١Dr. Samia Rouibah Introduction to Management Science

Table of ContentsChapter 4 (Linear Programming: Formulation and Applications)

Super Grain Corp. Advertising-Mix Problem (Section 4.1) 4.2–4.5Resource Allocation Problems & Think-Big Capital Budgeting (Section 4.2) 4.6–4.10Cost-Benefit-Trade-Off Problems & Union Airways (Section 4.3) 4.11–4.15

٢Dr. Samia Rouibah Introduction to Management Science

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?

٣Dr. Samia Rouibah Introduction to Management Science

Cost and Exposure Data

500,000600,0001,300,000Expected number of exposures

40,00030,00090,000Planning budget

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

EachSunday Ad

EachMagazine Ad

EachTV CommercialCost Category

Costs

٤Dr. Samia Rouibah Introduction to Management Science

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

٥Dr. Samia Rouibah Introduction to Management Science

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.

٦Dr. Samia Rouibah Introduction to Management Science

Solving the Model

• The optimal solution provides the following plan for the promotional campaign

– Do not run any TV ads– Run 20 advertisements in magazines– Run 10 advertisements in Sunday Suplement

٧Dr. Samia Rouibah Introduction to Management Science

Evaluation of the adequacy of the Model

• May not provide a perfect match to the problem– Approximations and assumptions generally are required to have workable model

A reasonable high correlation between the prediction of the model and what would actually happen in the real problem.

– Fractional solutions are allowed– All output cells and target cell expressed as a SUMPRODUCT of data cells and

changing cells– Is it reasonable to assume that the cost is proportional to the number of

advertisement in that medium?

٨Dr. Samia Rouibah Introduction to Management Science

Resource Allocation Problems

• Are LP problems involving the allocation of resources to activities. The identifying feature for any such problem is that each functional constraints in the LP model is a resource constraint, which has the form,

Amount of resource used ≤ Amount of resource available

for one of the resources

• Characteristics of Resource Allocation Problems– The amount available of each resource– The amount of each resource needed by each activity– The contribution per unit of each activity to the overall measure of performance

• The Super Grain Corp. Advertising Problem– 3 Activities: TV commercials, Magazines Ads, Sunday ads.– Decisions: the levels (number) of these activities– 3 resources: Advertising Budget, Planning Budget, TV spots available

• How many resource constraint we should have for this problem?

٩Dr. Samia Rouibah Introduction to Management Science

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?

١٠Dr. Samia Rouibah Introduction to Management Science

Financial Data for the Projects

$50 million$70 million$45 millionNet present value

60 million70 million10 million3

20 million80 million90 million2

50 million80 million60 million1

$90 million$80 million$40 million0

Shopping CenterHotelOffice BuildingYear

Investment Capital Requirements

The company funds availability:•$25 millions now•$20 millions after 1 year•$20 millions after 2 years•$15 millions after 3 years

١١Dr. Samia Rouibah Introduction to Management Science

Formulation as a Resource Allocation Problem

• Activities– Activity 1: Invest in the construction of an office building– Activity 2: Invest in the construction of a hotel– Activity 2: Invest in the construction of shopping center

• Resources– Resource 1: Total investment capital available now– Resource 2: Cumulative investment capital available by the end of one year– Resource 3: Cumulative investment capital available by the end of two years– Resource 4: Cumulative investment capital available by the end of three years

– Amount of resource 1 available: $25 million– Amount of resource 2 available: $45 million– Amount of resource 3 available: $65 million– Amount of resource 4 available: $80 million

١٢Dr. Samia Rouibah Introduction to Management Science

Spreadsheet Formulation

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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

١٣Dr. Samia Rouibah Introduction to Management Science

Algebraic Formulation

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.

١٤Dr. Samia Rouibah Introduction to Management Science

Summary of Formulation Procedure for Resource-Allocation Problems (see p.120)

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.

١٥Dr. Samia Rouibah Introduction to Management Science

Cost-Benefit-Trade-Off Problem

• Are LP problems where the mix of levels of various activities is chosen to achieve a minimum acceptable levels for various benefits at a minimum cost. The identifying features is that each functional constraint is a benefit constraint, which has the form

Level Achieved ≥ Minimum acceptable levelfor one of the benefits

• Characteristics of Cost-Benefit-Trade-Off problems– The minimum acceptable level for each benefit (managerial policy decision)– For each benefit, the contribution of each activity to that benefit (per unit of the

activity)– The cost per unit of each activity

١٦Dr. Samia Rouibah Introduction to Management Science

The profit &Gambit Co. Advertising-Mix Problem

• Activities– Activity 1: Advertise on TV– Activities 2: Advertise in the print media

• Benefits– Benefit 1: Increased sales for a spray prewash stain remover– Benefit 2: Increased sales for a liquid laundry detergent– Benefit 3: Increased sales for a powder laundry detergent

• Sales goals– Increased sales to be at least: 3%, 18% and 4%respectively

The problem is to determine how much to advertise in each medium to meet all the sales goals at a minimum total cost

• Is this problem a resource allocation problem?

١٧Dr. Samia Rouibah Introduction to Management Science

Comparison between Super Grain and Profit & Gambit

• Both are advertising-mix problems

• They lead to different LP models

• Differences in the managerial view of the key – Super Grain managers focused first on how much to spend on the advertising

campaign and then set limits that led to resource constraints

١٨Dr. Samia Rouibah Introduction to Management Science

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?

١٩Dr. Samia Rouibah Introduction to Management Science

Schedule Data

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

15√Midnight to 6 AM

52√√10 PM to midnight

43√8 PM to 10 PM

82√√6 PM to 8 PM

73√√4 PM to 6 PM

64√√2 PM to 4 PM

87√√√Noon to 2 PM

65√√10 AM to noon

79√√8 AM to 10 AM

48√6 AM to 8 AM

MinimumNumber of

Agents Needed54321Time Period

Time Periods Covered by Shift

٢٠Dr. Samia Rouibah Introduction to Management Science

Spreadsheet Formulation

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101112131415161718192021

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

٢١Dr. Samia Rouibah Introduction to Management Science

Algebraic Formulation

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

Minimize Cost = $170S1 + $160S2 + $175S3 + $180S4 + $195S5subject to

Total 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)

٢٢Dr. Samia Rouibah Introduction to Management Science

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.

٢٣Dr. Samia Rouibah Introduction to Management Science

Distribution-Network Problems

Are problems that deal with the distribution of goods through a distribution network at a minimum cost.

٢٤Dr. Samia Rouibah Introduction to Management Science

The Big M Distribution-Network 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?

٢٥Dr. Samia Rouibah Introduction to Management Science

Some Data

9 lathes8 lathes10 lathesOrder Size

15 lathes700900800Factory 2

12 lathes$800$900$700Factory 1

OutputFrom

Customer 3Customer 2Customer 1To

Shipping Cost for Each Lathe

٢٦Dr. Samia Rouibah Introduction to Management Science

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

٢٧Dr. Samia Rouibah Introduction to Management Science

Spreadsheet Formulation

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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

Equality constraints are called Fixed-requirements constraints

٢٨Dr. Samia Rouibah Introduction to Management Science

Algebraic Formulation

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).

٢٩Dr. Samia Rouibah Introduction to Management Science

Types of Functional Constraints

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

Resource-allocation problems and mixed problems

For some resource,Amount used ≤Amount available

LHS ≤ RHSResource constraint

Distribution-network problems and mixed problems

For some quantity,Amount provided =Required amount

LHS = RHSFixed-requirement constraint

Cost-benefit-trade-off problems and mixed problems

For some benefit,Level achieved ≥Minimum Acceptable

LHS ≥ RHSBenefit constraint

Main UsageTypical InterpretationForm*Type

٣٠Dr. Samia Rouibah Introduction to Management Science

Model Formulation From a Managerial Perspective

• Measure of performance must capture what management wants accomplished

• Limitations of resources should be expressed as resource constraints

• Minimum acceptable levels for benefits should be expressed as benefit constraints

• Fixed requirements for certain quantities should be expressed as Fixed-requirements constraints

٣١Dr. Samia Rouibah Introduction to Management Science

Model Formulation From a Managerial Perspective

• Spreadsheets help some managers to formulate/solve small LP themselves

• LP studies need strong managerial input and support (good communication)

• Model validation process is used to test initial versions of the model (errors, omissions)

• Model enrichment process is used to reach a more elaborate model and reasonably easy to solve

• What-if-analysis addresses some key questions that remain after formulating and solving

٣٢Dr. Samia Rouibah Introduction to Management Science

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