Linear Programming: Formulation and...

Linear Programming: Formulation and Applications

Prepared by Nguyễn Xuân Thọ -David (Da41g201)

Chapter 3

3.2

Outline of Contents

Section 3.1. Case 1: Super Grain Corp. Advertising-Mix Problem (Section3.1 in Textbook)

Section 3.2. Resource Allocation Problems (Section 3.2 in Textbook)

- Case 2. TBA Airlines Airplane Purchasing Problem - Case 3. Think-Big Development Co. Capital Budgeting Problem Section 3.3. Cost-Benefit-Trade-Off Problems (Section 3.3 in Textbook) - Case 4. Union Airways Personel Scheduling Problem Section 3.4. Mixed Problems (Section 3.4 in Textbook) - Case 4. Super Grain Corp. Advertising-Mix Problem (add more constraints and target) Section 3.5. Transportation Problems (Section 3.5 in Textbook) - Case 5: Big M Company Distribution Problem Section 3.6. Assignment Problems (Section 3.6 in Textbook) - Case 6: Sellmore Co. Assignment Problem Finally Section. Solution of Case study 3.4 “New Frontier” with 6 questions (Page 118)

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Learning Objectives

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1. Recognize various kinds of managerial problems to which linear programming can be applied. 2. Describe the five major (or key) categories of linear programming problems, including their identifying features 3. Formulate a linear programming model from a description of a problem in any of these categories. 4. Describe the difference between resource constraints and benefit constraints, including the difference in how they arise. 5. Describe Fixed-constraint and where they arise. 6. Identify the kinds of Excel functions that linear programming spreadsheet models use for the output cells, including objective cell. 7. Identify the four components of any linear programming model and the kinds of spreadsheet cells used for each component 8. Recognize managerial problems that can be formulated and analyzed as linear programming problems. 9. Understand the flexibility that managers have in prescribing key consideration that can be incorporated into programming model.

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3.1. 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?

3.5

Cost and Exposure Data

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Analysis of the problem

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Like any linear programming model, this model will have four component: 1. The data 2. The decision 3. The constraints 4. The measure of performance

3.7

Algebraic Formulation

Let TV = Number of commercials for separate spots on television

M = Number of advertisements in magazines.

SS = Number of advertisements in Sunday supplements.

Maximize Exposure = 1,300TV + 600M + 500SS

subject to

Ad Spending: 300TV + 150M + 100SS ≤ 4,000 ($thousand)

Planning Cost: 90TV + 30M + 30SS ≤ 1,000 ($thousand)

Number of TV Spots: TV ≤ 5

and

TV ≥ 0, M ≥ 0, SS ≥ 0. (Variables nonnegative)

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3.8

Spreadsheet Formulation

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B C D E F G H

TV Spots Magazine Ads SS Ads

Exposures per Ad 1,300 600 500

(thousands)

Budget Budget

Cost per Ad ($thousands) Spent Available

Ad Budget 300 150 100 4,000 <= 4,000

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

Total Exposures

TV Spots Magazine Ads SS Ads (thousands)

Number of Ads 0 20 10 17,000

<=

Max TV Spots 5

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Let’s put: 1. The data -> data cells 2. The decision -> Changing cells 3. The constraints-> Output cells 4. The measure of performance => Objective cell

3.2. Resource-Allocation Problems

Resource –allocation problems are linear programming problems involving the allocation of resources to activities. The identifying feature for any such problem is that each functional constraints in the programming model is a resource constraints, which has the form:

Amount of resource used <= Amount of resource available

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Characteristics of Resource-Allocation Problems

- Every problem of this type has a resource constraint for each resource. The amounts of resources used depend on the level of activities. -To deal with a resource allocation problem, the manager (or management science) needs to gather three kind of data: 1. Amount available of each resource 2. The amount of resource used per unit of activity must be estimated 3. The contribution per unit of each activity to the overall measure of

performance

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Da41g201 3.11

Resource-Allocation problem Example 2

• 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?

Case study: The TBA Airlines Problem

3.12

Data for the TBA Airlines Problem

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Analysis of case study

This is a Resource – Allocation problem. The activities under consideration are:

• Activity 1: Purchase small airplane • Activity 2: Purchase large airplane The decision to be made are the level of these activities, that is: • Number of small airplane to purchase • Number of large airplane to purchase These is a single resource constraints: Investment capital spent < = $250 million Another side constraints: Number of small airplane to purchase < = 5

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3.14

Integer Programming Formulation

Let S = Number of small airplanes to purchase

L = Number of large airplanes to purchase

Maximize Profit = 7S + 22L ($millions)

subject to

Capital Available: 25S + 75L ≤ 250 ($millions)

Max Small Planes: S ≤ 5

and

S ≥ 0, L ≥ 0

S, L are integers.

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3.15

Spreadsheet Model

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3.16

Case study. 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.

• The company currently has $25 million available now for investment, another $20 million in one year, another $20 million in two years, and another $15 million in three years

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

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Resource-Allocation problem Example 3

3.17

Financial Data for the Projects

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This is a resource –allocation problem. The activities under consideration are:

• Activity 1: Invest in the construction of an office building • Activity 2: Invest in the construction of a hotel • Activity 3: Invest in the construction of a shopping center The resource constraints for 4 investment point are cumulated as

below: • Amount of resource 1 available = 25$ million • Amount of resource 2 available = $(25 +20) million = $45 million • Amount of resource 3 available = $(45+20) million = $65 million • Amount of resource 4 available = $(65+15) million = $80 million

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3.19


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 + 50SC

subject 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)

and

OB ≥ 0, H ≥ 0, SC ≥ 0.

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3.21

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.

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3.22

Template for Resource-Allocation Problems

Activities

Unit Profit profit per unit of activity

Resources 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)

<=

Constr

ain

ts

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3.3. Cost-Benefit –Trade –Off Problems

Cost-Benefit-Trade-Off problems are linear programming problems where the mix of levels of various activities is chosen to achieve minimum acceptable levels for various benefits at a minimum cost. The identifying feature is that each functional constraints is Benefit Constraints, which has the form:

Three kinds of data are needed:

• The minimum acceptable level for each benefit

• For each benefit, the contribution of each activity to that benefit (per unit of activity)

• The cost per unit of each activity

Level of achieved >= Minimum acceptable level for one of the benefits

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3.24

Case study: Union Airways Problem (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?

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Cost- Benefit – Trade –Off Example 1

3.25

Schedule Data

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This problem is a pure cost-benefit-trade-off problem. To formulate the problem, we need to identify the activities and benefits involved. Activities correspond to shift • The level of each activity is the number of agents assigned to that shift • A unit of each activity is one agent assigned to that shift Benefit correspond to time periods • For each time period, the benefit provided by the activities is the

service that agent provide customers during that period. • The level of a benefit is measured by the number of agents on duty

during that time period.

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3.27


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

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

subject to

Total agents 6AM–8AM: S1 ≥ 48

Total agents 8AM–10AM: S1 + S2 ≥ 79

Total agents 10AM–12PM: S1 + S2 ≥ 65

Total agents 12PM–2PM: S1 + S2 + S3 ≥ 87

Total agents 2PM–4PM: S2 + S3 ≥ 64



Total agents 8PM–10PM: S4 ≥ 43

Total agents 10PM–12AM: S4 + S5 ≥ 52

Total agents 12AM–6AM: S5 ≥ 15

and

Si ≥ 0 (for i = 1 to 5) and Number of working shift =integer

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3.28


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B C D E F G H I J

6am-2pm 8am-4pm Noon-8pm 4pm-midnight 10pm-6am

Shift Shift Shift Shift Shift

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

Total Minimum

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

6am-8am 1 0 0 0 0 48 >= 48

8am-10am 1 1 0 0 0 79 >= 79

10am- 12pm 1 1 0 0 0 79 >= 65

12pm-2pm 1 1 1 0 0 118 >= 87

2pm-4pm 0 1 1 0 0 70 >= 64

4pm-6pm 0 0 1 1 0 82 >= 73

6pm-8pm 0 0 1 1 0 82 >= 82

8pm-10pm 0 0 0 1 0 43 >= 43

10pm-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-6am

Shift Shift Shift Shift Shift Total Cost

Number Working 48 31 39 43 15 $30,610

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3.29

Summary of Formulation Procedure for Cost-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.

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3.30

Template for Cost-Benefit Trade off Problems

Activities

Unit Cost cost per unit of activity

Benefit 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)

>=

Constr

ain

ts

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3.31

Types of Functional Constraints

Type Form* Typical Interpretation Main Usage

Resource constraint LHS ≤ RHS

For some resource,

Amount used ≤

Amount available

Resource-allocation problems

and mixed problems

Benefit constraint LHS ≥ RHS

For some benefit,

Level achieved ≥

Minimum Acceptable

Cost-benefit-trade-off

problems and mixed problems

Fixed-requirement constraint LHS = RHS

For some quantity,

Amount provided =

Required amount

Transportation problems and

mixed problems

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

RHS = Right-hand side (a constant).

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3.4. Mixed-Problems

3.32

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.

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Mixed-Problems Example 1

Benefit and Fixed-Requirement Data

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3.34


Let TV = Number of commercials for separate spots on television

M = Number of advertisements in magazines.

SS = Number of advertisements in Sunday supplements.

Maximize Exposure = 1,300TV + 600M + 500SS

subject 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)

and

TV ≥ 0, M ≥ 0, SS ≥ 0.

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3.35


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B C D E F G H

TV Spots Magazine Ads SS Ads

Exposures per Ad 1,300 600 500

(thousands)

Cost per Ad ($thousands) Budget Spent Budget Available

Ad Budget 300 150 100 3,775 <= 4,000

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

Number Reached per Ad (mil lions) Total Reached Minimum Acceptable

Young 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 Amount

Coupon 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

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Summary of the Formulation Procedure for Mixed Linear Programming Problems

1. Identify the activities for the problem. The decision to be made are the level of these activities. 2. Identify an appropriate overall measure of performance for solutions of the problem 3. For each activity, estimate the contribution per unit of the activity to this overall measure of

performance. 4. Identify any resources that must be allocated to the activities. For each one, identify the

amount available and then the amount used per unit of each activity 5. Identify any benefits to be obtained from activities. For each one, identify the minimum

acceptable level and then the benefit contribution per unit of each activity. 6. Identify any fixed requirements. For each fixed requirement, identify the required amount and

then the contribution toward this required amount per unit of each activity. 7. Enter the data gathered in step 3-6 into data cells in a spreadsheet. 8. Designate changing cells for displaying the decision on activity levels 9. Use output cells to specify the constraints on resources, benefits, and fixed requirements. 10. Designate and objective cell for displaying the overall measure of performance

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3.37

Template for Mixed Problems

Activities

Unit Profit or Cost profit/cost per unit of activity

Resources 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

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SUMPRODUCT(profit/cost per unit, changing cells)

<=

>=

=

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3.5. Transportation Problems

• One of the most common applications of programming involves optimizing a shipping plan for transporting goods. This type of linear programming problem is called a transportation problem.

• This kind of application normally needs two kind of functional constraints. One kind that the amount of the product produced at each plant must equal the total amount shipped to customers. The other kind that the total amount received from the plants by each customer must equal the amount of ordered. There are fixed-requirement constraints, which make the problem a fixed-requirements problem. However, there are also variation of this problem where resource constraints or benefit constraints are needed.

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3.39

Case study: 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?

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Transportation Problem Example 1

3.40

Some Data

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3.41

The Distribution Network

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3.42


Let Sij = Number of lathes to ship from factory i to customer j (i = F1, F2; j

= C1, C2, C3).

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

+ $800SF2-C1 + $900SF2-C2 + $700SF2-C3

subject to

Factory 1: SF1-C1 + SF1-C2 + SF1-C3 = 12

Factory 2: SF2-C1 + SF2-C2 + SF2-C3 = 15

Customer 1: SF1-C1 + SF2-C1 = 10



and

Sij ≥ 0 (i = F1, F2; j = C1, C2, C3).

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3.43


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B C D E F G H

Shipping Cost

(per Lathe) Customer 1 Customer 2 Customer 3

Factory 1 $700 $900 $800

Factory 2 $800 $900 $700

Total

Shipped

Units Shipped Customer 1 Customer 2 Customer 3 Out Output

Factory 1 10 2 0 12 = 12

Factory 2 0 6 9 15 = 15

Total To Customer 10 8 9

= = = Total Cost

Order Size 10 8 9 $20,500

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3.6. Assignment Problems

Assignment Problem is the kind of problem making assignment. Frequently, these are assignments of people to jobs. Other application might instead involve assigning machines, vehicles, or plants to tasks

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3.45

Case study: Sellmore Company 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?

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Assignment Problem Example 1

3.46

Data for the Sellmore Problem

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3.47

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.

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3.48


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Required Time Word Hourly

(Hours) Processing Graphics Packets Registrations Wage

Ann 35 41 27 40 $14

Assignee Ian 47 45 32 51 $12

Joan 39 56 36 43 $13

Sean 32 51 25 46 $15

Word

Cost Processing Graphics Packets Registrations

Ann $490 $574 $378 $560

Assignee Ian $564 $540 $384 $612

Joan $507 $728 $468 $559

Sean $480 $765 $375 $690

Word Total

Assignment Processing Graphics Packets Registrations Assignments Supply

Ann 0 0 1 0 1 = 1

Assignee Ian 0 1 0 0 1 = 1

Joan 0 0 0 1 1 = 1

Sean 1 0 0 0 1 = 1

Total Assigned 1 1 1 1

= = = = Total Cost

Demand 1 1 1 1 $1,957

Task

Task

Task

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THANKS FOR YOUR ATTENTION!

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Linear Programming: Formulation and...

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