A Net Present Value Approach to Safety Stocks in a Multi-level MRP

*Corresponding author. Tel.: #46 13 281 504; fax: #46 13288 975; e-mail: [email protected].

Int. J. Production Economics 59 (1999) 361—375

A net present value approach to safety stocks in a multi-level MRPsystem

Robert W. Grubbstrom*

Department of Production Economics, Linko( ping Institute of Technology, S-581 83 Linko( ping, Sweden

Abstract

The Laplace transform and input—output Analysis have been used for formulating a basic theory for materialrequirements planning (MRP) in a sequence of recent papers. The transform has been used for describing timedevelopments and lags of the relevant production, demand and inventory variables in a compact way including effects oforder flows and lead times. Secondly, the transform has functioned as a generating function, and thirdly, the transformhas been applied for assessing cash flows adopting the net present value (NPV) principle (or the annuity stream which isa variation of NPV). Input-output Analysis, in particular, the input matrix, has been applied for describing multi-levelproduct structures. This has made the analysis compact and distinct.

In the current paper the analysis of determining optimal safety stock levels in MRP systems is extended froma single-level model recently investigated to a multi-level system assuming production to take place according toa lot-for-lot (L4L) policy when applying the NPV as the objective criterion and disregarding the opportunities for jointset-ups. Relaxing the L4L assumption is also discussed. ( 1999 Elsevier Science B.V. All rights reserved.

Keywords: MRP; Input—output analysis; Laplace transform; Multi-period; Production; Safety stock

1. Introduction

This paper follows up a recent paper treatinga similar problem for a single-level MRP system[1]. The main question addressed, is to find theamounts to produce together with their timing con-cerning end items having multi-level product struc-tures, when the demand for these end items isstochastic. The objective is to choose a best produc-tion plan maximising the net present value (NPV) of

the cash flow associated with production and de-mand. By deciding on the produced amounts andtheir timing, safety stock decisions will be incorpor-ated automatically.

Production on different levels is assumed to takeplace in batches of different sizes at different pointsin time. Cumulative production of an item will thenfollow a staircase function. The safety stock is thendefined as the difference between cumulative pro-duction and expected cumulative demand immedi-ately prior to a replenishment. This difference maybe negative in such cases when a positive expectedbacklog is favourable. This might typically happenfor an early batch in the sequence. Safety stocks are

0925-5273/99/$ - see front matter ( 1999 Elsevier Science B.V. All rights reserved.PII: S 0 9 2 5 - 5 2 7 3 ( 9 8 ) 0 0 0 1 6 - 4

Fig. 1. Inventory and backlog relationships.

only considered for top-level items, i.e. for finishedgoods having an external demand.

2. Fundamental equations

Let there be N items in the system altogether.Demand D, stock S and production P (includingpurchases when appropriate) are represented byN-dimensional column vectors each being a func-tion of time. These vectors are rates with the dimen-sion units per time unit and they are turned intoLaplace transforms denoted by tildes or by £M ) NTime integrals (cumulative values) of functions aredenoted by bars and inverse transforms by £~1M ) N.The notation i is reserved for the imaginary unit

J!1.For the production (assembly) of one unit of item

j, there is a need in the amount of hkj

of item k, andthere is a lead time q

jahead of the completion of the

production at which the components are needed.The h

kjare arranged into the square input matrix

H describing the product structures of all relevantproducts. The lead times q

1, q

2, 2, q

N, create inter-

nal demands and are represented by a diagonalmatrix, qJ the lead time matrix, having esqj in its jthdiagonal position, where s is the complex Laplacefrequency. HI "HqJ is the generalised input matrixand it captures component requirements togetherwith their required timing.

Consider Fig. 1 explaining the main equations tofollow. The following definitions are adopted. ¹otal

inventory SI accounts for all items in the system andis initial stock plus cumulative net production(I!H)PI /s less cumulative deliveries FI (s)/s:

SI (s)"S(0)#(I!H)PI (s)!FI (s)

s, (1)

where S(0) is the vector of initial inventory and I theidentity matrix. The factor 1/s is an operator ac-counting for a time integration.

Deliveries concern items that are demanded ex-ternally. Deliveries from inventory occur as soon asthere is a demand of an item not previously satisfiedand there is a supply to meet this demand, i.e. thatthis demand can be covered from available inven-tory.

Available inventory RI (s) is cumulative productionPI (s)/s less cumulative deliveries FI (s)/s and cumulat-ive internal demand HPI (s)/s, which evaluated at thetime the components are reserved is HqJ PI (s)/s, andRI (s) (as a time function) must never be negative,otherwise the production plan will be infeasible:

RI (s)"R(0)#(I!HqJ )PI (s)!FI (s)

s. (2)

Internal demand must always be met, but if ex-ternal demand cannot be met, we assume that thisdemand is backlogged and satisfied at the timeavailable inventory starts to become positive onceagain (when a batch is completed). The non-negativity of the inverse transform of RI (s) in eachcomponent is the main set of constraints of thisclass of problems.

362 R.W. Grubbstrom/Int. J. Production Economics 59 (1999) 361—375

Allocated component stock AI (s) is defined as thedifference between (total) inventory and availableinventory. This stock is reserved for productionand is part of work-in-process.

AI (s)"SI (s)!RI (s)"A(0)#H(qJ!I)PI (s)

s, (3)

where A(0)"S(0)!R(0). Backlogs BI (s) onlyconcern external demand and are given by initialbacklogs plus cumulative external demand lessdeliveries:

BI (s)"B(0)#DI (s)!FI (s)

s. (4)

Backlogs are necessarily non-negative. If Eqs. (2)and (4) are subtracted eliminating FI (s), we obtain

RI (s)"R(0)!B(0)#(I!HqJ )PI (s)!DI (s)

s#BI (s). (5)

For any item, its available inventory and its back-log cannot be positive at the same time, since a de-livery takes place from available inventory as soonas there is an unsatisfied external demand. Hence, iffor any component R

j(t)'0 at time t, then

Bj(t)"0, and vice versa. Therefore R(t) and B(t),

both being non-negative, may be written as

R(t)"C£~1GR(0)!B(0)#(I!HqJ )PI (s)!DI (s)

s HD`

,

(6)

B(t)"C£~1GB(0)!R(0)!(I!HqJ )PI (s)#DI (s)

s HD`,

(7)

where [ ) ]` is the maximum operator MaxM0, ) Noperating on a time function, and in transformterms

RI (s)"CR(0)!B(0)#(I!HqJ )PI (s)!DI (s)

s D`

, (8)

BI (s)"CB(0)!R(0)!(I!HqJ )PI (s)#DI (s)

s D`

, (9)

where [ ) ]`

is the transform of the non-negativetruncation of the time function corresponding to

the argument, since cumulative production as wellas cumulative external demand are zero at timet"0. Total inventory, being the sum of availableinventory and allocated component stock thenobeys

SI (s)"CR(0)!B(0)#(I!HqJ )PI (s)!DI (s)

s D`

#

A(0)#H(qJ!I)PI (s)s

, (10)

or in the time domain

S(t)"C£~1GR(0)!B(0)#(I!HqJ )PI (s)!DI (s)

s HD`

#£~1GA(0)#H(qJ!I)PI (s)

s H"MaxG£~1G

A(0)#H(qJ!I)PI (s)s H,

£~1GS(0)!B(0)#(I!H)PI (s)!DI (s)

s HH. (11)

The equations above defining the development ofSI , RI , BI and AI , we call the fundamental equations.

3. Solution to stockout function

We assume demand to be a renewal process, i.e.demand is created by unit events separated byindependent stochastic time intervals, all havingthe same probability distribution f (t), t*0.

A basic problem is to find an expression forE(B(t)) and E(BI (s)) in terms of decision variablesand stochastic properties of demand.

Theorem. If external demand for an individual itemfollows a renewal process and cumulative productionat time t is PM (including initial stock), then, during thetime that cumulative production remains at PM , thelevel of expected stockouts for the item B(t) will bepart of the function

E(B(t))"£~1ME(BI (s))N"£~1GfI PM `1(s)

s(1!fI (s))H, (12)

for the item in this interval.

R.W. Grubbstrom/Int. J. Production Economics 59 (1999) 361—375 363

Fig. 2. Illustration to stockout function for four different levels of cumulative production PM .

Therefore, the expected stockout as a function oftime can be written as the sum

E(B(t))"+PM[£~1ME(BI (s))N]

PMsPM

"+PM C£~1G

fI PM `1(s)

s(1!fI (s))HDPMsPM

, (13)

where sPM is the characteristic function being unity

within the intervals where cumulative productionof the item is PM and zero elsewhere, and where thesummation takes place over all intervals.

Proof of this theorem is given in Refs. [2,3].As an illustration, we choose demand as a Pois-

son process having the parameter j (per time unit).The resulting stockout function E(B(t)) is shown inFig. 2 for four levels of cumulative production.

The height of the dashed curves above the hori-zontal lines in Fig. 2 indicates the expected level ofstockouts over time for the corresponding givencumulative production. For instance, if PM "10 dur-ing the interval from t"7 till t"12 (indicated bybroken vertical lines), then E(B(t)) follows the partof the curve pointed out with an arrow.

4. Lot-for-lot policy (L4L)

The L4L lot-sizing policy involves productiondecisions on lower levels in exactly the amounts

that are required by internal demand at the timethis demand occurs. If production of externallydemanded items is given, then so is internal de-mand creating additional internal demand on stilllower product structure levels.

The L4L policy requires that the product struc-ture is of the assembly type. By an assembly struc-ture we mean that the product structure is such thatfor each higher-level item, the items immediatelyrequired are all on a lower level (no items are splitinto several later-stage items and there is no feed-back from higher to lower levels). If the items aresuitably enumerated in such a case, then this en-ables the input matrix to have elements only be-neath its main diagonal (a triagonal matrix witha zero-valued diagonal). Higher-level items thenhave lower item indices than lower-level items.

Let us denote the number of end items by N@. Ifthe production of these end items is set to PI @(s)(which contains zeros in all other positions, i.e. thelower N—N@ positions of PI @(s)), then the immediaterequirements of components will be HqJ PI @(s). Ifthese are produced L4L, they will create an internaldemand (HqJ )2PI @(s), etc. Total production will thenamount to

PI (s)"(I#HqJ#(HqJ )2#2)PI @(s)

" +kw0

(HqJ )kPI @(s)

"(I!HqJ )~1PI @(s). (14)


The summation of the geometric series +kw0

(HqJ )kconverges if the absolute values of the eigenvaluesof the matrix are all less than unity (in this case theyare all zero). However, with H and therefore HqJbeing triagonal matrices with zeros along their di-agonals, the sum converges in a finite number ofterms (exactly N terms in this type of problem,including the identity matrix for k"0).

The inverse matrix (I!HqJ )~1 has the same prin-cipal structure as the Leontief inverse in input—out-put analysis. Since HqJ is a generalised input matrixcapturing required amounts as well as their timing,the inverse written HI *"(I!HqJ )~1 may be con-sidered to be a generalised ¸eontief inverse.

5. Extensions to capacity requirementconsiderations

By extending the input matrix H with additionalrows, we may also take capacity requirements intoconsideration, cf. Ref. [[4], Ch. 3]. Let there beM work stations which may be used for the produc-tion of the items considered. Units of capacity usedwhen manufacturing each item j"1, 2, 2, N maythen be described by elements h

kj, where

k"N#1, N#2, 2, N#M, now making theinput matrix rectangular rather than square havingthe dimension (N#M)]N. Let the originalN]N-dimensional input matrix be denoted by H @and the extended matrix by H. The direct materialsas well as capacity requirements for the productionof end items PI @ will then be HqJ PI @, of which H@qJ PI @are the direct materials requirements. These, intheir turn create an internal demand for materialsand capacity amounting to HqJ H @qJ PI @. Assuming anL4L lot-sizing policy, the total materials and capa-city requirements (including final end items) for theproduction of PI @ will then become

PI (s)"(I @#(HqJ#HqJ H @qJ#HqJ (H @qJ )2#2)PI @(s)

"AI @#HqJ +kw0

(H @qJ )kBPI @(s)

"(I @#HqJ (I!H @qJ )~1PI @(s), (15)

where I @ is an [(N#M)]N]-dimensional matrixconsisting of an (N]N)-dimensional identity

matrix extended with M rows of zeros. SinceI#H @qJ (I!H @qJ )~1"(I!H @qJ )~1, we may alsowrite Eq. (15) as

PI (s)"CI

H AqJ (s)D(I!H @qJ (s))~1PI @(s), (16)

where H A contains the lower M rows of H. Thecapacity requirements have here been allocated atthe beginning of the respective lead times. This canbe modified if necessary.

Available capacity may be included in the funda-mental equations as negative demand, cf. Ref. [[4],Ch. 3].

An example of a product structure taking capa-city requirements X and Y (two work stations) intoaccount is given in Fig. 3.

The extended input matrix applicable to thisproduct structure may be written as

H"

A

E

B

C

D

X

Y

A E B C D

0 0 0 0 0

0 0 0 0 0

4 1 0 0 0

2 0 1 0 0

0 3 2 2 0

0 0 1 0 0

0 0 0 0 1

.

Items A and E are end items, B, C and D aresubordinate items and X and Y are capacities. Wetherefore have N"5, N@"2 and M"2.

To illustrate Eqs. (15) and (16) we choose a hor-izon of 100, lead times of A, E, B, C and D to be 14,12, 10, 8, 6, and 10 units of end item A to beproduced at time 50 and 5 units of end item E to beproduced at time 60. We then have

qJ"

e14s 0 0 0 0

0 e12s 0 0 0

0 0 e10s 0 0

0 0 0 e8s 0

0 0 0 0 e6s

,


Fig. 3. Example of a product structure with two end items andtaking capacity requirements into account.

(I!H@qJ )~1"

1 0 0 0 0

0 1 0 0 0

4e14s e12s 1 0 0

2e14s#4e24s e22s e10s 1 0

8e24s#4e22s#8e32s 3e12s#2e22s#2e30s 2e10s#2e18s 2e8s 1

.

C4e24s e22s e10s 0 0

8e30s#4e28%#8e38s 3e18s#2e28s#2e36s 2e16s#2e24s 2e14s e6sD.

PI @(s)"

10e~50s

5e~60s

0

0

0

.

The generalised Leontief inverse will be

To obtain the whole coefficient matrix in Eq. (16),the generalised Leontief inverse is augmented withthe following two rows:

The resulting cumulative production/capacitylevels £~1MPI

k(s)/sN, k"1, 2, 2, N#M, are shown

in Fig. 4.

6. Assumptions in current study

We make the following assumptions in our cur-rent study:

1. All external demands are created by renewalprocesses, one for each end item.

2. There is no external demand for an item that ispart of a higher-level item (no external demandfor spare parts, etc.).

3. All product structures are given including ca-pacity requirements, these parameters beingrepresented by the (extended) H matrix. Alllead times are given and are represented by thematrix qJ .

4. Capacity constraints are not considered.5. A finite horizon ¹K is assumed.6. All end items are produced in batches of pos-

sibly varying size.7. The decisions to be made concern the end items

as to their amounts and timing. This includesdeciding on the number of batches to produceuntil the horizon.

8. The objective function is the net present value

(NPV) (or the annuity stream proportional tothe NPV). The payments involved are fixedout-payments K

j, j"1, 2, 2, N, for each set-

up assumed to occur at the beginning of thecorresponding lead time, a unit productionout-payment for each item produced located atthe completion time c

j, j"1, 2,2, N#M,

where the last M components refer to unitcapacity costs, and a unit in-payment r

j, j"1,

2,2, N#M, for each unit sold at the time ofcompletion. The r

jare zero for all items that are

not sold externally.


Fig. 4. Total production of seven items in example (includingtwo capacities used).

9. Backlogging is permitted only for end items.The penalty for a backlog is the delay in rev-enue in-payments. Other penalties could alsobe introduced easily.

10. All items on lower levels (only having internaldemand) are produced L4L. Relaxing thisassumption is discussed in Section 10.

11. The opportunities for joint set-ups for lower-level items of the same kind are disregarded.This means that the internal demand fora batch of each lower-level item is generated bythe production decision concerning each enditem individually. If two end items need thesame type of component at exactly the same

time, then two separate batches will be produc-ed simultaneously (with the consequence of twoset-up charges).

7. Decision variables

As stated above, the decisions to be made are theamounts of end items and the timing of their pro-duction, which is all captured in the vector PI @ (s).Each component of this vector we write as a sum ofDirac impulses multiplied by a batch quantity. ThenPI @ (s) will be written as

PI @(s)"

PM11

e~st11#PM12

est12#2#PM-/(1)

e~st-/(1)

PM21

e~st11#PM22

est22#2#PM2n(2)

e~st2n(2)

F

PMN{1

e~stN{1#PMN{2

e~stN{2#2#PMN{n(N{)

e~stN{n(N{)

0

F

0

,

(17)

where n(k) is the number of batches of end itemk produced during ¹K , PM

klthe size of the lth batch of

item k, l "1, 2, 2, n(k), tkl

the completion time oflth batch of end item k. Completion times of lower-level items (including capacities) are then given bythe L4L policy, see below.

Associated with the end-item production vectorPI @ (s) we define a vector of completion times tI (s) forend items as

tJ @(s)"

e~st11#e~st12#2#e~st-/(1)

e~st21#e~st22#2#e~st2n(2)

F

e~stN{1#e~stN{2#2#e~stN{n(N{)

0

F

0

. (18)

These are decision variables. Low-level completiontimes are then obtained from the relation (16). Ex-amining the N—N@ last rows of the generalised


Leontief inverse HI *"(I!H @qJ (s))~1, cf. theexample of Section 5, an element in row k andcolumn j may be written as the sum

hI *kj(s)"+

m

gkjm

esqkjm, (19)

where the gkjm

are constants and the exponents aremade up of cumulative lead times q

kjmformed as

sums of the original lead times. The internal de-mand and therefore the production of a lower-levelitem is then

PIk(s)"

N{+j/1

hI *kj(s)PI

j(s)

"

N{+j/1

+m

gkjm

+t

PMjl

es(qkjm~tjl), k'N@. (20)

Lower-level completion times for item k, due tointernal demand from end item j, are thus formedby relations of the type t

kl{( j)"t

jl!q

kjm, for posit-

ive coefficients gkjm

, where l @"1, 2,2. Whether ornot the index m takes on more than one value,depends on the structure of the input matrix H. Inthe example above for k"4, j"1 (item C as inter-nally demanded for the production of item A), wehave m"1, 2 and for k"5, j"1 (item D de-manded for A) then m "1, 2, 3. The greater thenumber of occurrences of an item in a productstructure tree, the greater the number of set-ups ofthis item will be when following an L4L policy.

Having determined the completion times of alllower-level items, when given the completion timesof end items, the vector tJ @(s) may have its N—N@lower components exchanged, forming the vectorof all completion times tJ (s):

tJ (s)"

e~st11#e~st12#2#e~st-/(1)

e~st21#e~st22#2#e~st2n(2)

F

e~stN1#e~stN2#2#e~stNn(N)

, (21)

where a lower-level component of tI (s) is determinedby

tJ (s)"N{+j/1

n(j)+l/1

+m

es(qkjm~tjl), k'N@. (22)

Both vectors tJ @(s) and tJ (s) contain the transformsof sequences of unit impulses corresponding to the

completion times of items (excluding capacities).The lower-level completion times are not decisionvariables, but depend on the end-item completiontimes through the L4L policy and the associatedlead times.

8. Objective function and available inventoryconstraints

We are now in a position to formulate theobjective function in terms of all decision variables.Taking the net present value as the objective andusing o for the continuous interest rate, we maywrite

E(NPV)"r(E(DI (o))!oE(BI (o))!E(B(¹I ))e~oTª )

!cPI (o)!KqJ (o)tJ (o), (23)

where r and K are N-dimensional row vectors con-taining the components r

j(sales prices) and K

j(set-

up costs), respectively, and c an (N#M)-dimen-sional row vector with components c

j(unit

incremental production costs). The lead-time matrixqJ (o) entering the last terms accounts for allocatingthe set-up costs at the beginning of the lead time,since tJ (o) is defined from the completion dates. Theproduction costs have been allocated at the com-pletion dates by using the production vector PI (o).A more detailed description might have been toallocate production costs somewhere within thelead time interval. This timing is only a question asto the definition of the components of c which maybe individually adjusted for taking care of suchtiming considerations.

In this NPV expression we have used the NPVshort-cut by exchanging the complex frequencys for the continuous discount rate o [5].

In Eq. (23), E(DI (o)) is an N-dimensional columnvector of components representing expected ex-ternal demand discounted to time zero with noconsideration to stockouts, oE(BI (o)) is a similarvector of discounted changes in backlogs, sE(BI (s))being the transform of the time derivative of E(B(t)),and E(B(¹K ))e~oTª contains correction terms for thesake of backlogs not being satisfied at the horizon¹K , since the process then terminates and no more


cPI (o)"c CI

H AqJ (o)D(I!H @qJ (o))~1PI @(o)

"c CI

HA,qJ (o)D (I!H @qJ (o))~1

PM11

e~ot11#PM12

e~ot12#2#PM-/(1)

e~ot-/(1)

PM21

e~ot21#PM22

e~ot22#2#PM2n(2)

e~ot2n(2)

F

pNN{1

e~otN{1#PMN{2

e~otN{2#2#PMN{n(N{)

e~otN{n(N{)

0

F

0

, (27)

production is planned (turning the stockouts thereinto lost sales). Since we are dealing with renewalprocesses for demand, the expression E(B(¹K ))e~oTªonly depends on the final cumulative productionlevels PM

kn(k)and on no other decision variables. For

items not externally demanded, the correspondingcomponents are zero-valued. Since we are notallowing any external demand for internally de-manded items, the vectors E(DI (o)) and E(BI (o)) havezero-valued elements in their N—N@ lower positions.

The (N#M)-dimensional column vector PI (o)contains discounted values of all production (in-cluding capacity usage) and is obtained fromEq. (16) inserting o instead of s, where PI @ containsthe decision variables concerning amounts andtheir timing regarding externally demanded items.

We evaluate the terms in Eq. (23), each in turn.For a renewal process, the external demand of

item k (which is truncated at the horizon t"¹K )

obeys

E(DIk(o))"

1

2pi

b`*=

Pw/b~*=

fIk(w)

1!fIk(w)

)1!e~(o~w)Tª

o!wdw,

(24)

where fIk(o) is the transform of the probability den-

sity function of the stochastic time interval betweenconsecutive demand events. This formula applies toany renewal process. The term is void of decisionvariables.

Using the stockout function (13), the expecteddiscounted backlog follows:

E(BIk(o))"

1

2pi

n(k)+l/0

b`*=

Pw/b~*=

fI PM klk

(w)

w(1!fIk(w))

)e~(o~w)tkl!e~(o~w)tkl`1

o!wdw. (25)

The final stockout is evaluated as

E(Bk(¹K ))"C£~1G

fI PM kn(k)`1k

(s)

s(1!fI (s))HDt/T

K

"

1

2pi

b`*=

Pw/b~*=

fI PM kn(k)`1k

(s)

s(1!fIk(s))

esTK ds. (26)

The discounted production cost term in vectorform may be written as

where all elements of the coefficient vector

cCI

H AqJ D(I!H @qJ )~1

are void of decision variables. Finally, the set-upcost term follows:

KqJ (o)tJ (o)"

K

e~o(t11~q1)#e~o(t12~q1)#2#e~o(t-/(1)~q1)e~o(t21~q2)#e~o(t22~q2)#2#e~o(t2n(2)~q2)Fe~o(tN1~qN)#e~o(tN2~qN)#2#e~o(tNn(N)~qN)

.

(28)


These equations show that when considering anL4L lot-sizing policy with no joint set-ups, therewill be a substantial degree of independence be-tween decisions regarding the amounts of end itemsand their timing. Hence, the optimisation of theproduction of any end item can be carried outcompletely independently of the optimisation ofevery other end item.

For the first batch of each item there might bea lead time constraint in this multi-level case. In theone-level case treated in Ref. [1] we have been ableto disregard the effects that lead times impose onthe feasible set of values of the decision variables. Inthe current multi-level system, lead times mightimpose constraints due to necessarily avoiding pro-duction decisions to be taken in past time. Suchconstraints are due the non-negativity of availableinventory.

Disregarding capacity requirements at first,available inventory is given by

£~1MRI (s)N

"£~1G CR(0)!B(0)#(I!H @qJ (s))PI (s)!DI (s)

s D`H ,

(29)

where PI (s) has its original definition as total pro-duction of all items excluding capacity. In thisexpression, all vectors are N-dimensional. Regard-ing end items (the N@ first components), the non-negativity is taken care of by the backlogging op-portunity. We may therefore confine our interest tothe intermediate set of indices, N@#1, N@#2, 2,N, which concern lower-level items. For this set ofindices, there is no external demand nor any back-log and the corresponding components of both DI (s)and B(0) are zero-valued. Furthermore, from theL4L policy we have PI (s)"(I!H @qJ (s))~1PI @(s),which implies

£~1MRI (s)N"£~1G CR(0)!B(0)#PI @(s)!DI (s)

s D`H,

(30)

with the consequence that there is no net produc-tion of lower-level items, i.e. that available inven-tory of each of these stays the same. Eq. (16) meansthat total production is the weighted sum of the N@first columns of the generalised Leontief inverse

(I!H @qJ (s))~1, the weights being end item produc-tion variables of the kind PM

jle~stjl for the jth end

item produced in its lth batch. As discussed inSection 7, the generalised Leontief inverse(I!H @qJ (s))~1"I#H @qJ#(H @qJ )2#2 will in-clude a sum of weighted exponential functions+

mgkjm

esqkjm where the gkjm

are constants and theqkjm

are cumulative lead times made up as sums ofvarious lead times in the exponents appearing inq8 (s). For production to take place in present orfuture time we must therefore require that t

jl*q

kjm,

for all k, j, l and m, which means thattjl*Max

k,mqkjm

"q*j, for all j, where q*

jis an ab-

breviation.This result means that the first batch of each item

cannot be completed before the cumulative leadtime (for all its components and sub-components)has elapsed. This sets minimum constraints on thetiming decisions of the first batches.

If we also wish to take the timing constraints forutilising capacity into account, the N-dimensionalPI (s) above is exchanged for the (N#M)-dimen-sional extended vector

PI (s)"CI

H AqJ (s)D(I!H @qJ (s))~1PI @(s)

and the same reasoning may be applied to theexponents entering the components of this vector,possibly prolonging the cumulative lead times com-pared to when not taking capacity requirementsinto account.

9. Necessary optimisation conditions

Forming a Lagrangean function

¸"E(NP»)#N{+k/1

jk(tk1!q*

k), (31)

where the jk

are non-negative multipliers, the ne-cessary optimisation conditions for a maximum ofthe NPV are derived as

L¸Lt

kl

"o e~otklA!rk([E(B

k(tkl))]

l~1

![E(Bk(tkl))]

l)#a

k(o)PM

kl#K

keoqK


Table 1Economic parameters

Item Revenue r Unit productioncost c

Set-upcost K

A 5000 100 50E 2000 200 100B 150 50C 50 20D 10 10

Table 2Demand parameters for end items

Item Mean of interval k Variance ofinterval p2

A 1 1E 2 4

Table 3Optimal production decisions for end items A and E

Item A Item E

Batch k PMk

tk

SSk

PMk

tk

SSk

0 0 0 !32.00 0 0 !15.001 49 32.00 0.96 27 30.00 !0.092 65 48.04 2.16 41 54.18 0.323 80 62.84 2.96 57 81.37 7.004 95 77.04 3.375 111 91.63 11.00

# +j;N{

Kjeoqj+

m

eoqjmB#jk"0,

k)N@, l)n(k), (32)

L¸Lt

kl

"!rkoA[E(BI (o))]

PM kl`1

![E(BIk(o))]

PMkl!

cle~oTK

o£~1G

fI PM kn(k)`1(s)

s Ht/TKB

!ak(o)(e~otkl!e~otkl`1#c

le~oTK ))0,

k)N@, l)n(k), (33)

where PMkl

denotes cumulative production after thelth batch has been completed, where j

kin Eq. (32) is

only included for l"1, where

ak(o)"Cc C

I

H AqJ (o)D(I!H @qJ (o))~1D#0-6./ k

,

where the inequality in Eq. (33) accounts for thecondition that the PM

klare integers, and where c

lis

unity for l"n(k) and zero otherwise. The basicexpressions in Eqs. (32) and (33) have been takenover from Ref. [1]. If j

k'0, then t

kl"q*

k.

The corresponding sufficient conditions becomerather involved. However, simple numerical exam-ples indicate that in normal cases the optimum willbe unique.

10. A numerical example

We use the major part of the previous examplefrom Section 5 (cf. Fig. 3) to illustrate the optimisa-tion procedure proposed. The capacity require-ments are ignored below. Numerical values for theeconomic parameters used in the example are listedin Table 1. Revenues are generated only from theend items A and E. Stochastic demand parametersare also assumed to be different for these two finish-ed items. The time interval of demand followsexponential distributions for A and E.

Using the economic and demand parameterslisted in Tables 1 and 2, we can solve the problemfrom Eqs. (31)—(33). Since only the production offinished items are decision variables, we first calcu-late the optimal production decision for items

A and E following a similar procedure to the oneproposed in Ref. [1]. The results are listed inTable 3. The expression SS

krefers to the optimal

safety stock immediately prior to receiving batchk#1.

The optimal batch number is 5 and 3 for A and E,respectively. As soon as Table 3 is obtained,production decisions for the lower-level items areimmediately calculated according to Eq. (16). Theresults are listed in Tables 4—6.

Since an L4L policy is applied for the lower-levelitems, the optimal decisions are independent be-tween different finished items. This brings simpli-city when solving the safety stock problem based


Table 4Optimal production decisions for item B

Source of requirementItem A Item E

Batch k PMk

tk

Batch k PMk

tk

1 196 18.00 2 27 18.003 260 34.04 4 41 42.185 320 48.846 380 63.04 7 57 69.378 444 77.63

Table 5Optimal production decisions for item C


Batch k PMk

tk

Batch k PMk

tk

1 196 8.00 2 27 8.003 98 18.004 260 24.04 5 41 32.186 130 34.047 320 38.848 160 48.849 380 53.04 10 57 59.37

11 190 63.0412 444 67.6313 222 77.63

Table 6Optimal production decisions for item D


Batch k PMk

tk

Batch k PMk

tk

1 392 0.00 2 54 0.003 392 8.00 4 54 8.005 196 10.006 520 16.04 7 81 18.00

8 82 24.009 520 24.04

10 260 26.0411 640 30.84 12 82 32.1813 640 38.8414 320 40.84 15 123 42.1816 760 45.04 17 114 51.3718 760 53.0419 380 55.04 20 114 59.3721 888 59.6322 888 67.63 23 171 69.3724 444 69.63

on the optimisation conditions (31)—(33). The solu-tion for the production decision for lower-levelitems is also straightforward. On the other hand,the application of an L4L policy prevents addi-tional possible improvements of the productiondecisions. For example, we lose the opportunity tocombine batches 8 and 9 of Item D (Table 6) whichwould reduce the combined set-up costs involved.

11. Relaxing the lot-for-lot policy

As a final topic discussed in this paper, we con-sider the relaxation of the L4L policy for producinglower-level items, assuming an assembly systemstructure. Internal demand for a lower-level itemmay be generated from one or several down-streamproducts. If the down-stream production is given,then the internal demand would be given as a stair-case function possibly made up as the weightedsum of staircase functions generated by eachdown-stream item acting as a source of these re-quirements.

Below we show that when internal demand isgiven, then there is a finite number of staircasestructures for the production of the lower-level itemthat need be investigated in order to cover allcandidates for an overall optimum when applyingthe NPV criterion. This result is alike, but slightlymore general than the similar result discussed inRef. [6].

In Ref. [6] the following remark was made forthe lower-level production in a serial two-level sys-tem applying an average cost principle:

“Whatever staircase the upper-level cumulativeproduction follows, at the optimum there cannever be a gap between the lower-level cumulat-ive function and the highest of the steps of theupper function within a particular lower-levelproduction interval. If there were such a gap, thelower-level production level could be loweredreducing inventory holding costs with no othereffects. Also, whatever staircase the upper-levelcumulative production follows, at the optimumthere can never be a gap in time between the nextreplenishment of the lower level and the nextreplenishment of the upper level, because if this


Fig. 5. A two-level serial system with the left staircase showingcumulative lower-level production and the right showing cumu-lative upper-level production (which generates internal de-mand"solid left staircase curve). “Inner corners” are markedwith circles.

Fig. 6. Lower-level production opportunities and internaldemand. Moving a to the right or b downwards creates in-crements to the net present value.

was the case, then the lower-level replenishmentcould be postponed reducing holding costs withno other effects. Therefore in the two-level serialcase, we note:In the optimal solution, the lower-level replenish-ments always must take place at the same time assome upper-level replenishment and at that time(immediately prior to the replenishment in ques-tion) both cumulative levels coincide’’.

Geometrically, this was described by consideringvariations from the upper-level staircase as illus-trated in Fig. 5. The two staircases fit into eachother at the ‘‘inner’’ corners when lower-level re-plenishments take place (indicated with circles in thefigure). Thus, the optimal number of batches on thelower-level cannot be higher than on the upper level.

We use the left-hand side of the same figure inour current multi-level case in order to ascertain thefollowing similar conclusion. The solid left-hand sidestaircase is interpreted as the sum of all internaldemand from given down-stream production.

In the optimal solution, the lower-level replenish-ments must always take place at the same time assome upper-level item generates internal demand andat that time (immediately prior to the replenishmentin question) cumulative production of the item con-sidered must coincide with cumulative internaldemand.

To prove this, we first note that cumulative pro-duction always must be to the left and above inter-nal demand according to the non-negativity of

available inventory with the possibility that thecurves follow each other as a whole or partly. Ifa certain replenishment were to take place beforea vertical step in internal demand (cf. a in Fig. 6),then the replenishment could be postponed untilthe same time as the step with the only consequencethat the discounted cost of production and set-updecreased. On the other hand, if the replenishmentwere to correspond to a step reaching a point in aninterval strictly between two cumulative levels ofinternal demand (b in Fig. 6), then lowering thebatch quantity would have no other consequencethan decreasing an earlier production cost and rais-ing a later production cost by the same nominalamount. Taking the discount rate into considera-tion, this would raise the resulting NPV.

Hence, the lower-level production staircase musthave inner corners that coincide with its internaldemand generated from the given down-streamproduction. The lower-level staircase structuresthat may act as candidates for an overall optimalproduction are therefore finite in number for eachset of given down-stream production staircases.For a two-level case, the number of lower-levelstructures to be considered when internal demandhas n steps will be 2(n~1) (as already shown in Ref.[6]) and in a three-level case, assuming the thirdlevel only to have the second level as a source forinternal demand (a serial structure in this respect),we obtain altogether 3(n~1) structures to examine.In the general serial system of m levels (the upmostlevel being internal demand generated from enditems), the total number of lower-level structureswill be m(n~1).


This principle also immediately results in an inte-grality condition between batch sizes on differentlevels for a serial system (which has been extensive-ly discussed in the literature). If consecutive batcheson an upper level are equally sized, then batches onthe next level down will be equally sized and willamount to an integral multiple of the upper level size.

12. Conclusions

Our main findings are that the single-level results[1] are applicable in all essential respects also in themulti-level case when the L4L lot-sizing policy isapplied to all subordinate items of an assemblysystem on condition that joint set-up opportunitieson lower levels are not taken into account. Theoptimisation conditions for the multi-level systemtreated above have been shown to be simple gener-alisations of those of the single-level system.

The next step in this line of work would be toexamine further numerical examples based on theoptimisation conditions derived. Some work in thisdirection has already been carried out in Ref. [21].As regards the relaxation of the L4L policy discussedin Section 11, some future interest could be devotedto whether there might exist myopic conditions thatcould be derived for rejecting various alternativeswith the aim to decrease the number of cases createdby the subordinate item lot-sizing alternatives. A fur-ther but more complex issue raised above concernsthe question of joint set-ups. This would first needto be handled in more theoretical detail.

13. Further Reading

The following references are also of interest tothe reader: [7—20] and [22—26].

Acknowledgements

The author is deeply indepted to Ou Tang, Deptof Production Economics, Linkoping Institute ofTechnology, for many valuable discussions heldwith him, and for his work in preparing the numer-ical example of Section 10.

References

[1] R.W. Grubbstrom, A net present value approach to safetystocks in planned production, International Journal ofProduction Economics 56/57 (1998) 213—229.

[2] R.W. Grubbstrom, Stochastic properties of a production-inventory process with planned production using trans-form methodology, International Journal of ProductionEconomics 45 (1996) 407—419.

[3] L.E. Andersson, R.W. Grubbstrom, Asymptotic Behaviourof a Stochastic Multi-Period Inventory Process with Plan-ned Production, Working paper WP-210, Dept. of Pro-duction Economics, Linkoping Institute of Technology,April 1994.

[4] A. Segerstedt, Multi-level production and inventory con-trol — related to MRP and cover-time planning, Produc-tion-Economic Research in Linkoping, PROFIL 14,Linkoping, 1995.

[5] R.W. Grubbstrom, On the application of the Laplacetransform to certain economic problems, ManagementScience 13 (7) (1967) 558—567.

[6] R.W. Grubbstrom, A. Molinder, Safety production plansin MRP-systems using transform methodology, Interna-tional Journal of Production Economics 46—47 (1996)297—309.

[7] J.A. Aseltine, Transform Method in Linear System Analy-sis, McGraw-Hill, New York, 1958.

[8] R.V. Churchill, Complex Variables and Applications,McGraw-Hill, New York, 1960.

[9] R.V. Churchill, Operational Mathematics, McGraw-Hill,New York, 1958.

[10] R.W. Grubbstrom, The z-transform of tk, The Mathemat-ical Scientist 16 (1991) 118—129.

[11] A. Vazsonyi, The use of mathematics in production andinventory control, Management Science 1 (1) (1955) 70—85.

[12] R.W. Grubbstrom, A principle for determining the correctcapital costs of work-in-progress and inventory, Interna-tional Journal of Production Research 18 (2) (1980)259—271.

[13] R.W. Grubbstrom, Y. Jiang, A survey and analysis of theapplication of the Laplace transform to present valueproblems, Revista di Matematica per le Scienze Eco-nomiche e Sociale 12 (1) (1990) 43—62.

[14] A. Thorstenson, Capital costs in inventory models — a dis-counted cash flow approach, Production-Economic Re-search in Linkoping, PROFIL 8, Linkoping, 1988.

[15] A. Thorstenson, R.W. Grubbstrom, Theoretical analysisof the difference between the traditional and theannuity stream principles applied to inventory evaluation,in: R. Flavell (Ed.), Modelling Reality and PersonalModelling, Physica-Verlag, Heidelberg, 1993, pp.296—326.

[16] R.W. Grubbstrom, O. Tang, Further developments onsafety stocks in an MRP system applying Laplace trans-forms and input—output analysis, Proceedings 14th Inter-national Conference on Production Research, Osaka, 4—8August 1997, pp. 1718—1721.


[17] L. Bogataj, L. Horvat, Stochastic considerations of Grubb-strom—Molinder model of MRP, input—output and multi-echelon inventory systems, International Journal of Pro-duction Economics 45 (1996) 329—336.

[18] L. Bogataj, Lead times and other delays in inventorysystems, in: L. Bogataj (Ed.), Inventory Modelling, LectureNotes of the International Postgraduate Summer School,vol. 1, ISIR, Budapest, 1995, pp. 23—35.

[19] L. Bogataj, L. Horvat, The application of game theory toMRP, input—output and multi-echelon inventory systemswith exponentially distributed external demand, Proceed-ings of the International Symposium on Operations Re-search SOR ‘95, Portorozy , Slovenia, 1995, pp. 25—36.

[20] R.W. Grubbstrom, Material requirements planning andmanufacturing resource planning, in: M. Warner (Ed.),International Encyclopedia of Business and Management,Routledge, London, 1996.

[21] R.W. Grubbstrom, A. Molinder, Further theoretical consid-erations on the relationship between MRP, Input—outputanalysis and multi-echelon inventory systems, InternationalJournal of Production Economics 35 (1994) 299—311.

[22] R.W. Grubbstrom, P. Ovrin, Intertemporal generalizationof the relationship between material requirements plann-ing and input—output analysis, International Journal ofProduction Economics 26 (1992) 311—318.

[23] L. Horvat, L. Bogataj, MRP, input—output analysis andmulti-echelon inventory systems with exponentially dis-tributed external demand, Proceedings of GLOCOSMConference, Bangalore, India, 1996, pp. 149—154.

[24] L. Horvat, L. Bogataj, Game theory aspects of MRP,input—output and multi-echelon inventory systems,Pre-prints, vol. 3 Ninth International Working Seminaron Production Economics, Igls/Innsbruck, 1996, pp.1—12.

[25] L. Horvat, L. Bogataj, Multi-attribute decision makingupgrading of Grubbstrom—Molinder model of the rela-tionship between MRP and input—output analysis, toappear.

[26] A. Molinder, Material requirements planning employinginput—output analysis and Laplace transforms, Produc-tion-Economic Research in Linkoping, PROFIL 14,Linkoping, 1995.


A Net Present Value Approach to Safety Stocks in a Multi-level MRP

Documents

Transcript of A Net Present Value Approach to Safety Stocks in a Multi-level MRP