40629075 Introduction to Management Science Hiller Hiller Chapter03

© The McGraw-Hill Companies, Inc., 2008

3.1McGraw-Hill/Irwin

Table of ContentsChapter 3 (Linear Programming: Formulation and Applications)

Super Grain Corp. Advertising-Mix Problem (Section 3.1) 3.2–3.5Resource Allocation Problems (Section 3.2) 3.6–3.16Cost-Benefit-Trade-Off Problems (Section 3.3) 3.17–3.22Mixed Problems (Section 3.4) 3.23–3.28Transportation Problems (Section 3.5) 3.29–3.33Assignment Problems (Section 3.6) 3.34–3.37

Applications of Linear Programming with Spreadsheets (UW Lecture) 3.38–3.57These slides are based upon lectures to first-year MBA students at the University of Washington that discuss the application and formulation of linear programming models (as taught by one of the authors).



Super Grain Corp. Advertising-Mix Problem

• Goal: Design the promotional campaign for Crunchy Start.

• The three most effective advertising media for this product are– Television commercials on Saturday morning programs for children.

– Advertisements in food and family-oriented magazines.

– Advertisements in Sunday supplements of major newspapers.

• The limited resources in the problem are– Advertising budget ($4 million).

– Planning budget ($1 million).

– TV commercial spots available (5).

• The objective will be measured in terms of the expected number of exposures.

Question: At what level should they advertise Crunchy Start in each of the three media?



Cost and Exposure Data

Costs

Cost CategoryEach

TV CommercialEach

Magazine AdEach

Sunday Ad

Ad Budget $300,000 $150,000 $100,000

Planning budget 90,000 30,000 40,000

Expected number of exposures

1,300,000 600,000 500,000



Spreadsheet Formulation

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B C D E F G HTV Spots Magazine Ads SS Ads

Exposures per Ad 1,300 600 500(thousands)

Budget BudgetCost per Ad ($thousands) Spent Available

Ad Budget 300 150 100 4,000 <= 4,000Planning Budget 90 30 40 1,000 <= 1,000

Total ExposuresTV Spots Magazine Ads SS Ads (thousands)

Number of Ads 0 20 10 17,000<=

Max TV Spots 5



Algebraic Formulation

Let TV = Number of commercials for separate spots on televisionM = Number of advertisements in magazines.SS = Number of advertisements in Sunday supplements.

Maximize Exposure = 1,300TV + 600M + 500SSsubject to

Ad Spending: 300TV + 150M + 100SS ≤ 4,000 ($thousand)Planning Cost: 90TV + 30M + 30SS ≤ 1,000 ($thousand)Number of TV Spots: TV ≤ 5

andTV ≥ 0, M ≥ 0, SS ≥ 0.



The TBA Airlines Problem

• TBA Airlines is a small regional company that specializes in short flights in small airplanes.

• The company has been doing well and has decided to expand its operations.

• The basic issue facing management is whether to purchase more small airplanes to add some new short flights, or start moving into the national market by purchasing some large airplanes, or both.

Question: How many airplanes of each type should be purchased to maximize their total net annual profit?



Data for the TBA Airlines Problem

SmallAirplane

LargeAirplane

CapitalAvailable

Net annual profit per airplane $1 million $5 million

Purchase cost per airplane 5 million 50 million $100 million

Maximum purchase quantity 2 —



Violates Divisibility Assumption of LP

• Divisibility Assumption of Linear Programming: Decision variables in a linear programming model are allowed to have any values, including fractional values, that satisfy the functional and nonnegativity constraints. Thus, these variables are not restricted to just integer values.

• Since the number of airplanes purchased by TBA must have an integer value, the divisibility assumption is violated.



Spreadsheet Model

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B C D E F GSmall Airplane Large Airplane

Unit Profit ($millions) 1 5

Capital CapitalSpent Available

Capital ($millions) 5 50 100 <= 100

Total ProfitSmall Airplane Large Airplane ($millions)

Units Produced 0 2 10<=

Maximum Small Airplanes 2

Capital Per Unit Produced



Integer Programming Formulation

Let S = Number of small airplanes to purchase

L = Number of large airplanes to purchase

Maximize Profit = S + 5L ($millions)

subject to

Capital Available: 5S + 50L ≤ 100 ($millions)

Max Small Planes: S ≤ 2

and

S ≥ 0, L ≥ 0

S, L are integers.



Think-Big Capital Budgeting Problem

• Think-Big Development Co. is a major investor in commercial real-estate development projects.

• They are considering three large construction projects– Construct a high-rise office building.

– Construct a hotel.

– Construct a shopping center.

• Each project requires each partner to make four investments: a down payment now, and additional capital after one, two, and three years.

Question: At what fraction should Think-Big invest in each of the three projects?



Financial Data for the Projects

Investment Capital Requirements

Year Office Building Hotel Shopping Center

0 $40 million $80 million $90 million

1 60 million 80 million 50 million



Net present value $45 million $70 million $50 million




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B C D E F G HOffice Shopping

Building Hotel CenterNet Present Value 45 70 50

($millions) Cumulative CumulativeCapital Capital

Cumulative Capital Required ($millions) Spent AvailableNow 40 80 90 25 <= 25

End of Year 1 100 160 140 44.757 <= 45End of Year 2 190 240 160 60.583 <= 65End of Year 3 200 310 220 80 <= 80

Office Shopping Total NPVBuilding Hotel Center ($millions)

Participation Share 0.00% 16.50% 13.11% 18.11




Let OB = Participation share in the office building,H = Participation share in the hotel,SC = Participation share in the shopping center.

Maximize NPV = 45OB + 70H + 50SCsubject to

Total invested now: 40OB + 80H + 90SC ≤ 25 ($million)Total invested within 1 year: 100OB + 160H + 140SC ≤ 45 ($million)Total invested within 2 years: 190OB + 240H + 160SC ≤ 65 ($million)Total invested within 3 years: 200OB + 310H + 220SC ≤ 80 ($million)

andOB ≥ 0, H ≥ 0, SC ≥ 0.



Template for Resource-Allocation Problems

Activities

Unit Profit profit per unit of activityResources Resources

Used Available

SUMPRODUCTresource used per unit of activity (resource used per unit,

changing cells)

Total ProfitLevel of Activity changing cells SUMPRODUCT(profit per unit, changing cells)

<=

Const

rain

ts



Summary of Formulation Procedure for Resource-Allocation Problems

1. Identify the activities for the problem at hand.

2. Identify an appropriate overall measure of performance (commonly profit).

3. For each activity, estimate the contribution per unit of the activity to the overall measure of performance.

4. Identify the resources that must be allocated.

5. For each resource, identify the amount available and then the amount used per unit of each activity.

6. Enter the data in steps 3 and 5 into data cells.

7. Designate changing cells for displaying the decisions.

8. In the row for each resource, use SUMPRODUCT to calculate the total amount used. Enter <= and the amount available in two adjacent cells.

9. Designate a target cell. Use SUMPRODUCT to calculate this measure of performance.



Union Airways Personnel Scheduling

• Union Airways is adding more flights to and from its hub airport and so needs to hire additional customer service agents.

• The five authorized eight-hour shifts are– Shift 1: 6:00 AM to 2:00 PM

– Shift 2: 8:00 AM to 4:00 PM

– Shift 3: Noon to 8:00 PM

– Shift 4: 4:00 PM to midnight

– Shift 5: 10:00 PM to 6:00 AM

Question: How many agents should be assigned to each shift?



Schedule Data

Time Periods Covered by Shift

Time Period 1 2 3 4 5

MinimumNumber of

Agents Needed

6 AM to 8 AM √ 48

8 AM to 10 AM √ √ 79

10 AM to noon √ √ 65

Noon to 2 PM √ √ √ 87

2 PM to 4 PM √ √ 64

4 PM to 6 PM √ √ 73

6 PM to 8 PM √ √ 82

8 PM to 10 PM √ 43

10 PM to midnight √ √ 52

Midnight to 6 AM √ 15

Daily cost per agent $170 $160 $175 $180 $195




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B C D E F G H I J6am-2pm 8am-4pm Noon-8pm 4pm-midnight 10pm-6am

Shift Shift Shift Shift ShiftCost per Shift $170 $160 $175 $180 $195

Total MinimumTime Period Shift Works Time Period? (1=yes, 0=no) Working Needed

6am-8am 1 0 0 0 0 48 >= 488am-10am 1 1 0 0 0 79 >= 79

10am- 12pm 1 1 0 0 0 79 >= 6512pm-2pm 1 1 1 0 0 118 >= 872pm-4pm 0 1 1 0 0 70 >= 644pm-6pm 0 0 1 1 0 82 >= 736pm-8pm 0 0 1 1 0 82 >= 82

8pm-10pm 0 0 0 1 0 43 >= 4310pm-12am 0 0 0 1 1 58 >= 52

12am-6am 0 0 0 0 1 15 >= 15

6am-2pm 8am-4pm Noon-8pm 4pm-midnight 10pm-6amShift Shift Shift Shift Shift Total Cost

Number Working 48 31 39 43 15 $30,610




Let Si = Number working shift i (for i = 1 to 5),

Minimize Cost = $170S1 + $160S2 + $175S3 + $180S4 + $195S5

subject toTotal agents 6AM–8AM: S1 ≥ 48Total agents 8AM–10AM: S1 + S2 ≥ 79Total agents 10AM–12PM: S1 + S2 ≥ 65Total agents 12PM–2PM: S1 + S2 + S3 ≥ 87Total agents 2PM–4PM: S2 + S3 ≥ 64Total agents 4PM–6PM: S3 + S4 ≥ 73Total agents 6PM–8PM: S3 + S4 ≥ 82Total agents 8PM–10PM: S4 ≥ 43Total agents 10PM–12AM: S4 + S5 ≥ 52Total agents 12AM–6AM: S5 ≥ 15

andSi ≥ 0 (for i = 1 to 5)



Template for Cost-Benefit Tradoff Problems

Activities

Unit Cost cost per unit of activityBenefit Benefit

Achieved Needed

SUMPRODUCTbenefit achieved per unit of activity (benefit per unit,

changing cells)

Total CostLevel of Activity changing cells SUMPRODUCT(cost per unit, changing cells)

>=

Const

rain

ts



Summary of Formulation Procedure forCost-Benefit-Tradeoff Problems

1. Identify the activities for the problem at hand.

2. Identify an appropriate overall measure of performance (commonly cost).

3. For each activity, estimate the contribution per unit of the activity to the overall measure of performance.

4. Identify the benefits that must be achieved.

5. For each benefit, identify the minimum acceptable level and then the contribution of each activity to that benefit.

6. Enter the data in steps 3 and 5 into data cells.

7. Designate changing cells for displaying the decisions.

8. In the row for each benefit, use SUMPRODUCT to calculate the level achieved. Enter >= and the minimum acceptable level in two adjacent cells.

9. Designate a target cell. Use SUMPRODUCT to calculate this measure of performance.



Types of Functional Constraints

Type Form* Typical Interpretation Main Usage

Resource constraint LHS ≤ RHSFor some resource, Amount used ≤ Amount available

Resource-allocation problems and mixed problems

Benefit constraint LHS ≥ RHSFor some benefit, Level achieved ≥ Minimum Acceptable

Cost-benefit-trade-off problems and mixed problems

Fixed-requirement constraint

LHS = RHSFor some quantity, Amount provided = Required amount

Transportation problems and mixed problems

* LHS = Left-hand side (a SUMPRODUCT function). RHS = Right-hand side (a constant).



Continuing the Super Grain Case Study

• David and Claire conclude that the spreadsheet model needs to be expanded to incorporate some additional considerations.

• In particular, they feel that two audiences should be targeted — young children and parents of young children.

• Two new goals– The advertising should be seen by at least five million young children.

– The advertising should be seen by at least five million parents of young children.

• Furthermore, exactly $1,490,000 should be allocated for cents-off coupons.



Benefit and Fixed-Requirement Data

Number Reached in Target Category (millions)

EachTV Commercial

EachMagazine Ad

EachSunday Ad

MinimumAcceptable

Level

Young children 1.2 0.1 0 5

Parents of young children 0.5 0.2 0.2 5

Contribution Toward Required Amount

EachTV Commercial

EachMagazine Ad

EachSunday Ad

RequiredAmount

Coupon redemption 0 $40,000 $120,000 $1,490,000




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B C D E F G HTV Spots Magazine Ads SS Ads

Exposures per Ad 1,300 600 500(thousands)

Cost per Ad ($thousands) Budget Spent Budget AvailableAd Budget 300 150 100 3,775 <= 4,000

Planning Budget 90 30 40 1,000 <= 1,000

Number Reached per Ad (millions) Total Reached Minimum AcceptableYoung Children 1.2 0.1 0 5 >= 5

Parents of Young Children 0.5 0.2 0.2 5.85 >= 5

TV Spots Magazine Ads SS Ads Total Redeemed Required AmountCoupon Redemption per Ad 0 40 120 1,490 = 1,490

($thousands)Total Exposures

TV Spots Magazine Ads SS Ads (thousands)Number of Ads 3 14 7.75 16,175

<=Maximum TV Spots 5




Let TV = Number of commercials for separate spots on televisionM = Number of advertisements in magazines.SS = Number of advertisements in Sunday supplements.

Maximize Exposure = 1,300TV + 600M + 500SSsubject to

Ad Spending: 300TV + 150M + 100SS ≤ 4,000 ($thousand)Planning Cost: 90TV + 30M + 30SS ≤ 1,000 ($thousand)Number of TV Spots: TV ≤ 5

Young children: 1.2TV + 0.1M ≥ 5 (millions)Parents: 0.5TV + 0.2M + 0.2SS ≥ 5 (millions)

Coupons: 40M + 120SS = 1,490 ($thousand)

andTV ≥ 0, M ≥ 0, SS ≥ 0.



Template for Mixed Problems

Activities

Unit Profit or Cost profit/cost per unit of activityResources Resources

Used Available

SUMPRODUCTresource used per unit of activity (resource used per unit,

changing cells)

Benefit BenefitAchieved Needed

SUMPRODUCTbenefit achieved per unit of activity (benefit per unit,

changing cells)

Total Profit or CostLevel of Activity changing cells

Const

rain

ts

SUMPRODUCT(profit/cost per unit, changing cells)

<=

>=

=



The Big M Transportation Problem

• The Big M Company produces a variety of heavy duty machinery at two factories. One of its products is a large turret lathe.

• Orders have been received from three customers for the turret lathe.

Question: How many lathes should be shipped from each factory to each customer?



Some Data

Shipping Cost for Each Lathe

To Customer 1 Customer 2 Customer 3

From Output

Factory 1 $700 $900 $800 12 lathes

Factory 2 800 900 700 15 lathes

Order Size 10 lathes 8 lathes 9 lathes



The Distribution Network

F1

C2

C3

C1

F2

12 latheproduced

15 lathesproduced

10 lathesneeded

8 lathesneeded

9 lathesneeded

$700/lathe

$900/lathe

$800/lathe

$800/lathe $900/lathe

$700/lathe




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B C D E F G HShipping Cost

(per Lathe) Customer 1 Customer 2 Customer 3Factory 1 $700 $900 $800Factory 2 $800 $900 $700

TotalShipped

Units Shipped Customer 1 Customer 2 Customer 3 Out OutputFactory 1 10 2 0 12 = 12Factory 2 0 6 9 15 = 15

Total To Customer 10 8 9= = = Total Cost

Order Size 10 8 9 $20,500




Let Sij = Number of lathes to ship from i to j (i = F1, F2; j = C1, C2, C3).

Minimize Cost = $700SF1-C1 + $900SF1-C2 + $800SF1-C3 + $800SF2-C1 + $900SF2-C2 + $700SF2-C3

subject toFactory 1: SF1-C1 + SF1-C2 + SF1-C3 = 12Factory 2: SF2-C1 + SF2-C2 + SF2-C3 = 15Customer 1: SF1-C1 + SF2-C1 = 10Customer 2: SF1-C2 + SF2-C2 = 8Customer 3: SF1-C3 + SF2-C3 = 9

andSij ≥ 0 (i = F1, F2; j = C1, C2, C3).



Sellmore Company Assignment Problem

• The marketing manager of Sellmore Company will be holding the company’s annual sales conference soon.

• He is hiring four temporary employees:– Ann

– Ian

– Joan

– Sean

• Each will handle one of the following four tasks:– Word processing of written presentations

– Computer graphics for both oral and written presentations

– Preparation of conference packets, including copying and organizing materials

– Handling of advance and on-site registration for the conference

Question: Which person should be assigned to which task?



Data for the Sellmore Problem

Required Time per Task (Hours)

TemporaryEmployee

WordProcessing Graphics Packets Registrations

HourlyWage

Ann 35 41 27 40 $14

Ian 47 45 32 51 12

Joan 39 56 36 43 13

Sean 32 51 25 46 15




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B C D E F G H I J

Required Time Word Hourly(Hours) Processing Graphics Packets Registrations Wage

Ann 35 41 27 40 $14Assignee Ian 47 45 32 51 $12

Joan 39 56 36 43 $13Sean 32 51 25 46 $15

WordCost Processing Graphics Packets Registrations

Ann $490 $574 $378 $560Assignee Ian $564 $540 $384 $612

Joan $507 $728 $468 $559Sean $480 $765 $375 $690

Word TotalAssignment Processing Graphics Packets Registrations Assignments Supply

Ann 0 0 1 0 1 = 1Assignee Ian 0 1 0 0 1 = 1

Joan 0 0 0 1 1 = 1Sean 1 0 0 0 1 = 1

Total Assigned 1 1 1 1= = = = Total Cost

Demand 1 1 1 1 $1,957

Task

Task

Task



The Model for Assignment Problems

Given a set of tasks to be performed and a set of assignees who are available to perform these tasks, the problem is to determine which assignee should be assigned to each task.

To fit the model for an assignment problem, the following assumptions need to be satisfied:

1. The number of assignees and the number of tasks are the same.

2. Each assignee is to be assigned to exactly one task.

3. Each task is to be performed by exactly one assignee.

4. There is a cost associated with each combination of an assignee performing a task.

5. The objective is to determine how all the assignments should be made to minimize the total cost.



Formulating an LP Spreadsheet Model

• Enter all of the data into the spreadsheet. Color code (blue).

• What decisions need to be made? Set aside a cell in the spreadsheet for each decision variable (changing cell). Color code (yellow with border).

• Write an equation for the objective in a cell. Color code (orange with heavy border).

• Put all three components (LHS, ≤/=/≥, RHS) of each constraint into three cells on the spreadsheet.

• Some Examples:– Production Planning

– Diet / Blending

– Workforce Scheduling

– Transportation / Distribution

– Assignment



LP Example #1 (Product Mix)

The Quality Furniture Corporation produces benches and picnic tables. The firm has a limited supply of two resources: labor and wood. 1,600 labor hours are available during the next production period. The firm also has a stock of 9,000 pounds of wood available. Each bench requires 3 labor hours and 12 pounds of wood. Each table requires 6 labor hours and 38 pounds of wood. The profit margin on each bench is $8 and on each table is $18.

Question: What product mix will maximize their total profit?




Let B = Number of benches to produce,T = Number of tables to produce.

Maximize Profit = $8B + $18Tsubject to

Labor: 3B + 6T ≤ 1,600 hoursWood: 12B + 38T ≤ 9,000 pounds

andB ≥ 0, T ≥ 0.




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B C D E F GBenches Tables

Profit $8 $18

Resources Total AvailableLabor 3 6 1,600 <= 1,600Wood 12 38 9,000 <= 9,000

Total CostUnits Produced 161.90 185.71 $4,638.10

Used per Unit Produced



LP Example #2 (Diet Problem)

A prison is trying to decide what to feed its prisoners. They would like to offer some combination of milk, beans, and oranges. Their goal is to minimize cost, subject to meeting the minimum nutritional requirements imposed by law. The cost and nutritional contents of each food, along with the minimum nutritional requirements are shown below.

Milk(gallons)

NavyBeans(cups)

Oranges(large Calif.Valencia)

MinimumDaily

Requirement

Niacin (mg) 3.2 4.9 0.8 13.0

Thiamin (mg) 1.12 1.3 0.19 1.5

Vitamin C (mg) 32 0 93 45

Cost ($) 2.00 0.20 0.25

Question: What should the diet for each prisoner be?




Let x1 = gallons of milk per prisoner,x2 = cups of beans per prisoner,x3 = number of oranges per prisoner.

Minimize Cost = $2.00x1 + $0.20x2 + $0.25x3

subject toNiacin: 3.2x1 + 4.9x2 + 0.8x3 ≥ 13 mgThiamin: 1.12x1 + 1.3x2 + 0.19x3 ≥ 1.5 mgVitamin C: 32x1 + 93x3 ≥ 45 mg

andx1 ≥ 0, x2 ≥ 0, x3 ≥ 0.




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B C D E F G HMilk Beans(gal.) (cups) Oranges

Cost $2.00 $0.20 $0.25Minimum

Total RequirementNiacin (mg) 3.2 4.9 0.8 13 >= 13

Thiamin (mg) 1.12 1.3 0.19 3.438 >= 1.5Vitamin C (mg) 32 0 93 45 >= 45

Quantity Total Cost(per prisoner) 0 2.574 0.484 $0.64

Nutritional Contents (mg)



George Dantzig’s Diet

• Stigler (1945) “The Cost of Subsistence”– heuristic solution. Cost = $39.93.

• Dantzig invents the simplex method (1947)– Stigler’s problem “solved” in 120 man days. Cost = $39.69.

• Dantzig goes on a diet (early 1950’s), applies diet model:– ≤ 1,500 calories

– objective: maximize (weight minus water content)

– 500 food types

• Initial solutions had problems– 500 gallons of vinegar

– 200 bouillon cubes

For more details, see July-Aug 1990 Interfaces article “The Diet Problem”



Least-Cost Menu Planning Models in Food Systems Management

• Used in many institutions with feeding programs: hospitals, nursing homes, schools, prisons, etc.

• Menu planning often extends to a sequence of meals or a cycle.

• Variety important (separation constraints).

• Preference ratings (related to service frequency).

• Side constraints (color, categories, etc.)

• Generally models have reduced cost about 10%, met nutritional requirements better, and increased customer satisfaction compared to traditional methods.

• USDA uses these models to plan food stamp allotment.

For more details, see Sept-Oct 1992 Interfaces article “The Evolution of the Diet Model in Managing Food Systems”



LP Example #3 (Scheduling Problem)

An airline reservations office is open to take reservations by telephone 24 hours per day, Monday through Friday. The number of reservation agents needed for each time period is shown below. A union contract requires that all employees work 8 consecutive hours.

Question: How many reservation agents should work each 8-hour shift?

Time PeriodNumber of

Agents Needed

12am – 4am 11

4am – 8am 15

8am – 12pm 31

12pm – 4pm 17

4pm – 8pm 25

8pm – 12am 19




Let x1 = agents who work 12am – 8am,x2 = agents who work 4am – 12pm,x3 = agents who work 8am – 4pm,x4 = agents who work 12pm – 8pm,x5 = agents who work 4pm – 12am,x6 = agents who work 8pm – 4am.

Minimize Number of agents = x1 + x2 + x3 + x4 + x5 + x6

subject to12am–4am: x1 + x6 ≥ 114am–8am: x1 + x2 ≥ 158am–12pm: x2 + x3 ≥ 3112pm–4pm: x3 + x4 ≥ 174pm–8pm: x4 + x5 ≥ 258pm–12am: x5 + x6 ≥ 19

andx1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0, x5 ≥ 0, x6 ≥ 0.




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A B C D E F G H

Reservation Agents Scheduling Problem

Time Number Minimum NumberPeriod Working Required Shift Working

12am Š 4am 11 >= 11 12am - 8am 04am Š 8am 15 >= 15 4am - 12pm 158am Š 12pm 31 >= 31 8am - 4pm 1612pm Š 4pm 33 >= 17 12pm - 8pm 174pm Š 8pm 25 >= 25 4pm - 12am 88pm ŠŹ12am 19 >= 19 8pm - 4am 11

Total 67



Workforce Scheduling at United Airlines

• United employs 5,000 reservation and customer service agents.

• Some part-time (2-8 hour shifts), some full-time (8-10 hour shifts).

• Workload varies greatly over day.

• Modeled problem as LP:– Decision variables: how many employees of each shift length should begin at each

potential start time (half-hour intervals).

– Constraints: minimum required employees for each half-hour.

– Objective: minimize cost.

• Saved United about $6 million annually, improved customer service, still in use today.

For more details, see Jan-Feb 1986 Interfaces article “United Airlines Station Manpower Planning System”



LP Example #4 (Transportation Problem)

A company has two plants producing a certain product that is to be shipped to three distribution centers. The unit production costs are the same at the two plants, and the shipping cost per unit is shown below. Shipments are made once per week. During each week, each plant produces at most 60 units and each distribution center needs at least 40 units.

Distribution Center

1 2 3

PlantA $4 $6 $4

B $6 $5 $2

Question: How many units should be shipped from each plant to each distribution center?




Let xij = units to ship from plant i to distribution center j (i = A, B; j = 1, 2, 3),

Minimize Cost = $4xA1 + $6xA2 + $4xA3 + $6xB1 + $5xB2 + $2xB3

subject toPlant A: xA1 + xA2 + xA3 ≤ 60Plant B: xB1 + xB2 + xB3 ≤ 60Distribution Center 1: xA1 + xB1 ≥ 40Distribution Center 2: xA2 + xB2 ≥ 40Distribution Center 3: xA3 + xB3 ≥ 40

andxij ≥ 0 (i = A, B; j = 1, 2, 3).




3

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B C D E F G HDistribution Distribution Distribution

Cost Center 1 Center 2 Center 3Plant A $4 $6 $4Plant B $6 $5 $2

Shipment Distribution Distribution Distribution

Quantities Center 1 Center 2 Center 3 Shipped AvailablePlant A 40 20 0 60 <= 60Plant B 0 20 40 60 <= 60Shipped 40 40 40 Cost = $460

>= >= >=Needed 40 40 40



Distribution System at Proctor and Gamble

• Proctor and Gamble needed to consolidate and re-design their North American distribution system in the early 1990’s.

– 50 product categories

– 60 plants

– 15 distribution centers

– 1000 customer zones

• Solved many transportation problems (one for each product category).

• Goal: find best distribution plan, which plants to keep open, etc.

• Closed many plants and distribution centers, and optimized their product sourcing and distribution location.

• Implemented in 1996. Saved $200 million per year.

For more details, see 1997 Jan-Feb Interfaces article, “Blending OR/MS, Judgement, and GIS: Restructuring P&G’s Supply Chain”



LP Example #5 (Assignment Problem)

The coach of a swim team needs to assign swimmers to a 200-yard medley relay team (four swimmers, each swims 50 yards of one of the four strokes). Since most of the best swimmers are very fast in more than one stroke, it is not clear which swimmer should be assigned to each of the four strokes. The five fastest swimmers and their best times (in seconds) they have achieved in each of the strokes (for 50 yards) are shown below.

Backstroke Breaststroke Butterfly Freestyle

Carl 37.7 43.4 33.3 29.2

Chris 32.9 33.1 28.5 26.4

David 33.8 42.2 38.9 29.6

Tony 37.0 34.7 30.4 28.5

Ken 35.4 41.8 33.6 31.1

Question: How should the swimmers be assigned to make the fastest relay team?




Let xij = 1 if swimmer i swims stroke j; 0 otherwisetij = best time of swimmer i in stroke j

Minimize Time = ∑ i ∑ j tij xij

subject to

each stroke swum: ∑ i xij = 1 for each stroke j

each swimmer swims 1: ∑ j xij ≤ 1 for each swimmer i

andxij ≥ 0 for all i and j.




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B C D E F G H I

Best Times Backstroke Breastroke Butterfly FreestyleCarl 37.7 43.4 33.3 29.2Chris 32.9 33.1 28.5 26.4David 33.8 42.2 38.9 29.6Tony 37.0 34.7 30.4 28.5Ken 35.4 41.8 33.6 31.1

Assignment Backstroke Breastroke Butterfly FreestyleCarl 0 0 0 1 1 <= 1Chris 0 0 1 0 1 <= 1David 1 0 0 0 1 <= 1Tony 0 1 0 0 1 <= 1Ken 0 0 0 0 0 <= 1

1 1 1 1 Time = 126.2= = = =1 1 1 1

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40629075 Introduction to Management Science Hiller Hiller Chapter03