mcgraw-hill/irwin © the mcgraw-hill companies, inc., 2003 9.1 integer programming
TRANSCRIPT
© The McGraw-Hill Companies, Inc., 20039.2McGraw-Hill/Irwin
Integer Programming
• When are “non-integer” solutions okay?– Solution is naturally divisible
• e.g., $, pounds, hours
– Solution represents a rate
• e.g., units per week
– Solution only for planning purposes
• When is rounding okay?– When numbers are large
• e.g., rounding 114.286 to 114 is probably okay.
• When is rounding not okay?– When numbers are small
• e.g., rounding 2.6 to 2 or 3 may be a problem.
– Binary variables
• yes-or-no decisions
© The McGraw-Hill Companies, Inc., 20039.3McGraw-Hill/Irwin
The Challenges of Rounding
• Rounded Solution may not be feasible.
• Rounded solution may not be close to optimal.
• There can be many rounded solutions.
– Example: Consider a problem with 30 variables that are non-integer in the LP-solution. How many possible rounded solutions are there?
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x1
x2
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How Integer Programs are Solved
1 2 3 4 5
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x1
x2
© The McGraw-Hill Companies, Inc., 20039.5McGraw-Hill/Irwin
How Integer Programs are Solved
1 2 3 4 5
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x1
x2
© The McGraw-Hill Companies, Inc., 20039.6McGraw-Hill/Irwin
Applications of Binary Variables
• Making “yes-or-no” type decisions– Build a factory?
– Manufacture a product?
– Do a project?
– Assign a person to a task?
• Set-covering problems– Make a set of assignments that “cover” a set of requirements.
• Fixed costs– If a product is produced, must incur a fixed setup cost.
– If a warehouse is operated, must incur a fixed cost.
© The McGraw-Hill Companies, Inc., 20039.7McGraw-Hill/Irwin
Example #1 (Capital Budgeting)
• Norwood Development is considering the potential of four different development projects.
• Each project would be completed in at most three years.
• The required cash outflow for each project is given in the table below, along with the net present value of each project to Norwood, and the cash that is available each year.
Cash Outflow Required ($million)
CashAvailable($million)Project 1 Project 2 Project 3 Project 4
Year 1 9 7 6 11 28
Year 2 6 4 3 0 13
Year 3 6 0 4 0 10
NPV 30 16 22 14
Question: Which projects should be undertaken?
© The McGraw-Hill Companies, Inc., 20039.8McGraw-Hill/Irwin
Algebraic Formulation
Let yi = 1 if project i is undertaken; 0 otherwise (i = 1, 2, 3, 4).
Maximize NPV = 30y1 + 16y2 + 22y3 + 14y4
subject to
Year 1: 9y1 + 7y2 + 6y3 + 11y4 ≤ 28 ($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).
© The McGraw-Hill Companies, Inc., 20039.9McGraw-Hill/Irwin
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)
© The McGraw-Hill Companies, Inc., 20039.10McGraw-Hill/Irwin
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?
© The McGraw-Hill Companies, Inc., 20039.11McGraw-Hill/Irwin
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 McGraw-Hill Companies, Inc., 20039.12McGraw-Hill/Irwin
The Counties of Washington State
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12
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2122
2325
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26 27 28
29 30
31 32
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3435
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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
© The McGraw-Hill Companies, Inc., 20039.13McGraw-Hill/Irwin
Algebraic Formulation
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).
© The McGraw-Hill Companies, Inc., 20039.14McGraw-Hill/Irwin
Spreadsheet Solution
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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
© The McGraw-Hill Companies, Inc., 20039.15McGraw-Hill/Irwin
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?
© The McGraw-Hill Companies, Inc., 20039.16McGraw-Hill/Irwin
Algebraic Formulation
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).
© The McGraw-Hill Companies, Inc., 20039.17McGraw-Hill/Irwin
Spreadsheet Solution
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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 $1,850
© The McGraw-Hill Companies, Inc., 20039.18McGraw-Hill/Irwin
Capital Budgeting with Contingency Constraints(Yes-or-No Decisions)
• A company is planning their capital budget over the next several years.
• There are 10 potential projects they are considering pursuing.
• They have calculated the expected net present value of each project, along with the cash outflow that would be required over the next five years.
• Also, suppose there are the following contingency constraints:– at least one of project 1, 2 or 3 must be done,
– project 4 and project 5 cannot both be done,
– project 7 can only be done if project 6 is done.
Question: Which projects should they pursue?
© The McGraw-Hill Companies, Inc., 20039.19McGraw-Hill/Irwin
Data for Capital Budgeting Problem
Cash Outflow Required ($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)
© The McGraw-Hill Companies, Inc., 20039.20McGraw-Hill/Irwin
Spreadsheet Solution
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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
© The McGraw-Hill Companies, Inc., 20039.21McGraw-Hill/Irwin
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)?
© The McGraw-Hill Companies, Inc., 20039.22McGraw-Hill/Irwin
Spreadsheet Solution
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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
© The McGraw-Hill Companies, Inc., 20039.23McGraw-Hill/Irwin
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?
© The McGraw-Hill Companies, Inc., 20039.24McGraw-Hill/Irwin
Spreadsheet Solution
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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
© The McGraw-Hill Companies, Inc., 20039.25McGraw-Hill/Irwin
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
© The McGraw-Hill Companies, Inc., 20039.26McGraw-Hill/Irwin
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.
© The McGraw-Hill Companies, Inc., 20039.27McGraw-Hill/Irwin
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?
© The McGraw-Hill Companies, Inc., 20039.28McGraw-Hill/Irwin
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
© The McGraw-Hill Companies, Inc., 20039.29McGraw-Hill/Irwin
Spreadsheet Solution
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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