 50 0.25 10 0.1 74 139 72 0.075 50 0.25 10 0.1 75 80 80 1.125 50 0.25 10 0.1 76 131 141 0.8 50 0.25 10 0.1 77 109 105 0.675 50 0.25 10 0.1 78 80 80 0.425 50 0.25 10 0.1 79 32 84 0.675 50 0.25 10 0.1 80 67 97 1.05 50 0.25 10 0.1 81 126 131 0.825 50 0.25 10 0.1

76



1 93 92 0.525 50 0.25 10 0.1 2 89 57 0.025 50 0.25 10 0.1 3 129 121 0.575 50 0.25 10 0.1 4 118 84 0.3 50 0.25 10 0.1 5 112 176 0.2 50 0.25 10 0.1 6 77 106 1.25 50 0.25 10 0.1 7 92 97 0.725 50 0.25 10 0.1 8 120 102 0.65 50 0.25 10 0.1 9 56 20 1.125 50 0.25 10 0.1

10 156 55 0.05 50 0.25 10 0.1 11 95 56 0.225 50 0.25 10 0.1 12 35 155 0.825 50 0.25 10 0.1 13 125 80 1.2 50 0.25 10 0.1 14 62 162 1.15 50 0.25 10 0.1 15 107 147 0.125 50 0.25 10 0.1 16 42 114 0.075 50 0.25 10 0.1 17 53 131 0.9 50 0.25 10 0.1 18 114 165 0.65 50 0.25 10 0.1 19 60 178 0.65 50 0.25 10 0.1 20 128 73 0.025 50 0.25 10 0.1 21 135 37 1.225 50 0.25 10 0.1 22 99 88 0.625 50 0.25 10 0.1 23 94 93 0.95 50 0.25 10 0.1 24 126 114 1 50 0.25 10 0.1 25 80 128 0.15 50 0.25 10 0.1 26 50 130 0.75 50 0.25 10 0.1 27 96 99 1.025 50 0.25 10 0.1 28 80 178 0.95 50 0.25 10 0.1 29 101 52 0.9 50 0.25 10 0.1 30 38 80 0.475 50 0.25 10 0.1 31 120 167 0.525 50 0.25 10 0.1 32 35 29 0.375 50 0.25 10 0.1 33 135 72 0.175 50 0.25 10 0.1 34 80 173 1.2 50 0.25 10 0.1 35 155 127 0.65 50 0.25 10 0.1 36 45 85 1.025 50 0.25 10 0.1 37 112 110 0.6 50 0.25 10 0.1 38 108 71 0.775 50 0.25 10 0.1 39 123 158 0.15 50 0.25 10 0.1 40 119 61 0.75 50 0.25 10 0.1 41 137 100 0.85 50 0.25 10 0.1 42 99 94 1.05 50 0.25 10 0.1

77

43 88 165 1 50 0.25 10 0.1 44 76 103 0.875 50 0.25 10 0.1 45 177 119 0.45 50 0.25 10 0.1 46 48 61 0.125 50 0.25 10 0.1 47 89 70 0.8 50 0.25 10 0.1 48 93 77 0.7 50 0.25 10 0.1 49 167 75 0.925 50 0.25 10 0.1 50 89 123 0.15 50 0.25 10 0.1 51 57 81 0.1 50 0.25 10 0.1 52 94 93 0.375 50 0.25 10 0.1 53 118 12 0.325 50 0.25 10 0.1 54 98 25 0.3 50 0.25 10 0.1 55 125 69 0.775 50 0.25 10 0.1 56 25 108 0.225 50 0.25 10 0.1 57 156 154 0.625 50 0.25 10 0.1 58 129 112 0.75 50 0.25 10 0.1 59 75 118 0.275 50 0.25 10 0.1 60 35 136 1.25 50 0.25 10 0.1 61 174 48 0.75 50 0.25 10 0.1 62 93 91 1.1 50 0.25 10 0.1 63 138 133 0.15 50 0.25 10 0.1 64 77 112 0.75 50 0.25 10 0.1 65 67 18 0.275 50 0.25 10 0.1 66 110 82 0.325 50 0.25 10 0.1 67 120 108 1.1 50 0.25 10 0.1 68 61 112 0.925 50 0.25 10 0.1 69 102 23 1.225 50 0.25 10 0.1 70 105 18 0.45 50 0.25 10 0.1 71 83 106 0.575 50 0.25 10 0.1 72 111 131 0.525 50 0.25 10 0.1 73 99 115 0.5 50 0.25 10 0.1 74 84 98 0.175 50 0.25 10 0.1 75 88 61 0.75 50 0.25 10 0.1 76 89 94 0.625 50 0.25 10 0.1 77 81 176 0.825 50 0.25 10 0.1 78 51 101 0.175 50 0.25 10 0.1 79 93 93 0.425 50 0.25 10 0.1 80 37 123 1 50 0.25 10 0.1 81 103 132 0.775 50 0.25 10 0.1 82 153 80 0.175 50 0.25 10 0.1 83 60 166 0.8 50 0.25 10 0.1 84 135 72 0.6 50 0.25 10 0.1 85 95 95 0.25 50 0.25 10 0.1 86 148 73 1.125 50 0.25 10 0.1 87 83 95 0.775 50 0.25 10 0.1 88 13 130 0.75 50 0.25 10 0.1 89 114 41 0.775 50 0.25 10 0.1 90 115 44 0.325 50 0.25 10 0.1 91 175 80 1 50 0.25 10 0.1 92 24 124 0.825 50 0.25 10 0.1 93 69 120 1.175 50 0.25 10 0.1

78



1 15 35 0.95 50 0.25 10 0.1 2 132 70 1 50 0.25 10 0.1 3 45 69 0.6 50 0.25 10 0.1 4 79 89 1.075 50 0.25 10 0.1 5 126 138 1.075 50 0.25 10 0.1 6 47 61 0.1 50 0.25 10 0.1 7 82 73 0.9 50 0.25 10 0.1 8 122 65 0.775 50 0.25 10 0.1 9 87 66 0.825 50 0.25 10 0.1

10 110 86 0.85 50 0.25 10 0.1 11 116 154 1.2 50 0.25 10 0.1 12 74 53 1.2 50 0.25 10 0.1 13 56 10 0.925 50 0.25 10 0.1 14 79 54 0.5 50 0.25 10 0.1 15 126 122 0.875 50 0.25 10 0.1 16 147 75 0.425 50 0.25 10 0.1 17 139 72 0.5 50 0.25 10 0.1 18 86 56 1.25 50 0.25 10 0.1 19 82 80 0.775 50 0.25 10 0.1 20 84 80 0.25 50 0.25 10 0.1 21 79 79 1.25 50 0.25 10 0.1 22 73 81 1.025 50 0.25 10 0.1 23 120 77 0.3 50 0.25 10 0.1 24 31 63 0.25 50 0.25 10 0.1 25 85 63 0.5 50 0.25 10 0.1 26 45 141 0.1 50 0.25 10 0.1 27 31 59 1 50 0.25 10 0.1 28 95 95 0.95 50 0.25 10 0.1 29 132 121 0.525 50 0.25 10 0.1 30 62 111 0.375 50 0.25 10 0.1 31 84 82 1 50 0.25 10 0.1 32 107 150 0.825 50 0.25 10 0.1 33 77 69 0.275 50 0.25 10 0.1 34 92 91 0.075 50 0.25 10 0.1 35 117 106 0.4 50 0.25 10 0.1 36 84 77 0.8 50 0.25 10 0.1 37 80 101 1.225 50 0.25 10 0.1 38 132 46 0.15 50 0.25 10 0.1 39 150 34 0.75 50 0.25 10 0.1 40 69 138 0.8 50 0.25 10 0.1 41 115 141 1.05 50 0.25 10 0.1 42 46 149 0.2 50 0.25 10 0.1 43 133 50 1.075 50 0.25 10 0.1

79

44 61 68 0.825 50 0.25 10 0.1 45 112 65 0.1 50 0.25 10 0.1 46 75 86 0.2 50 0.25 10 0.1 47 51 82 0.425 50 0.25 10 0.1 48 81 65 0.2 50 0.25 10 0.1 49 46 40 1 50 0.25 10 0.1 50 82 34 1.05 50 0.25 10 0.1 51 101 56 0.65 50 0.25 10 0.1 52 67 68 0.1 50 0.25 10 0.1 53 83 80 0.425 50 0.25 10 0.1 54 120 76 0.1 50 0.25 10 0.1 55 163 67 1.175 50 0.25 10 0.1 56 97 82 0.475 50 0.25 10 0.1 57 115 119 0.05 50 0.25 10 0.1 58 88 110 1.225 50 0.25 10 0.1 59 87 89 1.1 50 0.25 10 0.1 60 64 8 0.875 50 0.25 10 0.1 61 119 60 0.975 50 0.25 10 0.1 62 73 67 1.225 50 0.25 10 0.1 63 126 89 0.825 50 0.25 10 0.1 64 129 22 0.075 50 0.25 10 0.1 65 64 52 0.525 50 0.25 10 0.1 66 130 21 1.225 50 0.25 10 0.1 67 88 69 0.9 50 0.25 10 0.1 68 93 88 0.5 50 0.25 10 0.1 69 74 141 0.325 50 0.25 10 0.1 70 117 108 0.25 50 0.25 10 0.1 71 107 10 0.475 50 0.25 10 0.1 72 112 125 0.325 50 0.25 10 0.1 73 101 97 0.625 50 0.25 10 0.1 74 43 156 1.25 50 0.25 10 0.1 75 42 22 1.175 50 0.25 10 0.1 76 98 106 1.15 50 0.25 10 0.1 77 93 53 0.575 50 0.25 10 0.1 78 103 91 0.575 50 0.25 10 0.1 79 98 109 0.45 50 0.25 10 0.1 80 79 83 0.45 50 0.25 10 0.1 81 52 122 0.875 50 0.25 10 0.1 82 52 32 0.45 50 0.25 10 0.1 83 24 86 0.25 50 0.25 10 0.1 84 25 74 0.575 50 0.25 10 0.1 85 83 83 0.325 50 0.25 10 0.1 86 108 5 0.025 50 0.25 10 0.1 87 76 89 0.55 50 0.25 10 0.1 88 58 120 0.475 50 0.25 10 0.1 89 29 104 0.325 50 0.25 10 0.1 90 135 141 0.3 50 0.25 10 0.1 91 90 72 0.1 50 0.25 10 0.1 92 153 83 0.05 50 0.25 10 0.1 93 116 130 0.325 50 0.25 10 0.1 94 83 83 1.175 50 0.25 10 0.1 95 41 117 0.2 50 0.25 10 0.1 96 121 72 0.825 50 0.25 10 0.1

80

97 46 73 1.1 50 0.25 10 0.1 98 68 105 0.325 50 0.25 10 0.1



1 93 55 0.775 50 0.25 10 0.1 2 52 91 1.05 50 0.25 10 0.1 3 97 85 0.5 50 0.25 10 0.1 4 68 99 0.05 50 0.25 10 0.1 5 119 111 0.35 50 0.25 10 0.1 6 170 72 0.425 50 0.25 10 0.1 7 106 35 0.775 50 0.25 10 0.1 8 130 37 0.75 50 0.25 10 0.1 9 82 64 1.175 50 0.25 10 0.1

10 127 32 0.525 50 0.25 10 0.1 11 83 118 0.375 50 0.25 10 0.1 12 139 109 0.625 50 0.25 10 0.1 13 156 52 0.6 50 0.25 10 0.1 14 70 142 0.7 50 0.25 10 0.1 15 138 17 0.075 50 0.25 10 0.1 16 24 56 0.8 50 0.25 10 0.1 17 129 160 1.075 50 0.25 10 0.1 18 112 129 1 50 0.25 10 0.1 19 22 102 0.2 50 0.25 10 0.1 20 24 38 0.3 50 0.25 10 0.1 21 88 84 1.05 50 0.25 10 0.1 22 122 73 1.15 50 0.25 10 0.1 23 115 11 0.85 50 0.25 10 0.1 24 36 140 0.575 50 0.25 10 0.1 25 104 155 0.95 50 0.25 10 0.1 26 98 107 0.625 50 0.25 10 0.1 27 115 73 0.675 50 0.25 10 0.1 28 71 65 0.725 50 0.25 10 0.1 29 120 154 0.65 50 0.25 10 0.1 30 48 73 1.25 50 0.25 10 0.1 31 48 36 0.25 50 0.25 10 0.1 32 113 126 0.85 50 0.25 10 0.1 33 140 76 1.15 50 0.25 10 0.1 34 79 88 0.525 50 0.25 10 0.1 35 66 149 0.75 50 0.25 10 0.1 36 79 94 0.975 50 0.25 10 0.1 37 105 95 1.25 50 0.25 10 0.1 38 108 149 0.3 50 0.25 10 0.1 39 110 74 0.75 50 0.25 10 0.1 40 134 65 0.275 50 0.25 10 0.1

81

41 77 153 1.25 50 0.25 10 0.1 42 156 118 0.4 50 0.25 10 0.1 43 85 74 0.125 50 0.25 10 0.1 44 74 96 0.5 50 0.25 10 0.1 45 107 62 0.575 50 0.25 10 0.1 46 53 90 0.175 50 0.25 10 0.1 47 45 155 0.175 50 0.25 10 0.1 48 101 76 0.325 50 0.25 10 0.1 49 109 87 0.775 50 0.25 10 0.1 50 168 74 0.05 50 0.25 10 0.1 51 101 91 1.05 50 0.25 10 0.1 52 111 147 1.25 50 0.25 10 0.1 53 39 135 0.325 50 0.25 10 0.1 54 55 94 1.05 50 0.25 10 0.1 55 82 79 0.525 50 0.25 10 0.1 56 120 92 0.075 50 0.25 10 0.1 57 123 76 0.025 50 0.25 10 0.1 58 153 51 0.875 50 0.25 10 0.1 59 103 106 0.725 50 0.25 10 0.1 60 95 8 0.525 50 0.25 10 0.1 61 15 125 0.275 50 0.25 10 0.1 62 79 25 0.525 50 0.25 10 0.1 63 128 133 0.425 50 0.25 10 0.1 64 110 80 0.125 50 0.25 10 0.1 65 90 107 1.025 50 0.25 10 0.1 66 9 97 0.875 50 0.25 10 0.1 67 89 85 0.525 50 0.25 10 0.1 68 36 140 0.625 50 0.25 10 0.1 69 70 107 0.05 50 0.25 10 0.1 70 159 127 0.4 50 0.25 10 0.1 71 130 71 0.225 50 0.25 10 0.1 72 112 155 0.275 50 0.25 10 0.1 73 49 128 0.3 50 0.25 10 0.1 74 75 81 1 50 0.25 10 0.1 75 77 130 0.45 50 0.25 10 0.1 76 97 40 0.9 50 0.25 10 0.1 77 68 87 0.175 50 0.25 10 0.1 78 90 89 0.35 50 0.25 10 0.1 79 135 119 0.15 50 0.25 10 0.1 80 48 137 1.15 50 0.25 10 0.1 81 70 50 0.975 50 0.25 10 0.1 82 43 78 0.5 50 0.25 10 0.1 83 89 74 0.975 50 0.25 10 0.1 84 68 93 1.15 50 0.25 10 0.1 85 98 96 0.575 50 0.25 10 0.1 86 108 88 0.4 50 0.25 10 0.1 87 37 80 0.85 50 0.25 10 0.1 88 101 107 0.325 50 0.25 10 0.1 89 86 143 1.225 50 0.25 10 0.1 90 79 55 0.225 50 0.25 10 0.1 91 36 120 0.925 50 0.25 10 0.1 92 134 136 1.1 50 0.25 10 0.1 93 94 53 0.95 50 0.25 10 0.1

82

94 85 82 1 50 0.25 10 0.1 95 92 107 0.875 50 0.25 10 0.1 96 76 22 0.1 50 0.25 10 0.1 97 57 130 0.4 50 0.25 10 0.1 98 163 71 1.05 50 0.25 10 0.1 99 42 84 0.6 50 0.25 10 0.1

100 111 89 0.75 50 0.25 10 0.1 101 95 31 0.425 50 0.25 10 0.1 102 56 128 0.925 50 0.25 10 0.1 103 118 103 1.225 50 0.25 10 0.1 104 147 83 0.1 50 0.25 10 0.1 105 80 165 0.775 50 0.25 10 0.1 106 46 40 0.825 50 0.25 10 0.1 107 49 122 0.45 50 0.25 10 0.1 108 120 40 0.725 50 0.25 10 0.1 109 116 115 0.15 50 0.25 10 0.1 110 142 119 0.125 50 0.25 10 0.1



1 131 127 0.375 50 0.25 10 0.1 2 53 105 0.325 50 0.25 10 0.1 3 209 234 1 50 0.25 10 0.1 4 214 107 1.175 50 0.25 10 0.1 5 227 136 1.125 50 0.25 10 0.1 6 125 213 0.9 50 0.25 10 0.1 7 136 63 0.075 50 0.25 10 0.1 8 183 7 0.65 50 0.25 10 0.1 9 113 139 1.125 50 0.25 10 0.1

10 118 206 1.075 50 0.25 10 0.1 11 152 25 0.225 50 0.25 10 0.1 12 161 148 0.85 50 0.25 10 0.1 13 294 119 0.675 50 0.25 10 0.1 14 145 169 0.275 50 0.25 10 0.1 15 117 93 0.05 50 0.25 10 0.1 16 150 150 0.725 50 0.25 10 0.1 17 206 256 0.075 50 0.25 10 0.1 18 114 231 0.75 50 0.25 10 0.1 19 124 105 0.025 50 0.25 10 0.1 20 34 150 0.15 50 0.25 10 0.1 21 151 167 0.925 50 0.25 10 0.1 22 80 23 0.9 50 0.25 10 0.1 23 155 148 0.8 50 0.25 10 0.1 24 23 137 0.65 50 0.25 10 0.1 25 157 176 0.875 50 0.25 10 0.1

83

26 137 175 0.5 50 0.25 10 0.1 27 127 163 0.1 50 0.25 10 0.1 28 247 218 0.375 50 0.25 10 0.1 29 119 48 1.175 50 0.25 10 0.1 30 162 63 0.675 50 0.25 10 0.1 31 180 140 1.025 50 0.25 10 0.1 32 272 185 0.925 50 0.25 10 0.1 33 67 132 0.975 50 0.25 10 0.1 34 86 111 0.325 50 0.25 10 0.1 35 148 155 1 50 0.25 10 0.1 36 234 114 1.125 50 0.25 10 0.1 37 105 150 0.675 50 0.25 10 0.1 38 209 276 1.15 50 0.25 10 0.1 39 204 81 0.2 50 0.25 10 0.1 40 233 190 0.25 50 0.25 10 0.1 41 134 281 0.175 50 0.25 10 0.1 42 142 138 0.475 50 0.25 10 0.1 43 193 102 0.75 50 0.25 10 0.1 44 30 66 0.75 50 0.25 10 0.1 45 120 149 0.775 50 0.25 10 0.1 46 136 130 0.575 50 0.25 10 0.1 47 85 106 1.05 50 0.25 10 0.1 48 148 145 0.725 50 0.25 10 0.1 49 40 89 0.475 50 0.25 10 0.1 50 268 123 0.85 50 0.25 10 0.1 51 80 218 0.55 50 0.25 10 0.1 52 293 155 0.45 50 0.25 10 0.1 53 218 192 1.075 50 0.25 10 0.1 54 116 268 0.875 50 0.25 10 0.1 55 142 182 0.075 50 0.25 10 0.1 56 124 111 0.625 50 0.25 10 0.1 57 198 18 0.425 50 0.25 10 0.1 58 260 186 0.475 50 0.25 10 0.1 59 75 63 0.95 50 0.25 10 0.1 60 269 109 0.2 50 0.25 10 0.1 61 169 165 0.325 50 0.25 10 0.1 62 78 88 0.225 50 0.25 10 0.1 63 208 75 0.9 50 0.25 10 0.1 64 199 23 0.75 50 0.25 10 0.1 65 85 130 1.025 50 0.25 10 0.1 66 125 202 0.35 50 0.25 10 0.1 67 82 89 0.2 50 0.25 10 0.1 68 133 181 0.25 50 0.25 10 0.1 69 96 48 1.075 50 0.25 10 0.1 70 239 109 0.075 50 0.25 10 0.1 71 88 192 0.325 50 0.25 10 0.1 72 98 90 0.075 50 0.25 10 0.1 73 117 92 0.5 50 0.25 10 0.1 74 134 197 1.2 50 0.25 10 0.1 75 252 216 0.8 50 0.25 10 0.1 76 148 145 0.575 50 0.25 10 0.1 77 53 223 1.2 50 0.25 10 0.1 78 181 143 0.65 50 0.25 10 0.1

84

79 174 106 1 50 0.25 10 0.1 80 160 184 0.25 50 0.25 10 0.1 81 36 140 0.95 50 0.25 10 0.1 82 229 110 0.25 50 0.25 10 0.1 83 250 215 0.65 50 0.25 10 0.1 84 204 124 1.05 50 0.25 10 0.1 85 91 107 0.15 50 0.25 10 0.1 86 17 109 0.85 50 0.25 10 0.1 87 157 85 0.625 50 0.25 10 0.1 88 151 104 0.25 50 0.25 10 0.1 89 150 150 0.95 50 0.25 10 0.1 90 125 93 0.55 50 0.25 10 0.1 91 250 193 0.45 50 0.25 10 0.1 92 40 114 1.025 50 0.25 10 0.1 93 238 62 1.25 50 0.25 10 0.1 94 73 214 0.025 50 0.25 10 0.1 95 173 18 0.975 50 0.25 10 0.1 96 128 196 0.775 50 0.25 10 0.1 97 62 155 0.175 50 0.25 10 0.1 98 195 138 1.05 50 0.25 10 0.1 99 150 241 0.95 50 0.25 10 0.1

100 190 275 1.2 50 0.25 10 0.1 101 121 240 0.175 50 0.25 10 0.1 102 236 186 0.725 50 0.25 10 0.1 103 167 68 0.85 50 0.25 10 0.1 104 126 134 0.725 50 0.25 10 0.1 105 254 150 0.325 50 0.25 10 0.1 106 167 150 0.5 50 0.25 10 0.1 107 132 166 0.375 50 0.25 10 0.1 108 204 57 0.925 50 0.25 10 0.1 109 85 176 0.175 50 0.25 10 0.1 110 144 116 0.6 50 0.25 10 0.1 111 138 145 0.325 50 0.25 10 0.1 112 148 147 1.15 50 0.25 10 0.1 113 158 110 0.65 50 0.25 10 0.1 114 269 84 0.35 50 0.25 10 0.1 115 97 18 1.025 50 0.25 10 0.1 116 83 94 0.8 50 0.25 10 0.1 117 90 233 0.6 50 0.25 10 0.1 118 182 135 0.95 50 0.25 10 0.1 119 284 164 1.125 50 0.25 10 0.1 120 133 116 0.975 50 0.25 10 0.1

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Appendix E. VBA code for simulation

Reading the demands from the dat-file demrate = dem_rate_string(num, 1) demarray = Splits(demrate) demstring = Joins(demarray) deminteger = CDbl(demstring) ‘ demandconv = deminteger If demrate = 0 Then

demanconv = 0 End If If demandconv >= 1000 Then

demandconv = demandconv / 1000 End If End Function Public Function Splits(ByVal sIn As String, _ Optional sDelim As String = ".") _ As Variant Dim nC As Long Dim nPos As Long Dim nDelimLen As Long Dim sOut() As String If sDelim <> "" Then nDelimLen = Len(sDelim) nPos = InStr(1, sIn, sDelim, bCompare) Do While nPos ReDim Preserve sOut(nC) sOut(nC) = Left(sIn, nPos - 1) sIn = Mid(sIn, nPos + nDelimLen) nC = nC + 1 nPos = InStr(1, sIn, sDelim, bCompare) Loop End If ReDim Preserve sOut(nC) sOut(nC) = sIn Splits = sOut End Function

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Public Function Joins(Source() As String, _ Optional sDelim As String = ",") As String Dim nC As Long Dim sOut As String For nC = LBound(Source) To UBound(Source) - 1 sOut = sOut & Source(nC) & sDelim Next Joins = sOut & Source(nC) End Function Determining the parameters of the normal distributions for the stochastic process Function demandparoup(num) demrate = Demand_rate(num, 1) demrateoupweek = 40 * demrate demrateoupweekint = CInt(demrateoupweek) rate1 = demrateoupweekint * 70 / 100 rate2 = 2 * demrateoupweek - rate1 rate1int = CInt(rate1) rate2int = CInt(rate2) demarray(0) = rate1int demarray(1) = 1 demarray(2) = rate2int demarray(3) = 1 demandparoup = demarray Holding cost rate for EOQ-retailers Function holdingeoq(num) demrate = Demand_rate(num, 1) vehiclenr = Vehicle_nr(num, 1) kolomcycleT = 1 + (vehiclenr - 1) * 2

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cycleT = Tcycle(1, kolomcycleT) Hcost = Range("HC").Value holding = cycleT * Hcost * demrate / 2 holdingeoq = holding End Function Holding cost for the Order-Up-to retailers retailer = num demrate = Demand_rate(retailer, 1) vehiclenr = Vehicle_nr(retailer, 1) kolomcycleT = 1 + (vehiclenr - 1) * 2 cycleT = Tcycle(1, kolomcycleT) Hcost = Range("HC").Value numberhoursim = Range("nrhours").Value delivery = cycleT * demrate voorraad = delivery demandweek = demandoup(retailer) demand = demandweek / 40 For i = 1 To numberhoursim rest = i Mod cycleT If rest = 0 Then voorraad = voorraad + delivery End If voorraad = voorraad - demand If voorraad >= 0 Then stockage = stockage + voorraad Else voorraad = 0 End If rest2 = i Mod 40 If rest2 = 0 Then demandweek = demandoup(retailer) demand = demandweek / 40 End If Next i totalstockagecost = stockage * Hcost holdingrate = totalstockagecost / numberhoursim holdingoup = holdingrate End Function Function that generates weekly demand according to stochastic process

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Function demandoup(num) randomvariable = Rnd() randomvariable2 = Rnd() If randomvariable <= 0.5 Then standnorm = WorksheetFunction.NormSInv(randomvariable2) avg = mu1(num, 1) dev = sigma1(num, 1) dema = standnorm * Sqr(dev) + avg Else standnorm = WorksheetFunction.NormSInv(randomvariable2) avg = mu2(num, 1) dev = sigma2(num, 1) dema = standnorm * Sqr(dev) + avg End If If dema > 0 Then demandoup = dema Else demandoup = 0 End If End Function Function to assign the appropriated vehicle to the retailer Function vehicleretailer(num As Integer) retailer = num numbervehicles = Range("nrvehicles").Value For i = 1 To numbervehicles Column = 2 * (i - 1) + 1 numbersubtours = nrsubtours(1, Column) For j = 1 To numbersubtours numberveh = i numbersubt = j arraymaking = arraymaken(numberveh, numbersubt) numbersalespoints = UBound(arraymaking) For k = 0 To numbersalespoints - 1 salespoint = arraymaking(k) salespointint = Val(salespoint) If salespointint = retailer Then voertuig = numberveh End If Next k

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Next j Next i vehicleretailer = voertuig End Function Function arraymaken(num As Integer, nsubtour As Integer) kolom = (num - 1) * 2 + 1 rij = nsubtour subtourke = subtours(rij, kolom) tussenresultaat = Split(subtourke) arraymaken = tussenresultaat End Function Calculating the shortage cost rate Function penalty(num) retailer = num demrate = Demand_rate(retailer, 1) vehiclenr = Vehicle_nr(retailer, 1) kolomcycleT = 1 + (vehiclenr - 1) * 2 cycleT = Tcycle(1, kolomcycleT) Hcost = Range("HC").Value numberhoursim = Range("nrhours").Value delivery = cycleT * demrate voorraad = delivery demandweek = demandoup(retailer) demand = demandweek / 40 For i = 1 To numberhoursim rest = i Mod cycleT If rest = 0 Then voorraad = voorraad + delivery End If voorraad = voorraad - demand If voorraad >= 0 Then stockage = stockage + voorraad Else shortage = shortage - voorraad voorraad = 0 End If rest2 = i Mod 40 If rest2 = 0 Then

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demandweek = demandoup(retailer) demand = demandweek / 40 End If Next i penaltycost = Range("PC").Value penaltycostrate = penaltycost * shortage / (numberhoursim) penalty = penaltycostrate End Function Calculating service level P1 Function service1(num) Dim breach As Integer retailer = num demrate = Demand_rate(retailer, 1) vehiclenr = Vehicle_nr(retailer, 1) kolomcycleT = 1 + (vehiclenr - 1) * 2 cycleT = Tcycle(1, kolomcycleT) Hcost = Range("HC").Value numberhoursim = Range("nrhours").Value delivery = cycleT * demrate voorraad = delivery demandweek = demandoup(retailer) demand = demandweek / 40 delivery = cycleT * demrate voorraad = delivery demandweek = demandoup(retailer) demand = demandweek / 40 For i = 1 To numberhoursim rest = i Mod cycleT If rest = 0 Then voorraad = voorraad + delivery numbreaches = numbreaches + breach breach = 0 End If voorraad = voorraad - demand If voorraad >= 0 Then stockage = stockage + voorraad Else breach = 1 voorraad = 0 End If rest2 = i Mod 40

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If rest2 = 0 Then demandweek = demandoup(retailer) demand = demandweek / 40 End If Next i P = numbreaches / (numbercycles) service1 = 1 - P End Function Calculating service level 2 Function service2(num) retailer = num demrate = Demand_rate(retailer, 1) vehiclenr = Vehicle_nr(retailer, 1) kolomcycleT = 1 + (vehiclenr - 1) * 2 cycleT = Tcycle(1, kolomcycleT) Hcost = Range("HC").Value numberhoursim = Range("nrhours").Value delivery = cycleT * demrate voorraad = delivery demandweek = demandoup(retailer) demand = demandweek / 40 For i = 1 To numberhoursim rest = i Mod cycleT If rest = 0 Then voorraad = voorraad + delivery End If voorraad = voorraad - demand Totaldemand = Totaldemand + demand If voorraad >= 0 Then stockage = stockage + voorraad Else shortage = shortage - voorraad voorraad = 0 End If rest2 = i Mod 40 If rest2 = 0 Then demandweek = demandoup(retailer) demand = demandweek / 40 End If Next i tussen = shortage / Totaldemand

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P = 1 - tussen service2 = P End Function Calculating breaches Function breaches(num) retailer = num demrate = Demand_rate(retailer, 1) vehiclenr = Vehicle_nr(retailer, 1) kolomcycleT = 1 + (vehiclenr - 1) * 2 cycleT = Tcycle(1, kolomcycleT) Hcost = Range("HC").Value numberhoussim = Range("nrhours").Value delivery = cycleT * demrate voorraad = delivery demandweek = demandoup(retailer) demand = demandweek / 40 interhours = 0 For i = 1 To numberhoursim interhours = interhours + 1 rest = i Mod cycleT If rest = 0 Then voorraad = voorraad + delivery End If voorraad = voorraad - demand Totaldemand = Totaldemand + demand If voorraad >= 0 Then stockage = stockage + voorraad Else breach = breach + 1 suminter = suminter + interhours interhours = 0 voorraad = 0 End If rest2 = i Mod 40 If rest2 = 0 Then demandweek = demandoup(retailer) demand = demandweek / 40 End If Next i breaches = suminter / breach End Function

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