Timing and Inventory

Topic 11:

Supply Chain and Coordination

1. Stochastic EOQ models 2. The reorder point & the order up to model3. Simulation of inventory models4. Supply chain coordination with LP 6. Aggregate planning7. The aggregate planning formulation with LP

There are four commonly employed ways in whichmanagement may express its willingness or tolerance for

running out of inventory:

1. Cycle Service Level: of the probability of not runningout of stock in one cycle.

2. Fill Rate: probability that demand may be filled fromstock

3. Number of stockouts per year

4. Proportion of days in stock

Criteria often used as cost surrogates

Continuous Review of Inventory (ROP or Q System)

R

Inve

ntor

y

Time

L

LL

Place an order for Q units whenever inventory withdrawal brings inventory

position to R

BDR L

The reorder point (R) is normally calculated by adding some level of safety stock (B) to the expected demand over leadtime

Numerical Example of a Q system

Demand over leadtime = N(36,15) and CSL = 90%

36

15

1-CSL

B B=1.28(15)=19.2

R R=36+19.2=55.2

Periodic Review of Inventory (PRS or P System)In

vent

ory

Time

LL

Review an item’s inventory position every P time periods. At that time,

place and order to replenish to T units

BDT LP

The up-to order level(T) is normally calculated by adding some level of safety stock (B) to the expected demand over the protection interval (L + P)

T

P

Numerical Example of a P System

D = N(40,15) per weekL = 3 weeksCSL = 80%

What P is required to approximate the cost tradeoffs of a 400 unit EOQ?

What is the desired level of T?

weeks1040

400

56545520)54(84.)13(40

54)310(*152

TLP

Numerical Example

• D = N(15,6) units per weekA = $50 per order

• h = $12 per unit per year L = 2 weeks CSL = 80%

• Assume Continuous review System

• A. What is the EOQ?

• B. What is the desired B?

• C. What is the desired R ?

6.8012

50))52(15(2

18.7)5.8(845.

5.82*6* 22

B

LtL

18.3718.7)2(15

Numerical Example (cont.)

• D = N(15,6) units per weekA = $50 per order

• h = $12 per unit per year L = 2 weeks CSL = 80%

• Assume Periodic Review

• A. What value of P provides approximate EOQ tradeoffs?

• B. What is the desired

• B and T?

weeksweeks 54.552780

6.80

4.1184.13)15(7

4.13)9.15(845.

9.157*6

7

2

T

B

weeksLP

LP

Advantages of P and Q Systems

• Continuous (Q) systems

– Carry less safety stock

– Order size is constant

– Individualize replenishment intervals

– Suited to quantity discounts and capacity limitations

• Periodic (P) systems

– Less need to take additional physical inventory

– Fixed replenishment intervals

– Can coordinate replenishment of multiple items

A Reorder Point (ROP) Inventory Model

• GEN;• LIMITS,10,,,10;• EQUIVALENCE,{ { INVPOS, XX[1] } };• EQUIVALENCE,{ { ONHAND, XX[2] } };• EQUIVALENCE,{ { BACKORD, XX[3] } };• EQUIVALENCE,{ { LOST, XX[4] } };• EQUIVALENCE,{ { ORDERS, XX[6] } };• EQUIVALENCE,{ { ROPOINT, XX[8] } };• EQUIVALENCE,{{QUANT,XX[9]}};• INITIALIZE,0.0,10000,YES;• INTLC,{{INVPOS,6},{ONHAND,6},{ROPOINT,4},{QUANT,12}};• TIMST,,INVPOS,"INVENTORY POSITION",0,0.0,1.0;• TIMST,,ONHAND,"INVENTORY ON HAND",0,0.0,1.0;• TIMST,,BACKORD,"BACKORDERS",0,0.0,1.0;• TIMST,,LOST,"DEMAND LOST",0,0.0,1.0;• TIMST,,ORDERS,"ORDERS PLACED",0,0.0,1.0;• NET;• FIN;

An Order-Up-To Inventory Model

• GEN,"UPTO MODEL",,,,YES,YES;• LIMITS,10,,,10;• EQUIVALENCE,{ { INVPOS, XX[1] } };• EQUIVALENCE,{ { ONHAND, XX[2] } };• EQUIVALENCE,{ { BACKORD, XX[3] } };• EQUIVALENCE,{ { LOST, XX[4] } };• EQUIVALENCE,{ { UPTO, XX[5] } };• EQUIVALENCE,{ { PERIOD, XX[7] } };• EQUIVALENCE,{ { QUANT, ATRIB[1] } };• INITIALIZE,0.0,10000,YES;• INTLC,{{INVPOS,6},{ONHAND,6},{UPTO,12},{PERIOD,6}};• TIMST,,INVPOS,"INVENTORY POSITION",0,0.0,1.0;• TIMST,,ONHAND,"INVENTORY ON HAND",0,0.0,1.0;• TIMST,,BACKORD,"BACKORDERS",0,0.0,1.0;• TIMST,,LOST,"DEMAND LOST",0,0.0,1.0;• NET;• FIN;

What is the bullwhip effect?

• Demand variability increases as you move up the supply chain from customers towards supply

Customer

RetailerDistributorFactoryTier 1 SupplierEquipment

Bullwhip effect in autos to machine tools

-80%

-60%

-40%

-20%

0%

20%

40%

60%

80%

% c

hang

e in

dem

and

GDP = solid line

Source:Anderson, Fine and Parker (1996)

Autos Machine tools

Bullwhip effect in the US PC supply chain

Semiconductor

1995 1996 1997 1998 1999 2000 2001

-40%

-20%

0%

20%

40%

60%

80%

PC

SemiconductorEquipment

Changes indemand

Semiconductor

1995 1996 1997 1998 1999 2000 2001

-40%

-20%

0%

20%

40%

60%

80%

PC

SemiconductorEquipment

Changes indemand

Annual percentage changes in demand (in $s) at three levels of the semiconductor supply chain: personal computers, semiconductors and semiconductor manufacturing equipment.

Consequences of the bullwhip effect

• Inefficient production or excessive inventory.

• Low utilization of the distribution channel.

• Necessity to have capacity far exceeding average demand.

• High transportation costs.

• Poor customer service due to stockouts.

Causes of the bullwhip effect

• Order synchronization– Multiple retailers who tend to order around the same time period– Manufacturers responding to an MRP system that place raw material orders at the

beginning of the month

• Order batching– In order to save on shipping or ordering costs, firms order a full pallet or full truck load

• Trade promotions and forward buying– Supplier offers a discount on product ordered in a specific time period– Supplier offers a quantity discount– A retailer orders a large quantity intending to take advantage of a discount and sells

excess product to a second retailer (this strategy is called diversion)

• Reactive and over-reactive ordering– A retailer who is not sure that demand is stable over time may act aggressively when

faced with periods of lower or higher than expected demand

• Shortage gaming– A retailer who wants to insure product from an under-capacitated supplier may over

order expecting to only receive a portion of the ordered quantity

Strategies to combat the bullwhip effect

• Information sharing:

– Collaborative Planning, Forecasting and Replenishment (CPFR)

• Smooth the flow of products

– Coordinate with retailers to spread deliveries evenly.

– Reduce minimum batch sizes.

– Smaller and more frequent replenishments (EDI).

• Eliminate pathological incentives

– Every day low price

– Restrict returns and order cancellations

– Order allocation based on past sales in case of shortages

• Vendor Managed Inventory (VMI): delegation of stocking decisions

Supply Chain Synchronization and Linear Programming

The Transportation Problem: a general formulation of a class of problemsrelated to the supply and distribution of goods and services across a network. Generally, the transportation problem is concerned with the most cost effective(or cost minimizing) way to supply several demand locations (nodes) from morethan one supply location (nodes)

Example

Special transportation concerns:

Route (or arcs) that have a maximum capacity

Routes that cannot be traversed

The Transshipment Problem: a more generalized version of thetransportation problem in which intermediate, transshipment, nodes are addedto the network. Transshipment nodes are often used to model warehouses,material transfer locations, or junctions for mixed mode delivery of goods andservices.

Example

Special transshipment concerns:

Backwards or sidewards movement in the network

Capacity limitations of the transshipment nodes

Special transshipment concerns:

Quaker Oats has begun manufacturing, in two of its plants, a new granola productmade of three parts oats, two parts raisins and one part almonds. Two oat vendorsand two almond vendors have been identified, but only one reliable vendor of raisinscould be found. The supply of raw materials and the shipped costs are provided

Vendor Supply intons

Cost to Plant 1 Cost to Plant 2

Oat 1 25,000 $100 $110

Oat 2 30,000 $105 $95

Raisin 50,000 $550 $525

Almond 1 9,000 $1,050 $1,150

Almond 2 10,000 $1,200 $1,100

Quaker ships to three distribution facilities. The shipping cost of completed (6-ton)pallets of product and the demand at each distribution facility are provided

Hannaford Quaker WalMart Plant Capacity

Plant 1 $100 $65 $90 9,500

Plant 2 $95 $70 $90 8,500

Demand 2,500 5,000 10,000

Quaker Oats has begun manufacturing, in two of its plants, a new granola productmade of three parts oats, two parts raisins and one part almonds. Two oat vendors

EXAMPLE

O1

O2

R

A1

A2

Q1

Q2

H

D

W

25,000

30,000

50,000

9,000

10,000

2,500

5,000

10,000

9,500

8,500

Formulation

Minimize Z= 100O1Q1+110O1Q2+105O2Q1+95O2Q2+550RQ1+525RQ2+1050A1Q1+1150A1Q2+1200A2Q1+1100A2Q2+100Q1H+65Q1D+90Q1W+95Q2H+70Q2D+90Q2W

Subject To: C1 1O1Q1+1O1Q2<=25000C2 1O2Q1+1O2Q2<=30000C3 1RQ1+1RQ2<=50000C4 1A1Q1+1A1Q2<=9000C5 1A2Q1+1A2Q2<=10000C6 1Q1H+1Q2H>=2500C7 1Q1D+1Q2D>=5000C8 1Q1W+1Q2W>=10000C9 1O1Q1+1O2Q1-3Q1H-3Q1D-3Q1W=0C10 1O1Q2+1O2Q2-3Q2H-3Q2D-3Q2W=0C11 1RQ1-2Q1H-2Q1D-2Q1W=0C12 1RQ2-2Q2H-2Q2D-2Q2W=0C13 1A1Q1+1A2Q1-1Q1H-1Q1D-1Q1W=0C14 1A1Q2+1A2Q2-1Q2H-1Q2D-1Q2W=0C15 Q1H+Q1D+Q1W<=9500C16 Q2H+Q2D+Q2W<=8500

Combined Report for GranolaDecision Solution Unit Cost Total Reduced Basis Allowable AllowableVariable Value Profit c(j) Contribution Cost Status Min. c(j) Max. c(j)

1 O1Q1 25,000.00 100.0000 2,500,000.00 0 basic -M 105.00002 O1Q2 0 110.0000 0 20.0000 at bound 90.00 M3 O2Q1 2,000.00 105.0000 210,000.00 0 basic 100.00 M4 O2Q2 25,500.00 95.0000 2,422,500.00 0 basic -M 105.00005 RQ1 18,000.00 550.0000 9,900,000.00 0 basic 535.00 M6 RQ2 17,000.00 525.0000 8,925,000.00 0 basic -M 540.00007 A1Q1 9,000.00 1,050.00 9,450,000.00 0 basic 1,020.00 1,200.00008 A1Q2 0 1,150.00 0 50.0000 at bound 1,100.00 M9 A2Q1 0 1,200.00 0 150.0000 at bound 1,050.00 M10 A2Q2 8,500.00 1,100.00 9,350,000.00 0 basic -M 1,130.000011 Q1H 0 100.0000 0 5.0000 at bound 95.00 M12 Q1D 5,000.00 65.0000 325,000.00 0 basic -2,465.00 70.000013 Q1W 4,000.00 90.0000 360,000.00 0 basic 85.0000 95.000014 Q2H 2,500.00 95.0000 237,500.00 0 basic -2,465.00 100.000015 Q2D 0 70.0000 0 5.0000 at bound 65.00 M16 Q2W 6,000.00 90.0000 540,000.00 0 basic 85.00 95.0000

Objective Function (Min.) = 44,220,000.0000

Left Hand Right Hand Slack Shadow Allowable AllowableConst Side Direction Side Surplus Price Min. RHS Max. RHS

1 C1 25,000.00 <= 25,000.00 0 -5.00 22,500.0000 27,000.00002 C2 27,500.00 <= 30,000.00 2,500.00 0 27,500.0000 M3 C3 35,000.00 <= 50,000.00 15,000.00 0 35,000.0000 M4 C4 9,000.00 <= 9,000.00 0 0 9,000.0000 M5 C5 8,500.00 <= 10,000.00 1,500.00 0 8,500.0000 M6 C6 2,500.00 >= 2,500.00 0 2,560.00 1,833.3330 2,500.00007 C7 5,000.00 >= 5,000.00 0 2,530.00 4,333.3340 5,000.00008 C8 10,000.00 >= 10,000.00 0 2,555.00 9,333.3330 10,000.00009 C9 0 = 0 0 105.00 -2,000.0000 2,500.000010 C10 0 = 0 0 95.00 -25,500.0000 2,500.000011 C11 0 = 0 0 550.00 -18,000.0000 15,000.000012 C12 0 = 0 0 525.00 -17,000.0000 15,000.000013 C13 0 = 0 0 1,050.00 -9,000.0000 014 C14 0 = 0 0 1,100.00 -8,500.0000 1,500.000015 C15 9,000.00 <= 9,500.00 500.00 0 9,000.0000 M16 C16 8,500.00 <= 8,500.00 0 -30.00 8,500.0000 9,166.6670

Combined Report for GranolaDecision Solution Unit Cost Total Reduced Basis Allowable AllowableVariable Value Profit c(j) Contribution Cost Status Min. c(j) Max. c(j)

1 O1Q1 25,000.00 100.0000 2,500,000.00 0 basic -M 105.00002 O1Q2 0 110.0000 0 20.0000 at bound 90.00 M3 O2Q1 2,000.00 105.0000 210,000.00 0 basic 100.00 M4 O2Q2 25,500.00 95.0000 2,422,500.00 0 basic -M 105.00005 RQ1 18,000.00 550.0000 9,900,000.00 0 basic 535.00 M6 RQ2 17,000.00 525.0000 8,925,000.00 0 basic -M 540.00007 A1Q1 9,000.00 1,050.00 9,450,000.00 0 basic 1,020.00 1,200.00008 A1Q2 0 1,150.00 0 50.0000 at bound 1,100.00 M9 A2Q1 0 1,200.00 0 150.0000 at bound 1,050.00 M10 A2Q2 8,500.00 1,100.00 9,350,000.00 0 basic -M 1,130.000011 Q1H 0 100.0000 0 5.0000 at bound 95.00 M12 Q1D 5,000.00 65.0000 325,000.00 0 basic -2,465.00 70.000013 Q1W 4,000.00 90.0000 360,000.00 0 basic 85.0000 95.000014 Q2H 2,500.00 95.0000 237,500.00 0 basic -2,465.00 100.000015 Q2D 0 70.0000 0 5.0000 at bound 65.00 M16 Q2W 6,000.00 90.0000 540,000.00 0 basic 85.00 95.0000

Objective Function (Min.) = 44,220,000.0000

Left Hand Right Hand Slack Shadow Allowable AllowableConst Side Direction Side Surplus Price Min. RHS Max. RHS

1 C1 25,000.00 <= 25,000.00 0 -5.00 22,500.0000 27,000.00002 C2 27,500.00 <= 30,000.00 2,500.00 0 27,500.0000 M3 C3 35,000.00 <= 50,000.00 15,000.00 0 35,000.0000 M4 C4 9,000.00 <= 9,000.00 0 0 9,000.0000 M5 C5 8,500.00 <= 10,000.00 1,500.00 0 8,500.0000 M6 C6 2,500.00 >= 2,500.00 0 2,560.00 1,833.3330 2,500.00007 C7 5,000.00 >= 5,000.00 0 2,530.00 4,333.3340 5,000.00008 C8 10,000.00 >= 10,000.00 0 2,555.00 9,333.3330 10,000.00009 C9 0 = 0 0 105.00 -2,000.0000 2,500.000010 C10 0 = 0 0 95.00 -25,500.0000 2,500.000011 C11 0 = 0 0 550.00 -18,000.0000 15,000.000012 C12 0 = 0 0 525.00 -17,000.0000 15,000.000013 C13 0 = 0 0 1,050.00 -9,000.0000 014 C14 0 = 0 0 1,100.00 -8,500.0000 1,500.000015 C15 9,000.00 <= 9,500.00 500.00 0 9,000.0000 M16 C16 8,500.00 <= 8,500.00 0 -30.00 8,500.0000 9,166.6670

1

2

3

4

5

6

7

8

9

1

2

3

4

5

6

7

8

9

Mon

day

Tu

esd

ayA Multi-Period Transshipment

Problem

1. Inventories and multi-period planning2. Limitations on shipping quantities3. Changes in demand 4. Multimode shipping5. Returns6. Reverse logistics

Aggregate Production Planning (APP)

A macro approach to operational planning that focuses on overall capacity. APP links higher-level facility planning to lower level scheduling decisions within a medium-term planning horizon (2 to 18 months).

Aggregate Production Planning (APP)

Material RequirementsPlanning (MRP)

Master ProductionScheduling (MPS)

Shop FloorControl

Purchasing

DemandPlanning

and Forecasting

CapacityPlanning

(CRP) Material and Capacity Plans

APP (cont)

• Role of Aggregate Planning– Long-term planning function

– Strategic preparation for tactical actions

• Aggregate Planning Issues

– Staffing -hiring, firing, training

– Procurement - supplier contracts for materials, components

– Sub-Contracting - capacity vendoring

– Marketing - promotional activities

Basic Aggregate Planning (cont.)

CottonShirts

Men’sWomen’s Boy’s Girl’s

StyleA

StyleB

StyleC

Week1 Week2 Week3 Week4 Week5

Level of Aggregation

- product - time

Typical managerial objectives Minimize production costsMaximize profitsMaximize customer serviceMinimize inventoryMinimize changes in production rates Minimize changes in work-force levelsMaximize utilization of plant and equipment

Reactive alternativesAdjusting the workforce (hire, fire and layoff)Overtime and undertimeVacation schedules Anticipation inventorySubcontractorsBackorder, backlogs and stockouts

Aggressive alternatives (adjusting the demand patterns)Complementary productsCreative pricing

Anticipation Inventory

Pure APP Planning Strategies

Time

Un

its

Demand

Production

Pure APP Planning Strategies

Time

Un

its

Demand & Production

Tools and Methods Used to Develop Aggregate Plans

1. Trial and error (most commonly used technique)

2. Mathematical Techniques

3. Simulation (more complex trial and error?)

A Basic Linear Programming Formulation

• The heart of the APPLP formulation is the production constraint

• (Beginning Inventory) + Production – (Ending Inventory) = Demand

• Or I1 + P - I2 = D

• Chained over multiple periods by inventory flow

• I1 + P1 - I2 = D1

• I2 + P2 - I3 = D2

• I3 + P3 - I4 = D3

• etc…

I B P O S F I B

F Dt t t t t t t t

t t

1 1

Where:t= Number of Discrete Planning PeriodI = Units of InventoryP= Units of Production in Regular TimeO= Units of Production in OvertimeS= Units of Production SubcontractedD= Units of DemandF= Units of Filled DemandB= Units Backordered

A More Complete Production Constraint

LP Example 1:

Maximize -1I1-1I2-1I3-1I4-1I5-3B1-3B2-3B3-3B4-3B5-12P1-14P2-14P3-16P4-18O1- 20O2-20O3-22O4- 20S1-20S2-20S3-20S4+80F1+80F2+80F3+80F4

Subject to: 1I1-1I2-1B1+1B2+1P1+1O1+1S1-1F1=01I2-1I3-1B2+1B3+1P2+1O2+1S2-1F2=01I3-1I4-1B3+1B4+1P3+1O3+1S3-1F3=01I4-1I5-1B4+1B5+1P4+1O4+1S4-1F4=01F1<=2001F2<=2501F3<=3501F4<=250

Combined Report

Decision Solution Unit Cost Total Reduced Basis Allowable AllowableVariable Value Profit c(j) Contribution Cost Status Min. c(j) Max. c(j)

1 I1 800.0000 -1.0000 -800.0000 0 basic -4.0000 02 I2 600.0000 -1.0000 -600.0000 0 basic -4.0000 03 I3 350.0000 -1.0000 -350.0000 0 basic -4.0000 04 I4 0 -1.0000 0 -1.0000 at bound -M 05 I5 0 -1.0000 0 -4.0000 at bound -M 3.00006 B1 0 -3.0000 0 -4.0000 at bound -M 1.00007 B2 0 -3.0000 0 -4.0000 at bound -M 1.00008 B3 0 -3.0000 0 -4.0000 at bound -M 1.00009 B4 0 -3.0000 0 -3.0000 at bound -M 010 B5 250.0000 -3.0000 -750.0000 0 basic -4.0000 011 P1 0 -12.0000 0 -11.0000 at bound -M -1.000012 P2 0 -14.0000 0 -12.0000 at bound -M -2.000013 P3 0 -14.0000 0 -11.0000 at bound -M -3.000014 P4 0 -16.0000 0 -13.0000 at bound -M -3.000015 O1 0 -18.0000 0 -17.0000 at bound -M -1.000016 O2 0 -20.0000 0 -18.0000 at bound -M -2.000017 O3 0 -20.0000 0 -17.0000 at bound -M -3.000018 O4 0 -22.0000 0 -19.0000 at bound -M -3.000019 S1 0 -20.0000 0 -19.0000 at bound -M -1.000020 S2 0 -20.0000 0 -18.0000 at bound -M -2.000021 S3 0 -20.0000 0 -17.0000 at bound -M -3.000022 S4 0 -20.0000 0 -17.0000 at bound -M -3.000023 F1 200.0000 80.0000 16,000.0000 0 basic 1.0000 M24 F2 250.0000 80.0000 20,000.0000 0 basic 2.0000 M25 F3 350.0000 80.0000 28,000.0000 0 basic 3.0000 M26 F4 250.0000 80.0000 20,000.0000 0 basic 3.0000 M

Objective Function (Max.) = 81,500.0000


1 I1 800.0000 -1.0000 -800.0000 0 basic -4.0000 02 I2 600.0000 -1.0000 -600.0000 0 basic -4.0000 03 I3 350.0000 -1.0000 -350.0000 0 basic -4.0000 04 I4 0 -1.0000 0 -1.0000 at bound -M 05 I5 0 -1.0000 0 -4.0000 at bound -M 3.00006 B1 0 -3.0000 0 -4.0000 at bound -M 1.00007 B2 0 -3.0000 0 -4.0000 at bound -M 1.00008 B3 0 -3.0000 0 -4.0000 at bound -M 1.00009 B4 0 -3.0000 0 -3.0000 at bound -M 010 B5 250.0000 -3.0000 -750.0000 0 basic -4.0000 011 P1 0 -12.0000 0 -11.0000 at bound -M -1.000012 P2 0 -14.0000 0 -12.0000 at bound -M -2.000013 P3 0 -14.0000 0 -11.0000 at bound -M -3.000014 P4 0 -16.0000 0 -13.0000 at bound -M -3.000015 O1 0 -18.0000 0 -17.0000 at bound -M -1.000016 O2 0 -20.0000 0 -18.0000 at bound -M -2.000017 O3 0 -20.0000 0 -17.0000 at bound -M -3.000018 O4 0 -22.0000 0 -19.0000 at bound -M -3.000019 S1 0 -20.0000 0 -19.0000 at bound -M -1.000020 S2 0 -20.0000 0 -18.0000 at bound -M -2.000021 S3 0 -20.0000 0 -17.0000 at bound -M -3.000022 S4 0 -20.0000 0 -17.0000 at bound -M -3.000023 F1 200.0000 80.0000 16,000.0000 0 basic 1.0000 M24 F2 250.0000 80.0000 20,000.0000 0 basic 2.0000 M25 F3 350.0000 80.0000 28,000.0000 0 basic 3.0000 M26 F4 250.0000 80.0000 20,000.0000 0 basic 3.0000 M

Objective Function (Max.) = 81,500.0000

Combined Report (cont.)

Left Hand Right Hand Slack Shadow Allowable AllowableConstraint Side Direction Side or Surplus Price Min. RHS Max. RHS

1 C1 0 = 0 0 -1.0000 -800.0000 M2 C2 0 = 0 0 -2.0000 -600.0000 M3 C3 0 = 0 0 -3.0000 -350.0000 M4 C4 0 = 0 0 -3.0000 -250.0000 M5 C5 200.0000 <= 200.0000 0 79.0000 0 M6 C6 250.0000 <= 250.0000 0 78.0000 0 M7 C7 350.0000 <= 350.0000 0 77.0000 0 M8 C8 250.0000 <= 250.0000 0 77.0000 0 M


1 C1 0 = 0 0 -1.0000 -800.0000 M2 C2 0 = 0 0 -2.0000 -600.0000 M3 C3 0 = 0 0 -3.0000 -350.0000 M4 C4 0 = 0 0 -3.0000 -250.0000 M5 C5 200.0000 <= 200.0000 0 79.0000 0 M6 C6 250.0000 <= 250.0000 0 78.0000 0 M7 C7 350.0000 <= 350.0000 0 77.0000 0 M8 C8 250.0000 <= 250.0000 0 77.0000 0 M

LP Example 2:

Maximize -1I1-1I2-1I3-1I4-1I5-3B1-3B2-3B3-3B4-3B5-12P1-14P2- 14P3-16P4-18O1-20O2-20O3-22O4-20S1-20S2-20S3-20S4+80F1+80F2+80F3+80F4

Subject to: 1I1-1I2-1B1+1B2+1P1+1O1+1S1-1F1=01I2-1I3-1B2+1B3+1P2+1O2+1S2-1F2=01I3-1I4-1B3+1B4+1P3+1O3+1S3-1F3=01I4-1I5-1B4+1B5+1P4+1O4+1S4-1F4=01F1<=2001F2<=2501F3<=3501F4<=250

I1=0

Combined Report


1 I1 0 -1.0000 0 0 basic -M M2 I2 0 -1.0000 0 -4.0000 at bound -M 3.00003 I3 0 -1.0000 0 -4.0000 at bound -M 3.00004 I4 0 -1.0000 0 -4.0000 at bound -M 3.00005 I5 0 -1.0000 0 -4.0000 at bound -M 3.00006 B1 0 -3.0000 0 -15.0000 at bound -M 12.00007 B2 200.0000 -3.0000 -600.0000 0 basic -3.0000 1.00008 B3 450.0000 -3.0000 -1,350.0000 0 basic -3.0000 1.00009 B4 800.0000 -3.0000 -2,400.0000 0 basic -3.0000 1.000010 B5 1,050.0000 -3.0000 -3,150.0000 0 basic -3.0000 1.000011 P1 0 -12.0000 0 0 at bound -M -12.000012 P2 0 -14.0000 0 -5.0000 at bound -M -9.000013 P3 0 -14.0000 0 -8.0000 at bound -M -6.000014 P4 0 -16.0000 0 -13.0000 at bound -M -3.000015 O1 0 -18.0000 0 -6.0000 at bound -M -12.000016 O2 0 -20.0000 0 -11.0000 at bound -M -9.000017 O3 0 -20.0000 0 -14.0000 at bound -M -6.000018 O4 0 -22.0000 0 -19.0000 at bound -M -3.000019 S1 0 -20.0000 0 -8.0000 at bound -M -12.000020 S2 0 -20.0000 0 -11.0000 at bound -M -9.000021 S3 0 -20.0000 0 -14.0000 at bound -M -6.000022 S4 0 -20.0000 0 -17.0000 at bound -M -3.000023 F1 200.0000 80.0000 16,000.0000 0 basic 12.0000 M24 F2 250.0000 80.0000 20,000.0000 0 basic 9.0000 M25 F3 350.0000 80.0000 28,000.0000 0 basic 6.0000 M26 F4 250.0000 80.0000 20,000.0000 0 basic 3.0000 M

Objective Function (Max.) = 76,500.0000 (Note: Alternate Solution Exists!!)


1 I1 0 -1.0000 0 0 basic -M M2 I2 0 -1.0000 0 -4.0000 at bound -M 3.00003 I3 0 -1.0000 0 -4.0000 at bound -M 3.00004 I4 0 -1.0000 0 -4.0000 at bound -M 3.00005 I5 0 -1.0000 0 -4.0000 at bound -M 3.00006 B1 0 -3.0000 0 -15.0000 at bound -M 12.00007 B2 200.0000 -3.0000 -600.0000 0 basic -3.0000 1.00008 B3 450.0000 -3.0000 -1,350.0000 0 basic -3.0000 1.00009 B4 800.0000 -3.0000 -2,400.0000 0 basic -3.0000 1.000010 B5 1,050.0000 -3.0000 -3,150.0000 0 basic -3.0000 1.000011 P1 0 -12.0000 0 0 at bound -M -12.000012 P2 0 -14.0000 0 -5.0000 at bound -M -9.000013 P3 0 -14.0000 0 -8.0000 at bound -M -6.000014 P4 0 -16.0000 0 -13.0000 at bound -M -3.000015 O1 0 -18.0000 0 -6.0000 at bound -M -12.000016 O2 0 -20.0000 0 -11.0000 at bound -M -9.000017 O3 0 -20.0000 0 -14.0000 at bound -M -6.000018 O4 0 -22.0000 0 -19.0000 at bound -M -3.000019 S1 0 -20.0000 0 -8.0000 at bound -M -12.000020 S2 0 -20.0000 0 -11.0000 at bound -M -9.000021 S3 0 -20.0000 0 -14.0000 at bound -M -6.000022 S4 0 -20.0000 0 -17.0000 at bound -M -3.000023 F1 200.0000 80.0000 16,000.0000 0 basic 12.0000 M24 F2 250.0000 80.0000 20,000.0000 0 basic 9.0000 M25 F3 350.0000 80.0000 28,000.0000 0 basic 6.0000 M26 F4 250.0000 80.0000 20,000.0000 0 basic 3.0000 M


LP Example 3:

Maximize -1I1-1I2-1I3-1I4-1I5-3B1-3B2-3B3-3B4-3B5-12P1-14P2-14P3-16P4-18O1-20O2-20O3-22O4-20S1-20S2-20S3-20S4+80F1+80F2+80F3+80F4

Subject to: 1I1-1I2-1B1+1B2+1P1+1O1+1S1-1F1=01I2-1I3-1B2+1B3+1P2+1O2+1S2-1F2=01I3-1I4-1B3+1B4+1P3+1O3+1S3-1F3=01I4-1I5-1B4+1B5+1P4+1O4+1S4-1F4=01F1<=2001F2<=2501F3<=3501F4<=250I1=0

B5<=10

Combined Report


1 I1 0 -1.0000 0 0 basic -M M2 I2 840.0000 -1.0000 -840.0000 0 basic -1.0000 3.00003 I3 590.0000 -1.0000 -590.0000 0 basic -1.0000 3.00004 I4 240.0000 -1.0000 -240.0000 0 basic -2.0000 3.00005 I5 0 -1.0000 0 -16.0000 at bound -M 15.00006 B1 0 -3.0000 0 -15.0000 at bound -M 12.00007 B2 0 -3.0000 0 -4.0000 at bound -M 1.00008 B3 0 -3.0000 0 -4.0000 at bound -M 1.00009 B4 0 -3.0000 0 -4.0000 at bound -M 1.000010 B5 10.0000 -3.0000 -30.0000 0 basic -15.0000 M11 P1 1,040.0000 -12.0000 -12,480.0000 0 basic -12.0000 012 P2 0 -14.0000 0 -1.0000 at bound -M -13.000013 P3 0 -14.0000 0 0 at bound -M -14.000014 P4 0 -16.0000 0 -1.0000 at bound -M -15.000015 O1 0 -18.0000 0 -6.0000 at bound -M -12.000016 O2 0 -20.0000 0 -7.0000 at bound -M -13.000017 O3 0 -20.0000 0 -6.0000 at bound -M -14.000018 O4 0 -22.0000 0 -7.0000 at bound -M -15.000019 S1 0 -20.0000 0 -8.0000 at bound -M -12.000020 S2 0 -20.0000 0 -7.0000 at bound -M -13.000021 S3 0 -20.0000 0 -6.0000 at bound -M -14.000022 S4 0 -20.0000 0 -5.0000 at bound -M -15.000023 F1 200.0000 80.0000 16,000.0000 0 basic 12.0000 M24 F2 250.0000 80.0000 20,000.0000 0 basic 13.0000 M25 F3 350.0000 80.0000 28,000.0000 0 basic 14.0000 M26 F4 250.0000 80.0000 20,000.0000 0 basic 15.0000 M



1 I1 0 -1.0000 0 0 basic -M M2 I2 840.0000 -1.0000 -840.0000 0 basic -1.0000 3.00003 I3 590.0000 -1.0000 -590.0000 0 basic -1.0000 3.00004 I4 240.0000 -1.0000 -240.0000 0 basic -2.0000 3.00005 I5 0 -1.0000 0 -16.0000 at bound -M 15.00006 B1 0 -3.0000 0 -15.0000 at bound -M 12.00007 B2 0 -3.0000 0 -4.0000 at bound -M 1.00008 B3 0 -3.0000 0 -4.0000 at bound -M 1.00009 B4 0 -3.0000 0 -4.0000 at bound -M 1.000010 B5 10.0000 -3.0000 -30.0000 0 basic -15.0000 M11 P1 1,040.0000 -12.0000 -12,480.0000 0 basic -12.0000 012 P2 0 -14.0000 0 -1.0000 at bound -M -13.000013 P3 0 -14.0000 0 0 at bound -M -14.000014 P4 0 -16.0000 0 -1.0000 at bound -M -15.000015 O1 0 -18.0000 0 -6.0000 at bound -M -12.000016 O2 0 -20.0000 0 -7.0000 at bound -M -13.000017 O3 0 -20.0000 0 -6.0000 at bound -M -14.000018 O4 0 -22.0000 0 -7.0000 at bound -M -15.000019 S1 0 -20.0000 0 -8.0000 at bound -M -12.000020 S2 0 -20.0000 0 -7.0000 at bound -M -13.000021 S3 0 -20.0000 0 -6.0000 at bound -M -14.000022 S4 0 -20.0000 0 -5.0000 at bound -M -15.000023 F1 200.0000 80.0000 16,000.0000 0 basic 12.0000 M24 F2 250.0000 80.0000 20,000.0000 0 basic 13.0000 M25 F3 350.0000 80.0000 28,000.0000 0 basic 14.0000 M26 F4 250.0000 80.0000 20,000.0000 0 basic 15.0000 M


Work Force Capacity Constraints

• The APPLP can incorporate short-term capacity adjustments

• For example, a work force can be increased or diminished

• (Workers in p1) = (Workers in p0) + (p0 hires) – (p0 layoffs)

• or Wi – Hi + Fi = W0

• and chained over multiple periods by work force level

• W1 - H1 + L1 = W0

• W2 - W1 - H2 + L2 = 0

• W3 - W2 - H3 + L3 = 0

LP Example 4:

Maximize -1I1-1I2-1I3-1I4-1I5-3B1-3B2-3B3-3B4-5B5- 12P1-14P2-14P3-16P4-18O1-20O2-20O3-22O4- 20S1-20S2-20S3-20S4+80F1+80F2+80F3+80F4-250W1- 250W2-250W3-250W4-500H1-500H2-500H3-500H4-1000L1-1000L2- 1000L3-1000L4

Subject to: 1I1-1I2-1B1+1B2+1P1+1O1+1S1-1F1=01I2-1I3-1B2+1B3+1P2+1O2+1S2-1F2=01I3-1I4-1B3+1B4+1P3+1O3+1S3-1F3=01I4-1I5-1B4+1B5+1P4+1O4+1S4-1F4=0

1F1<=2001F2<=2501F3<=3501F4<=250

1I1=01B5<=10

1W1-1H1+1L1=20-1W1+1W2-1H2+1L2=0-1W2+1W3-1H3+1L3=0-1W3+1W4-1H4+1L4=0

1P1-10W1<=01P2-10W2<=01P3-10W3<=01P4-10W4<=01O1-2W1<=01O2-2W2<=01O3-2W3<=01O4-2W4<=0

Combined Report

Decision Solution Unit Cost or Total Reduced Basis Allowable AllowableVariable Value Profit c(j) Contribution Cost Status Min. c(j) Max. c(j)

1 I1 0 -1.0000 0 0 basic -M M2 I2 40.0000 -1.0000 -40.0000 0 basic -2.0000 03 I3 0 -1.0000 0 -1.0000 at bound -M 04 I4 0 -1.0000 0 -1.0000 at bound -M 0.00005 I5 0 -1.0000 0 -21.0000 at bound -M 20.00006 B1 0 -3.0000 0 -22.0000 at bound -M 19.00007 B2 0 -3.0000 0 -4.0000 at bound -M 1.00008 B3 0 -3.0000 0 -3.0000 at bound -M 09 B4 0 -3.0000 0 -3.0000 at bound -M 0.000010 B5 10.0000 -5.0000 -50.0000 0 basic -20.0000 M11 P1 200.0000 -12.0000 -2,400.0000 0 basic -19.0000 114.800012 P2 200.0000 -14.0000 -2,800.0000 0 basic -20.0000 95.000013 P3 200.0000 -14.0000 -2,800.0000 0 basic -20.0000 76.000014 P4 200.0000 -16.0000 -3,200.0000 0 basic -20.0000 55.000015 O1 40.0000 -18.0000 -720.0000 0 basic -19.0000 616.000016 O2 10.0000 -20.0000 -200.0000 0 basic -20.0000 -19.000017 O3 40.0000 -20.0000 -800.0000 0 basic -20.0000 430.000018 O4 0 -22.0000 0 -2.0000 at bound -M -20.000019 S1 0 -20.0000 0 -1.0000 at bound -M -19.000020 S2 0 -20.0000 0 0 at bound -M -20.000021 S3 110.0000 -20.0000 -2,200.0000 0 basic -21.0000 -20.000022 S4 40.0000 -20.0000 -800.0000 0 basic -21.0000 -17.000023 F1 200.0000 80.0000 16,000.0000 0 basic 19.0000 M24 F2 250.0000 80.0000 20,000.0000 0 basic 20.0000 M25 F3 350.0000 80.0000 28,000.0000 0 basic 20.0000 M26 F4 250.0000 80.0000 20,000.0000 0 basic 20.0000 M27 W1 20.0000 -250.0000 -5,000.0000 0 basic -482.0000 1,018.000028 W2 20.0000 -250.0000 -5,000.0000 0 basic -482.0000 840.000029 W3 20.0000 -250.0000 -5,000.0000 0 basic -482.0000 650.000030 W4 20.0000 -250.0000 -5,000.0000 0 basic -482.0000 460.000031 H1 0 -500.0000 0 -1,268.0000 at bound -M 768.000032 H2 0 -500.0000 0 -1,090.0000 at bound -M 590.000033 H3 0 -500.0000 0 -900.0000 at bound -M 400.000034 H4 0 -500.0000 0 -710.0000 at bound -M 210.000035 L1 0 -1,000.0000 0 -232.0000 at bound -M -768.000036 L2 0 -1,000.0000 0 -410.0000 at bound -M -590.000037 L3 0 -1,000.0000 0 -600.0000 at bound -M -400.000038 L4 0 -1,000.0000 0 -790.0000 at bound -M -210.0000


Decision Solution Unit Cost or Total Reduced Basis Allowable AllowableVariable Value Profit c(j) Contribution Cost Status Min. c(j) Max. c(j)

1 I1 0 -1.0000 0 0 basic -M M2 I2 40.0000 -1.0000 -40.0000 0 basic -2.0000 03 I3 0 -1.0000 0 -1.0000 at bound -M 04 I4 0 -1.0000 0 -1.0000 at bound -M 0.00005 I5 0 -1.0000 0 -21.0000 at bound -M 20.00006 B1 0 -3.0000 0 -22.0000 at bound -M 19.00007 B2 0 -3.0000 0 -4.0000 at bound -M 1.00008 B3 0 -3.0000 0 -3.0000 at bound -M 09 B4 0 -3.0000 0 -3.0000 at bound -M 0.000010 B5 10.0000 -5.0000 -50.0000 0 basic -20.0000 M11 P1 200.0000 -12.0000 -2,400.0000 0 basic -19.0000 114.800012 P2 200.0000 -14.0000 -2,800.0000 0 basic -20.0000 95.000013 P3 200.0000 -14.0000 -2,800.0000 0 basic -20.0000 76.000014 P4 200.0000 -16.0000 -3,200.0000 0 basic -20.0000 55.000015 O1 40.0000 -18.0000 -720.0000 0 basic -19.0000 616.000016 O2 10.0000 -20.0000 -200.0000 0 basic -20.0000 -19.000017 O3 40.0000 -20.0000 -800.0000 0 basic -20.0000 430.000018 O4 0 -22.0000 0 -2.0000 at bound -M -20.000019 S1 0 -20.0000 0 -1.0000 at bound -M -19.000020 S2 0 -20.0000 0 0 at bound -M -20.000021 S3 110.0000 -20.0000 -2,200.0000 0 basic -21.0000 -20.000022 S4 40.0000 -20.0000 -800.0000 0 basic -21.0000 -17.000023 F1 200.0000 80.0000 16,000.0000 0 basic 19.0000 M24 F2 250.0000 80.0000 20,000.0000 0 basic 20.0000 M25 F3 350.0000 80.0000 28,000.0000 0 basic 20.0000 M26 F4 250.0000 80.0000 20,000.0000 0 basic 20.0000 M27 W1 20.0000 -250.0000 -5,000.0000 0 basic -482.0000 1,018.000028 W2 20.0000 -250.0000 -5,000.0000 0 basic -482.0000 840.000029 W3 20.0000 -250.0000 -5,000.0000 0 basic -482.0000 650.000030 W4 20.0000 -250.0000 -5,000.0000 0 basic -482.0000 460.000031 H1 0 -500.0000 0 -1,268.0000 at bound -M 768.000032 H2 0 -500.0000 0 -1,090.0000 at bound -M 590.000033 H3 0 -500.0000 0 -900.0000 at bound -M 400.000034 H4 0 -500.0000 0 -710.0000 at bound -M 210.000035 L1 0 -1,000.0000 0 -232.0000 at bound -M -768.000036 L2 0 -1,000.0000 0 -410.0000 at bound -M -590.000037 L3 0 -1,000.0000 0 -600.0000 at bound -M -400.000038 L4 0 -1,000.0000 0 -790.0000 at bound -M -210.0000


Combined Report (cont.)


1 C1 0 = 0 0 -19.0000 -10.0000 30.00002 C2 0 = 0 0 -20.0000 -10.0000 30.00003 C3 0 = 0 0 -20.0000 -110.0000 M4 C4 0 = 0 0 -20.0000 -40.0000 M5 C5 200.0000 <= 200.0000 0 61.0000 190.0000 230.00006 C6 250.0000 <= 250.0000 0 60.0000 240.0000 280.00007 C7 350.0000 <= 350.0000 0 60.0000 240.0000 M8 C8 250.0000 <= 250.0000 0 60.0000 210.0000 M9 C9 0 = 0 0 18.0000 0 10.000010 C10 10.0000 <= 10.0000 0 15.0000 0 50.000011 C11 20.0000 = 20.0000 0 -768.0000 18.7500 20.454512 C12 0 = 0 0 -590.0000 -2.5000 1.000013 C13 0 = 0 0 -400.0000 -20.0000 4.000014 C14 0 = 0 0 -210.0000 -20.0000 4.000015 C15 0 <= 0 0 7.0000 -30.0000 10.000016 C16 0 <= 0 0 6.0000 -30.0000 10.000017 C17 0 <= 0 0 6.0000 -200.0000 110.000018 C18 0 <= 0 0 4.0000 -200.0000 40.000019 C19 0 <= 0 0 1.0000 -30.0000 10.000020 C20 -30.0000 <= 0 30.0000 0 -30.0000 M21 C21 0 <= 0 0 0 -40.0000 110.000022 C22 -40.0000 <= 0 40.0000 0 -40.0000 M


1 C1 0 = 0 0 -19.0000 -10.0000 30.00002 C2 0 = 0 0 -20.0000 -10.0000 30.00003 C3 0 = 0 0 -20.0000 -110.0000 M4 C4 0 = 0 0 -20.0000 -40.0000 M5 C5 200.0000 <= 200.0000 0 61.0000 190.0000 230.00006 C6 250.0000 <= 250.0000 0 60.0000 240.0000 280.00007 C7 350.0000 <= 350.0000 0 60.0000 240.0000 M8 C8 250.0000 <= 250.0000 0 60.0000 210.0000 M9 C9 0 = 0 0 18.0000 0 10.000010 C10 10.0000 <= 10.0000 0 15.0000 0 50.000011 C11 20.0000 = 20.0000 0 -768.0000 18.7500 20.454512 C12 0 = 0 0 -590.0000 -2.5000 1.000013 C13 0 = 0 0 -400.0000 -20.0000 4.000014 C14 0 = 0 0 -210.0000 -20.0000 4.000015 C15 0 <= 0 0 7.0000 -30.0000 10.000016 C16 0 <= 0 0 6.0000 -30.0000 10.000017 C17 0 <= 0 0 6.0000 -200.0000 110.000018 C18 0 <= 0 0 4.0000 -200.0000 40.000019 C19 0 <= 0 0 1.0000 -30.0000 10.000020 C20 -30.0000 <= 0 30.0000 0 -30.0000 M21 C21 0 <= 0 0 0 -40.0000 110.000022 C22 -40.0000 <= 0 40.0000 0 -40.0000 M

APP Example

Frank-N-Stuff Manufacturing makes light fixtures. The production manager needs help preparing an aggregateproduction plan for the year 2002. Expected demand and the number of weekends in each period is provided in thetable below.

Projected demand in 2002 Number of weekendsPeriod 1 2000 13Period 2 1800 12Period 3 4500 13Period 4 1200 11

It takes 4 labor hours to produce one fixture. The contract with the Labor Union specifies that all employees arepaid 8 hours a day each workday and management assumes that there are 55 regular workdays available eachperiod. No workday overtime is allowed. However, each employee can be asked to work up to 8 hours of overtimeeach weekend if needed. All demand must be satisfied by year’s end (end of year backlog is not allowed). Ifnecessary, subcontracting can be used. Currently there are 10 employees working in the plant.

Inventory Holding Cost (product/period) $8Backlog Cost (product/period) $16Subcontracting Cost (product) $150Hiring & Training Cost (worker) $2500Layoff Cost (worker) $1000Regular Time Cost (worker/hour) $15Overtime Cost (worker/hour) $20

a. Prepare an aggregate plan that will minimize total costs and meet the demand.

b. Prepare an aggregate plan that will minimize total costs, meet the demand and keep regular productionconstant throughout the year (i.e., the number of chairs produced in regular time must be the same for allfour periods)

Timing and Inventory

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Transcript of Timing and Inventory