5-1 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Integer Programming Chapter 5

5-2 Chapter Topics Integer Programming (IP) Models Computer Solution of Integer Programming Problems With Excel 0-1 Integer Programming Modeling Examples

5-3 Integer Programming Models Types of Models Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 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.

5-4 A Total Integer Model (1 of 3)

5-5 A Total Integer Model (2 of 3) ■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:

5-6 A Total Integer Model (3 of 3) Integer Programming Model: Maximize Z = $100x 1 + $150x 2 subject to: $8,000x 1 + 4,000x 2  $40,000 15x x 2  200 ft 2 x 1, x 2  0 and integer x 1 = number of presses x 2 = number of lathes Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

5-7 ■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. Integer Programming Graphical Solution Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

5-8 Integer Programming Example Graphical Solution of Machine Shop Model Maximize Z = $100x 1 + $150x 2 subject to: 8,000x 1 + 4,000x 2  $40,000 15x x 2  200 ft 2 x 1, x 2  0 and integer Optimal Solution: Z = $1, x 1 = 2.22 presses x 2 = 5.55 lathes Figure 5.1 Feasible Solution Space with Integer Solution Points Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

5-9 Branch and Bound Method ■Traditional approach to solving integer programming problems.  Feasible solutions can be partitioned into smaller subsets  Smaller subsets evaluated until best solution is found.  Method is a tedious and complex mathematical process. ■Excel and QM for Windows used in this book. ■See book’s companion website – “Integer Programming: the Branch and Bound Method” for detailed description of this method. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

5-10 Computer Solution of IP Problems Total Integer Model with Excel (1 of 5) Integer Programming Model of Machine Shop: Maximize Z = $100x 1 + $150x 2 subject to: 8,000x 1 + 4,000x 2  $40,000 15x x 2  200 ft 2 x 1, x 2  0 and integer Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

5-11 Exhibit 5.8 Computer Solution of IP Problems Total Integer Model with Excel (2 of 5) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

5-12 Exhibit 5.10 Computer Solution of IP Problems Total Integer Model with Excel (3 of 5) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

5-13 Exhibit 5.9 Computer Solution of IP Problems Total Integer Model with Excel (4 of 5) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

5-14 Exhibit 5.11 Computer Solution of IP Problems Total Integer Model with Excel (5 of 5) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

5-15 A Integer Model (1 of 2)

5-16 Integer Programming Model: x 1 = construction of a swimming pool x 2 = construction of a tennis center x 3 = construction of an athletic field x 4 = construction of a gymnasium Maximize Z = 300x x x x 4 subject to: 35,000x ,000x ,000x ,000x 4  120,000 4x 1 + 2x 2 + 7x 3 + 3x 4  12 acres x 1 + x 2  1 facility x 1, x 2, x 3, x 4 = 0 or 1 A Integer Model (2 of 2) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

5-17 Recreational Facilities Example: Maximize Z = 300x x x x 4 subject to: $35,000x ,000x ,000x ,000x 4  $120,000 4x 1 + 2x 2 + 7x 3 + 3x 4  12 acres x 1 + x 2  1 facility x 1, x 2, x 3, x 4 = 0 or 1 Computer Solution of IP Problems 0 – 1 Model with Excel (1 of 5) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

5-18 Exhibit 5.2 Computer Solution of IP Problems 0 – 1 Model with Excel (2 of 5) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

5-19 Exhibit 5.3 Computer Solution of IP Problems 0 – 1 Model with Excel (3 of 5) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

5-20 Exhibit 5.4 Computer Solution of IP Problems 0 – 1 Model with Excel (4 of 5) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

5-21 Exhibit 5.5 Computer Solution of IP Problems 0 – 1 Model with Excel (5 of 5)

– 1 Integer Programming Modeling Examples Fixed Charge and Facility Example (1 of 4)

– 1 Integer Programming Modeling Examples Fixed Charge and Facility Example (1 of 4)

– 1 Integer Programming Modeling Examples Fixed Charge and Facility Example (1 of 4)

5-25 y i = 0 if farm i is not selected, and 1 if farm i is selected;i = 1,2,3,4,5,6 x ij = potatoes (1000 tons) shipped from farm I to plant j;j = A,B,C. Minimize Z =18x 1A + 15x 1B + 12x 1C + 13x 2A + 10x 2B + 17x 2C + 16x x 3B +18x 3C + 19x 4A + 15x 4b + 16x 4C + 17x 5A + 19x 5B +12x 5C + 14x 6A + 16x 6B + 12x 6C + 405y y y y y y 6 subject to: x 1A + x 1B + x 1C y 1 ≤ 0x 2A + x 2B + x 2C -10.5y 2 ≤ 0 x 3A + x 3B + x 3C y 3 ≤ 0x 4A + x 4b + x 4C - 9.3y 4 ≤ 0 x 5A + x 5B + x 5C y 5 ≤ 0x 6A + x 6B + X 6C - 9.6y 6 ≤ 0 x 1A + x 2A + x 3A + x 4A + x 5A + x 6A = 12 x 1B + x 2B + x 3B + x 4B + x 5B + x 6B = 10 x 1C + x 2C + x 3C + x 4C + x 5C + x 6C = 14 x ij ≥ 0 y i = 0 or 1 0 – 1 Integer Programming Modeling Examples Fixed Charge and Facility Example (2 of 4)

5-26 Exhibit – 1 Integer Programming Modeling Examples Fixed Charge and Facility Example (3 of 4)

5-27 Exhibit – 1 Integer Programming Modeling Examples Fixed Charge and Facility Example (4 of 4)

– 1 Integer Programming Modeling Examples Set Covering Example (1 of 4)

5-29 x i = city i, i = 1 to 12; x i = 0 if city is not selected as a hub and x i = 1 if it is. Minimize Z = x 1 + x 2 + x 3 + x 4 + x 5 + x 6 + x 7 + x 8 + x 9 + x 10 + x 11 + x 12 subject to:Atlanta:x 1 + x 3 + x 8  1 Boston:x 2 + x 10  1 Charlotte:x 1 + x 3 + x 11  1 Cincinnati:x 4 + x 5 + x 6 + x 8 + x 10  1 Detroit:x 4 + x 5 + x 6 + x 7 + x 10  1 Indianapolis: x 4 + x 5 + x 6 + x 7 + x 8 + x 12  1 Milwaukee:x 5 + x 6 + x 7  1 Nashville: x 1 + x 4 + x 6 + x 8 + x 12  1 New York:x 2 + x 9 + x 11  1 Pittsburgh:x 4 + x 5 + x 10 + x 11  1 Richmond: x 3 + x 9 + x 10 + x 11  1 St Louis: x 6 + x 8 + x 12  1 x ij = 0 or 1 0 – 1 Integer Programming Modeling Examples Set Covering Example (2 of 4)

5-30 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 0 – 1 Integer Programming Modeling Examples Set Covering Example (3 of 4) Exhibit 5.20

5-31 Exhibit – 1 Integer Programming Modeling Examples Set Covering Example (4 of 4)