Integer Programming Chapter Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 5-1 Chapter Topics Integer Programming (IP) Models Integer Programming Graphical Solution Computer Solution of Integer Programming Problems With Excel and QM for Windows 0-1 Integer Programming Modeling Examples Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 5-2 Integer Programming Models Types of Models Total Integer Model: All decision variables required to have integer solution values 0-1 Integer Model: All decision variables required to have integer values of zero or one Mixed Integer Model: Some of the decision variables (but not all) required to have integer values Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 5-3 A Total Integer Model (1 of 2) ■ Machine shop obtaining new presses and lathes ■ Marginal profitability: each press $100/day; each lathe $150/day ■ Resource constraints: $40,000 budget, 200 sq ft floor space ■ Machine purchase prices and space requirements: Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 5-4 A Total Integer Model (2 of 2) Integer Programming Model: Maximize Z = $100x1 + $150x2 subject to: $8,000x1 + 4,000x2 $40,000 15x1 + 30x2 200 ft2 x1, x2 and integer x1 = number of presses x2 = number of lathes Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 5-5 A - Integer Model (1 of 2) ■ Recreation facilities selection to maximize daily usage by residents ■ Resource constraints: $120,000 budget; 12 acres of land ■ Selection constraint: either swimming pool or tennis center (not both) Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 5-6 A - Integer Model (2 of 2) Integer Programming Model: Maximize Z = 300x1 + 90x2 + 400x3 + 150x4 subject to: $35,000x1 + 10,000x2 + 25,000x3 + 90,000x4 $120,000 4x1 + 2x2 + 7x3 + 3x4 12 acres x1 + x2 facility x1, x2, x3, x4 = or x1 = construction of a swimming pool x2 = construction of a tennis center x3 = Inc construction of an athletic field Copyright © 2010 Pearson Education, Publishing as Prentice Hall 5-7 A Mixed Integer Model (1 of 2) ■ $250,000 available for investments providing greatest return after one year ■ Data: Condominium cost $50,000/unit; $9,000 profit if sold after one year Land cost $12,000/ acre; $1,500 profit if sold after one year Municipal bond cost $8,000/bond; $1,000 profit if sold after one year Only condominiums, 15 acres of land, and 20 municipal bonds available Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 5-8 A Mixed Integer Model (2 of 2) Integer Programming Model: Maximize Z = $9,000x1 + 1,500x2 + 1,000x3 subject to: 50,000x1 + 12,000x2 + 8,000x3 $250,000 x1 condominiums x2 15 acres x3 20 bonds x2 x1, x3 and integer x1 = condominiums purchased x2 = acres of land purchased x3 =Inc bonds Copyright © 2010 Pearson Education, Publishing purchased as Prentice Hall 5-9 Integer Programming Graphical Solution ■ Rounding non-integer solution values up to the nearest integer value can result in an infeasible solution ■ A feasible solution is ensured by rounding down non-integer solution values but may result in a less than optimal (sub-optimal) solution Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 5-10 – Integer Programming Modeling Examples Capital Budgeting Example (2 of 4) x1 = selection of web site project x2 = selection of warehouse project x3 = selection clothing department project x4 = selection of computer department project x5 = selection of ATM project xi = if project “i” is selected, if project “i” is not selected Maximize Z = $120x1 + $85x2 + $105x3 + $140x4 + $70x5 subject to: 55x1 + 45x2 + 60x3 + 50x4 + 30x5 150 40x1 + 35x 25x3 +as 35x4 + 30x5 110 + Copyright © 2010 Pearson Education, Inc Publishing Prentice Hall 5-32 – Integer Programming Modeling Examples Capital Budgeting Example (3 of 4) Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Exhibit 5.16 5-33 – Integer Programming Modeling Examples Capital Budgeting Example (4 of 4) Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Exhibit 5.17 5-34 – Integer Programming Modeling Examples WhichCharge of six farms should be purchased that will (1 Fixed and Facility Example of meet 4) current production capacity at minimum total cost, including annual fixed costs and shipping costs? Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 5-35 – Integer Programming Modeling Examples y = if farm i is not selected, if farm i isExample selected; i = (2 Fixed Charge andand Facility 1,2,3,4,5,6 of 4) i xij = potatoes (1000 tons) shipped from farm I to plant j; j = A,B,C Minimize Z = 18x1A+ 15x1B+ 12x1C+ 13x2A+ 10x2B+ 17x2C+ 16x3+ 14x3B +18x3C+ 19x4A+ 15x4b+ 16x4C+ 17x5A+ 19x5B+12x5C+ 14x6A + 16x6B+ 12x6C+ 405y1+ 390y2+ 450y3+ 368y4+ 520y5+ 465y6 subject to: x1A + x1B + x1B - 11.2y1 ≤ 0 x3A + x3A + x3C - 12.8y3 ≤ x5A + x5B + x5B - 10.8y5 ≤ Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall x2A + x2B + x2C -10.5y2 ≤ x4A + x4b + x4C - 9.3y4 ≤ x6A + x6B + X6C - 9.6y6 ≤ 5-36 – Integer Programming Modeling Examples Fixed Charge and Facility Example (3 of 4) Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Exhibit 5.18 5-37 – Integer Programming Modeling Examples Fixed Charge and Facility Example (4 of 4) Exhibit 5.19 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 5-38 – Integer Programming Modeling Examples APSCovering wants to construct the minimum Set Example (1 ofset4)of new hubs in these twelve cities such that there is a hub within Cities Cities within 300 miles 300 miles of every city: Atlanta Atlanta, Charlotte, Nashville Boston Boston, New York Charlotte Atlanta, Charlotte, Richmond Cincinnati Cincinnati, Detroit, Indianapolis, Nashville, Pittsburgh Detroit Cincinnati, Detroit, Indianapolis, Milwaukee, Pittsburgh Indianapolis Cincinnati, Detroit, Indianapolis, Milwaukee, Nashville, St Louis Milwaukee Detroit, Indianapolis, Milwaukee Nashville Atlanta, Cincinnati, Indianapolis, Nashville, St Louis New York Boston, New York, Richmond 10 Pittsburgh Cincinnati, Detroit, Pittsburgh, Richmond 11 Richmond Charlotte, New York, Pittsburgh, Richmond 12 St Louis Indianapolis, Nashville, St Louis Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 5-39 – Integer Programming Modeling Examples x = city i, i = to 12; Example x = if city is not Set Covering (2 selected of 4) as a hub and x i i i = if it is Minimize Z = x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10 + x11 + x12 subject to: Atlanta: x1 + x3 + x8 Boston: x2 + x10 Charlotte: x1 + x3 + x11 Cincinnati: x4 + x5 + x6 + x8 + x10 Detroit: x4 + x5 + x6 + x7 + x10 Indianapolis: x4 + x5 + x6 + x7 + x8 + x12 Milwaukee: x5 + x6 + x7 Nashville: x1 + x4 + x6+ x8 + x12 New York: x2 + x9+ x11 + x10 + x11 Copyright © 2010Pittsburgh: Pearson Education, Inc.xPublishing + x5as Prentice Hall 5-40 – Integer Programming Modeling Examples Set Covering Example (3 of 4) Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Exhibit 5-41 – Integer Programming Modeling Examples Set Covering Example (4 of 4) Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Exhibit 5-42 Total Integer Programming Modeling Example Problem Statement (1 of 3) ■ Textbook company developing two new regions ■ Planning to transfer some of its 10 salespeople into new regions ■ Average annual expenses for sales person: ▪ Region - $10,000/salesperson ▪ Region - $7,500/salesperson ■ Total annual expense budget is $72,000 ■ Sales generated each year: ▪ Region - $85,000/salesperson ▪ Region - $60,000/salesperson ■ How many salespeople should be transferred into each region in order to maximize increased sales? Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 5-43 Total Integer Programming Modeling Example Model Formulation (2 of 3) Step 1: Formulate the Integer Programming Model Maximize Z = $85,000x1 + 60,000x2 subject to: x1 + x2 10 salespeople $10,000x1 + 7,000x2 $72,000 expense budget x1, x2 or integer Step 2: Solve the Model using QM for Windows Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 5-44 Total Integer Programming Modeling Example Solution with QM for Windows (3 of 3) Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 5-45 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 5-46 ... Programming Models Types of Models Total Integer Model: All decision variables required to have integer solution values 0-1 Integer Model: All decision variables required to have integer values of zero.. .Chapter Topics Integer Programming (IP) Models Integer Programming Graphical Solution Computer... of the decision variables (but not all) required to have integer values Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 5-3 A Total Integer Model (1 of 2) ■ Machine shop obtaining