McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.1 Integer Programming.

© The McGraw-Hill Companies, Inc., 20039.1McGraw-Hill/Irwin

Integer Programming


Integer Programming

• When are “non-integer” solutions okay?– Solution is naturally divisible

• e.g., $, pounds, hours

– Solution represents a rate

• e.g., units per week

– Solution only for planning purposes

• When is rounding okay?– When numbers are large

• e.g., rounding 114.286 to 114 is probably okay.

• When is rounding not okay?– When numbers are small

• e.g., rounding 2.6 to 2 or 3 may be a problem.

– Binary variables

• yes-or-no decisions


The Challenges of Rounding

• Rounded Solution may not be feasible.

• Rounded solution may not be close to optimal.

• There can be many rounded solutions.

– Example: Consider a problem with 30 variables that are non-integer in the LP-solution. How many possible rounded solutions are there?

1 2 3 4 5

1

2

3

4

5

x1

x2


How Integer Programs are Solved

1 2 3 4 5

1

2

3

4

5

x1

x2


Applications of Binary Variables

• Making “yes-or-no” type decisions– Build a factory?

– Manufacture a product?

– Do a project?

– Assign a person to a task?

• Set-covering problems– Make a set of assignments that “cover” a set of requirements.

• Fixed costs– If a product is produced, must incur a fixed setup cost.

– If a warehouse is operated, must incur a fixed cost.


Example #1 (Capital Budgeting)

• Norwood Development is considering the potential of four different development projects.

• Each project would be completed in at most three years.

• The required cash outflow for each project is given in the table below, along with the net present value of each project to Norwood, and the cash that is available each year.

Cash Outflow Required ($million)

CashAvailable($million)Project 1 Project 2 Project 3 Project 4

Year 1 9 7 6 11 28

Year 2 6 4 3 0 13

Year 3 6 0 4 0 10

NPV 30 16 22 14

Question: Which projects should be undertaken?


Algebraic Formulation

Let yi = 1 if project i is undertaken; 0 otherwise (i = 1, 2, 3, 4).

Maximize NPV = 30y1 + 16y2 + 22y3 + 14y4

subject to

Year 1: 9y1 + 7y2 + 6y3 + 11y4 ≤ 28 ($million)

Year 2 (cumulative): 15y1 + 11y2 + 9y3 + 11y4 ≤ 41 ($million)

Year 3 (cumulative): 21y1 + 11y2 + 13y3 + 11y4 ≤ 51 ($million)

and

yi are binary (i = 1, 2, 3, 4).


Spreadsheet Solution

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A B C D E F G H I

Norwood Development Capital Budgeting

Project 1 Project 2 Project 3 Project 4NPV ($million) 30 16 22 14

Cumulative CumulativeOutflow Available

Year 1 9 7 6 11 22 <= 28Year 2 15 11 9 11 35 <= 41Year 3 21 11 13 11 45 <= 51

Total NPVProject 1 Project 2 Project 3 Project 4 ($million)

Undertake? 1 1 1 0 68

Cumulative Outflow Required ($million)


Additional Considerations(Logic and Dependency Constraints)

• At least one of projects 1, 2, or 3

• Project 2 can’t be done unless project 3 is done

• Either project 3 or project 4, but not both

• No more than two projects total

Question: What constraints would need to be added for each of these additional considerations?


Example #2 (Set Covering Problem)

• The Washington State legislature is trying to decide on locations at which to base search-and-rescue teams.

• The teams are expensive, so they would like as few as possible.

• Response time is critical, so they would like every county to either have a team located in that county or in an adjacent county.

Question: Where should search-and-rescue teams be located?


The Counties of Washington State

1

2

3

4

7

6

9

10

11

12

8

5

13

14

15

16

17

18

19

20

2122

2325

24

26 27 28

29 30

31 32

33

3435

36

37

1. Clallum2. J efferson3. Grays Harbor4. Pacific5. Wahkiakum6. Kitsap7. Mason8. Thurston9. Whatcom10. Skagit11. Snohomish12. King13. Pierce14. Lewis15. Cowlitz16. Clark17. Skamania18. Okanogan

19. Chelan20. Douglas21. Kittitas22. Grant23. Yakima24. Klickitat25. Benton26. Ferry27. Stevens28. Pend Oreille29. Lincoln30. Spokane31. Adams32. Whitman33. Franklin34. Walla Walla35. Columbia36. Garfield37. Asotin

Counties



Let yi = 1 if a team is located in county i; 0 otherwise (i = 1, 2, … , 37).

Minimize Number of Teams = y1 + y2 + … + y37

subject to

County 1 covered: y1 + y2 ≥ 1

County 2 covered: y1 + y2 + y3 + y6 + y7 ≥ 1

County 3 covered: y2 + y3 + y4 + y7 + y8 + y14 ≥ 1

:

County 37 covered: y32 + y36 + y37 ≥ 1

and

yi are binary (i = 1, 2, … , 37).



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A B C D E F G H I J K L M N

Search & Rescue Location

# Teams # TeamsCounty Team? Nearby County Team? Nearby

1 Clallam 0 1 >= 1 19 Chelan 0 2 >= 12 Jefferson 1 1 >= 1 20 Douglas 0 1 >= 13 Grays Harbor 0 2 >= 1 21 Kittitas 1 1 >= 14 Pacific 0 1 >= 1 22 Grant 0 1 >= 15 Wahkiakum 0 1 >= 1 23 Yakima 0 3 >= 16 Kitsap 0 1 >= 1 24 Klickitat 0 1 >= 17 Mason 0 1 >= 1 25 Benton 0 1 >= 18 Thurston 0 1 >= 1 26 Ferry 0 1 >= 19 Whatcom 0 1 >= 1 27 Stevens 1 1 >= 110 Skagit 1 1 >= 1 28 Pend Oreille 0 1 >= 111 Snohomish 0 1 >= 1 29 Lincoln 0 1 >= 112 King 0 1 >= 1 30 Spokane 0 1 >= 113 Pierce 0 2 >= 1 31 Adams 0 1 >= 114 Lewis 1 2 >= 1 32 Whitman 0 2 >= 115 Cowlitz 0 2 >= 1 33 Franklin 1 1 >= 116 Clark 0 1 >= 1 34 Walla Walla 0 1 >= 117 Skamania 1 2 >= 1 35 Columbia 0 1 >= 118 Okanogan 0 1 >= 1 36 Garfield 1 1 >= 1

37 Asotin 0 1 >= 1Total Teams: 8


Example #3 (Fixed Costs)

• Woodridge Pewter Company is a manufacturer of three pewter products: platters, bowls, and pitchers.

• The manufacture of each product requires Woodridge to have the appropriate machinery and molds available. The machinery and molds for each product can be rented at the following rates: for the platters, $400/week; for the bowls, $250/week; for the pitcher, $300/week.

• Each product requires the amounts of labor and pewter given in the table below. The sales price and variable cost are also given in the table.

LaborHours

Pewter(pounds)

SalesPrice

VariableCost

Platter 3 5 $100 $60

Bowl 1 4 85 50

Pitcher 4 3 75 40

Available 130 240

Question: Which products should be produced, and in what quantity?



Let x1 = Number of platters produced,x2 = Number of bowls produced,x3 = Number of pitchers produced,yi = 1 if lease machine and mold for product i; 0 otherwise (i = 1, 2, 3).

Maximize Profit = ($100–$60)x1 + ($85–$50)x2 + ($75–$40)x3 – $400y1 – $250y2 – $300y3

subject toLabor: 3x1 + x2 + 4x3 ≤ 130 hoursPewter: 5x1 + 4x2 + 3x3 ≤ 240 poundsAllow production only if machines and molds are purchased:

x1 ≤ 99y1

x2 ≤ 99y2

x3 ≤ 99y3

andxi ≥ 0, and yi are binary (i = 1, 2, 3).



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A B C D E F G H

Woodridge Pewter Company

Platters Bowls PitchersSales Price $100 $85 $75

Variable Cost $60 $50 $40Fixed Cost $400 $250 $300

Constraint Usage (per unit produced) Total AvailableLabor (hrs.) 3 1 4 60 <= 130

Pewter (lbs.) 5 4 3 240 <= 240

Lease Equipment? 0 1 0Revenue $5,100

Production Quantity 0 60 0 Variable Cost $3,000<= <= <= Fixed Cost $250

Produce only if Lease 0 99 0 Profit $1,850


Capital Budgeting with Contingency Constraints(Yes-or-No Decisions)

• A company is planning their capital budget over the next several years.

• There are 10 potential projects they are considering pursuing.

• They have calculated the expected net present value of each project, along with the cash outflow that would be required over the next five years.

• Also, suppose there are the following contingency constraints:– at least one of project 1, 2 or 3 must be done,

– project 4 and project 5 cannot both be done,

– project 7 can only be done if project 6 is done.

Question: Which projects should they pursue?


Data for Capital Budgeting Problem

Cash Outflow Required ($million)

CashAvailable($million)

Project

1 2 3 4 5 6 7 8 9 10

Year 1 1 4 0 4 4 3 2 8 2 6 25

Year 2 2 2 2 2 2 4 2 3 3 6 25

Year 3 3 2 5 2 4 2 3 4 8 2 25

Year 4 4 4 5 4 5 3 1 2 1 1 25

Year 5 1 1 0 6 5 5 5 1 1 2 25

NPV 20 25 22 30 42 25 18 35 28 33 ($million)



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A B C D E F G H I J K L M N O

Capital Budgeting with Contingency Constraints

Project Project Project Project Project Project Project Project Project Project1 2 3 4 5 6 7 8 9 10

NPV ($million) 20 25 22 30 42 25 18 35 28 33Cumulative Cumulative

Cumulative Cash Outflow Required ($million) Total Outflow AvailableYear 1 1 4 0 4 4 3 2 8 2 6 22 <= 25Year 2 3 6 2 6 6 6 4 11 5 12 44 <= 50Year 3 6 8 7 8 10 8 7 15 13 14 73 <= 75Year 4 10 12 12 12 15 11 8 17 14 15 97 <= 100Year 5 11 13 12 18 20 16 13 18 15 17 117 <= 125

Project Project Project Project Project Project Project Project Project Project Total NPV1 2 3 4 5 6 7 8 9 10 ($million)

Undertake? 1 1 1 0 1 1 1 0 1 1 213

Contingency ConstraintsProject 1,2,3 3 >= 1Project 4,5 1 <= 1Project 7 1 <= 1 Project 6


Electrical Generator Startup Planning (Fixed Costs)

• An electrical utility company owns five generators.

• To generate electricity, a generator must be started up, and associated with this is a fixed startup cost.

• All of the generators are shut off at the end of each day.

Generator

A B C D E

Fixed Startup Cost $2,450 $1,600 $1,000 $1,250 $2,200

Variable Cost (per MW) $3 $4 $6 $5 $4

Capacity (MW) 2,000 2,800 4,300 2,100 2,000

Question: Which generators should be started up to meet the total capacity needed for the day (6000 MW)?



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A B C D E F G H I J

Electrical Utility Generator Startup Planning

Generator A Generator B Generator C Generator D Generator EFixed Startup Cost $2,450 $1,600 $1,000 $1,250 $2,200Cost per Megawatt $3 $4 $6 $5 $4Max Capacity (MW) 2,000 2,800 4,300 2,100 2,000

Startup? 1 1 0 1 0Total MW MW Needed

MW Generated 2,100 3,000 0 900 0 6000 >= 6,000<= <= <= <= <=

Capacity 2,000 2,800 0 2,100 0

Fixed Cost $5,300Variable Cost $22,800

Total Cost $28,100


Quality Furniture (Either-Or Constraints)

• Reconsider the Quality Furniture Problem:– 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.

• Now suppose that they would not produce any fewer than 200 units of either product (i.e., either produce 0 or at least 200).

Question: What product mix will maximize their total profit?



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A B C D E F G

Quality Furniture (with either-or constraints)

Benches TablesProfit $8.00 $18.00

Min Production (if any) 200 200

Resources ResourcesUsed Available

Labor 3 6 1600 <= 1,600Wood 12 38 6400 <= 9,000

Produce? 1 0

Min Production 200 0<= <= Total Profit

Production Quantities 533.33 0 $4,266.67<= <=

Max Production 533 0Max Possible 533 237

Use of Resources


Meeting a Subset of Constraints

• Consider a linear programming model with the following constraints, and suppose that meeting 3 out of 4 of these is good enough

– 12x1 + 24x2 + 18x3 ≥ 2,400

– 15x1 + 32x2 + 12x3 ≥ 1,800

– 20x1 + 15x2 + 20x3 ≤ 2,000

– 18x1 + 21x2 + 15x3 ≤ 1,600


Meeting a Subset of Constraints

Let yi = 1 if constraint i is enforced; 0 otherwise.

Constraints:

y1 + y2 + y3 + y4 ≥ 3

12x1 + 24x2 + 18x3 ≥ 2,400y1

15x1 + 32x2 + 12x3 ≥ 1,800y2

20x1 + 15x2 + 20x3 ≤ 2,000 + M (1 – y3)

18x1 + 21x2 + 15x3 ≤ 1,600 + M (1 – y4)

where M is a large number.


Facility Location

• Consider a company that operates 5 plants and 3 warehouses that serve customers in 4 different regions.

• To lower costs, they are considering streamlining by closing one or more plants and warehouses.

• Associated with each plant are fixed costs, shipping costs, and production costs. Each plant has a limited capacity.

• Associated with each warehouse are fixed costs and shipping costs. Each warehouse has a limited capacity.

Questions:Which plants should they keep open?

Which warehouses should they keep open?

How should they divide production among the open plants?

How much should be shipped from each plant to each warehouse, and from each warehouse to each customer?


Data for Facility Location Problem

FixedCost

(per month)

(Shipping + Production) Cost(per unit)

Capacity(units per

month)WH #1 WH #2 WH #3

Plant 1 $42,000 $650 $750 $850 400

Plant 2 50,000 500 350 550 300

Plant 3 45,000 450 450 350 300

Plant 4 50,000 400 500 600 350

Plant 5 47,000 550 450 350 375

Fixed Cost(per month)

Shipping Cost (per unit)

Capacity(per month)Cust. 1 Cust. 2 Cust. 3 Cust. 4

WH #1 $45,000 $25 $65 $70 $35 600

WH #2 25,000 50 25 40 60 400

WH #3 65,000 60 20 40 45 900

Demand: 250 225 200 275



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A B C D E F G H I J K L M

Plant to WarehouseShipping + Production FixedCost Warehouse 1 Warehouse 2 Warehouse 3 Cost Capacity

Plant 1 $650 $750 $850 $42,000 400Plant 2 $500 $350 $550 $50,000 300Plant 3 $450 $450 $350 $45,000 300Plant 4 $400 $500 $600 $50,000 350Plant 5 $550 $450 $350 $47,000 375

Shipment Total ActualQuantities Warehouse 1 Warehouse 2 Warehouse 3 Shipped Capacity Open? Total Costs

Plant 1 0 0 0 0 <= 0 0 Shipping Cost (P-->W) $332,500Plant 2 0 300 0 300 <= 300 1 Shipping Cost (W-->C) $37,375Plant 3 0 0 275 275 <= 300 1 Fixed Cost (P) $142,000Plant 4 0 0 0 0 <= 0 0 Fixed Cost (W) $90,000Plant 5 0 0 375 375 <= 375 1 Total Cost $601,875

Total Shipped 0 300 650

Warehouse to CustomerShipping FixedCost Customer 1 Customer 2 Customer 3 Customer 4 Cost Capacity

Warehouse 1 $25 $65 $70 $35 $45,000 600Warehouse 2 $50 $25 $40 $60 $25,000 400Warehouse 3 $60 $20 $40 $45 $65,000 900

Shipment Shipped Shipped ActualQuantities Customer 1 Customer 2 Customer 3 Customer 4 Out In Capacity Open?

Warehouse 1 0 0 0 0 0 <= 0 <= 0 0Warehouse 2 250 0 50 0 300 <= 300 <= 400 1Warehouse 3 0 225 150 275 650 <= 650 <= 900 1Total Shipped 250 225 200 275

>= >= >= >=Needed 250 225 200 275

McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.1 Integer Programming.

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Transcript of McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.1 Integer Programming.