Chapter 6 Integer and Goal Programming Models Jason C. H. Chen, Ph.D. Professor of MIS School of Business Administration Gonzaga University Spokane, WA 99223 chen@jepson.gonzaga.edu

Variations of Basic Linear Programming Integer Programming Goal Programming Nonlinear Programming (skip)

Integer Programming (IP) Where some or all decision variables are required to be whole numbers. General Integer Variables (0,1,2,3,etc.) Values that count how many Binary Integer Variables (0 or 1) Usually represent a Yes/No decision

General Integer Example: Harrison Electric Co. Produce 2 products (lamps and ceiling fans) using 2 limited resources Decision: How many of each product to make? (must be integers) Objective: Maximize profit

Decision Variables L = number of lamps to make F = number of ceiling fans to make Lamps (per lamp) Fans (per fan) Hours Available Profit Contribution $600 $700 Wiring Hours 2 hrs 3 hrs 12 Assembly Hours 6 hrs 5 hr 30

LP Model Summary Max 600 L + 700 F ($ of profit) Subject to the constraints: 2L + 3F < 12 (wiring hours) 6L + 5F < 30 (assembly hours) L, F > 0

Graphical Solution

Properties of Integer Solutions Rounding off the LP solution might not yield the optimal IP solution The IP objective function value is usually worse than the LP value IP solutions are usually not at corner points

Using Solver for IP IP models are formulated in Excel in the same way as LP models The additional integer restriction is entered like an additional constraint int - Means general integer variables bin - Means binary variables Go to file 6-1.xls

Harrison Electric (General Integer) F   Lamps Fans Number of units 3.00 2.00 Profit $600 $700 $3,200.00 Constraints: Wiring hours 2 3 12.00 <= 12 Assembly hours 6 5 28.00 30 LHS Sign RHS Go to file 6-1.xls

Binary Integer Example: Portfolio Selection Choosing stocks to include in portfolio Decision: Which of 7 stocks to include? Objective: Maximize expected annual return (in $1000’s)

Stock Data

Decision Variables Use the first letter of each stock’s name Example for Trans-Texas Oil: T = 1 if Trans-Texas Oil is included T = 0 if not included

Restrictions Invest up to $3 million Include at least 2 Texas companies Include no more than 1 foreign company Include exactly 1 California company If British Petro is included, then Trans-Texas Oil must also be included

Objective Function (in $1000’s return) Max 50T + 80B + 90D + 120H + 110L + 40S + 75C Subject to the constraints: Invest up to $3 Million 480T + 540B + 680D + 1000H + 700L + 510S + 900C < 3000

Include At Least 2 Texas Companies T + H + L > 2 Include No More Than 1 Foreign Company B + D < 1 Include Exactly 1 California Company S + C = 1

If British Petro is included (B=1), then Trans-Texas Oil must also be included (T=1) T=0 T=1 B=0 ok B=1 not ok Combinations of B and T allows the 3 acceptable combinations and prevents the unacceptable one B < T

IP Model for Portfolio Selection Max $50T + $80B + $90D + $120H + $110L + $40S + $75C Subject to the constraints: 480T + 540B + 680D + 1000H + 700L + 510S + 900C < 3000 (investment limit) T + H + L > 2 (Texas companies) B + D < 1 (foreign companies) S + C = 1 (California companies) B < T (Trans-Texas and British petro) All variables = 0 or 1 Go to file 6-3.xls

Go to file 6-3.xls Simkin and Steinberg (Binary) T B D H L S C   Trans-Texas Oil British Petro Dutch Shell Houston Oil Lone Star Petro San Diego Oil Calif Petro Invest? (1 = Yes, 0 = No) 1 Exp annual return ('000) $50 $80 $90 $120 $110 $40 $75 $360 Constraints: Investment limit 480 540 680 1000 700 510 900 2890 <= 3000 Foreign companies British & Trans-Texas Texas companies 2 >= California companies = LHS Sign RHS Go to file 6-3.xls

Goal Programming Models Permit multiple objectives Try to “satisfy” goals rather than optimize Objective is to minimize underachievement of goals

Goal Programming Example: Wilson Doors Co. Makes 3 types of doors from 3 limited resources Decision: How many of each of 3 types of doors to make? Objective: Minimize total underachievement of goals

Data

LP Model Maximize $70E+ $110I + $110C St. 4E + 3I + 7 C < 9,000 (steel usage) 2E + 4I + 3C < 6,000 (forming time) 2E + 3I + 4C < 5,200 (assembly time) E, I, C > 0 Go to file 6-6.xls

E: 1400, I=800, and C=0 with a total sales of $186,000 LP Solution (File: 6-6.xls) Wilson Doors (LP)   E I C Exterior doors Interior doors Comm doors Number of units 1400.00 800.00 0.00 Revenue $70 $110 $186,000.00 Constraints: Steel usage 4 3 7 8000.00 <= 9000 Forming time 2 6000.00 6000 Assembly time 5200.00 5200 LHS Sign RHS E: 1400, I=800, and C=0 with a total sales of $186,000

Goals Total sales at least $180,000 Exterior door sales at least $70,000 Interior door sales at lest $60,000 Commercial door sales at least $35,000

Regular Decision Variables E = number of exterior doors made I = number of interior doors made C = number of commercial doors made Deviation Variables di+ = amount by which goal i is overachieved di- = amount by which goal i is underachieved

Goal Constraints Goal 1: Total sales at least $180,000 70E + 110I + 110C + dT- - dT+ = 180,000 Goal 2: Exterior door sales at least $70,000 70E + dE- - dE+ = 70,000 Note: Each highlighted deviation variable measures goal underachievement

Goal 3: Interior door sales at least $60,000 110 I + dI- - dI+ = 60,000 Goal 4: Commercial door sales at least $35,000 110C + dC- - dC+ = 35,000

Goals Total sales at least $180,000 Exterior door sales at least $70,000 Interior door sales at lest $60,000 Commercial door sales at least $35,000 Goal 1: 70E + 110I + 110C + dT- - dT+ = 180,000 Goal 2: 70E + dE- - dE+ = 70,000 Goal 3: 110 I + dI- - dI+ = 60,000 Goal 4: 110C + dC- - dC+ = 35,000

Objective Function Minimize total goal underachievement Min dT- + dE- + dI- + dC- Subject to the constraints: The 4 goal constraints The “regular” constraints (3 limited resources) nonnegativity

Objective Function Minimize dT- + dE- + dI- + dC- Subject to the constraints: 70E + 110I + 110C + dT- - dT+ = 180,000 (total sales goal) 70E + dE- - dE+ = 70,000 (exterior door sales goal) 110 I + dI- - dI+ = 60,000 (interior door sales goal) 110C + dC- - dC+ = 35,000 (comm. door sales goal) 4E + 3I + 7 C < 9,000 (steel usage) 2E + 4I + 3C < 6,000 (forming time) 2E + 3I + 4C < 5,200 (assembly time) E, I, C, dT-, dT+, dE- , dE+, dI- , dI+, dC- , dC+ > 0 Go to file 6-6.xls

Go to file 6-6.xls, sheet:6-6A Weighted Goals When goals have different priorities, weights can be used Suppose that Goal 1 is 5 times more important than each of the others Objective Function Min 5dT- + dE- + dI- + dC- Go to file 6-6.xls, sheet:6-6A

Go to file 6-6.xls, sheet:6-6A GP#1 Wilson Doors (Weighted GP #1)   E I C dT- dT+ dE- dE+ dI- dI+ dC- dC+ Exterior doors Interior doors Comm doors Under ach total sales Over ach total sales Under ach exter doors Over ach exter doors Under ach inter doors Over ach inter doors Under ach comm doors Over ach comm doors Solution value 1000.00 800.00 200.00 0.00 28000.00 13000.00 Goal weights 5 1 Constraints: Achieved Total sales goal 70 110 -1 180000.00 = 180000 Exterior doors goal 70000.00 70000 Interior doors goal 60000.00 60000 88000.00 Comm doors goal 35000.00 35000 22000.00 Steel usage 4 3 7 7800.00 <= 9000 Forming time 2 5800.00 6000 Assembly time 5200.00 5200 LHS Sign RHS Go to file 6-6.xls, sheet:6-6A GP#1

Properties of Weighted Goals Solution may differ depending on the weights used Appropriate only if goals are measured in the same units What if Goal 1 is only 2.5 times important than each of the others? Objective Function Min 2.5dT- + dE- + dI- + dC- Go to file 6-6.xls, sheet:6-6B GP#2, 6-6B IP

Go to file 6-6.xls, sheet:6-6B GP#2 Wilson Doors (Weighted GP #2)   E I C dT- dT+ dE- dE+ dI- dI+ dC- dC+ Exterior doors Interior doors Comm doors Under ach total sales Over ach total sales Under ach exter doors Over ach exter doors Under ach inter doors Over ach inter doors Under ach comm doors Over ach comm doors Solution value 1000.00 642.42 318.18 4333.33 0.00 10666.67 0.0 Goal weights 2.5 1 10833.33 Constraints: Achieved Total sales goal 70 110 -1 180000.00 = 180000 175666.67 Exterior doors goal 70000.00 70000 Interior doors goal 60000.00 60000 70666.67 Comm doors goal 35000.00 35000 Steel usage 4 3 7 8154.55 <= 9000 Forming time 2 5524.24 6000 Assembly time 5200.00 5200 LHS Sign RHS Go to file 6-6.xls, sheet:6-6B GP#2

Go to file 6-6.xls, sheet:6-6B IP Wilson Doors (Weighted GP #2 - IP)   E I C dT- dT+ dE- dE+ dI- dI+ dC- dC+ Exterior doors Interior doors Comm doors Under ach total sales Over ach total sales Under ach exter doors Over ach exter doors Under ach inter doors Over ach inter doors Under ach comm doors Over ach comm doors Solution value 1000.00 644.00 317.00 4290.00 0.00 10840.00 130.00 0.0 Goal weights 2.5 1 10855.00 Constraints: Achieved Total sales goal 70 110 -1 180000.00 = 180000 175710.00 Exterior doors goal 70000.00 70000 Interior doors goal 60000.00 60000 70840.00 Comm doors goal 35000.00 35000 34870.00 Steel usage 4 3 7 8151.00 <= 9000 Forming time 2 5527.00 6000 Assembly time 5200.00 5200 LHS Sign RHS Go to file 6-6.xls, sheet:6-6B IP

Ranked Goals Lower ranked goals are considered only if all higher ranked goals are achieved Suppose they added a 5th goal Goal 5: Steel usage as close to 9000 lb as possible 4E + 3I + 7C + dS- = 9000 (lbs steel) (no dS+ is needed because we cannot exceed 9000 pounds)

Rank R1: Goal 1 Rank R2: Goal 5 Rank R3: Goals 2, 3, and 4 A series of LP models must be solved Solve for the R1 goal while ignoring the other goals Objective Function: Min dT-

Objective Function Objective Function: Min dT- Subject to the constraints: 70E + 110I + 110C + dT- - dT+ = 180,000 (total sales goal) 4E + 3I + 7C + dS- = 9000 (steel usage goal) 70E + dE- - dE+ = 70,000 (exterior door sales goal) 110 I + dI- - dI+ = 60,000 (interior door sales goal) 110C + dC- - dC+ = 35,000 (comm. door sales goal) 4E + 3I + 7 C < 9,000 (steel usage) 2E + 4I + 3C < 6,000 (forming time) 2E + 3I + 4C < 5,200 (assembly time) E, I, C, dT-, dT+, dE- , dE+, dI- , dI+, dC- , dC+ > 0 Go to file 6-7.xls

Go to file 6-7A R1.xls Wilson Doors (Rank R1 Goals Only) E I C dT- dT+   E I C dT- dT+ dS- dE- dE+ dI- dI+ dC- dC+ Exterior doors Interior doors Comm doors Under ach total sales Over ach total sales Under ach steel usage Under ach exter doors Over ach exter doors Under ach inter doors Over ach inter doors Under ach comm doors Over ach comm doors Solution value 1000.00 800.00 200.00 0.00 1200.00 28000.00 13000.00 Objective coeff 1 Constraints: Achieved Total sales goal 70 110 -1 180000.00 = 180000 Steel usage goal 4 3 7 9000.00 9000 7800.00 Exterior doors goal 70000.00 70000 Interior doors goal 60000.00 60000 88000.00 Comm doors goal 35000.00 35000 22000.00 Forming time 2 5800.00 <= 6000 Assembly time 5200.00 5200 LHS Sign RHS Go to file 6-7A R1.xls

Objective Function: Min dS- 2) If the R1 goal can be achieved (dT- = 0), then this is added as a constraint and we attempt to satisfy the R2 goal (Goal 5) Objective Function: Min dS- 3) If the R2 goal can be achieved (dS- = 0), then this is added as a constraint and we solve for the R3 goals (Goals 2, 3, and 4) Objective Function: Min dE- + dI- + dC- Go to file 6-7.xls

Go to file 6-7B R2.xls Wilson Doors (Rank R2 Goals Only) E I C dT- dT+   E I C dT- dT+ dS- dE- dE+ dI- dI+ dC- dC+ Exterior doors Interior doors Comm doors Under ach total sales Over ach total sales Under ach steel usage Under ach exter doors Over ach exter doors Under ach inter doors Over ach inter doors Under ach comm doors Over ach comm doors Solution value 1900.00 466.67 0.00 4333.33 63000.00 8666.67 35000.00 Objective coeff 1 Constraints: Achieved Total sales goal 70 110 -1 180000.00 = 180000 184333.33 Steel usage goal 4 3 7 9000.00 9000 Exterior doors goal 70000.00 70000 133000.00 Interior doors goal 60000.00 60000 51333.33 Comm doors goal 35000 Forming time 2 5666.67 <= 6000 Assembly time 5200.00 5200 LHS Sign RHS Go to file 6-7B R2.xls

Go to file 6-7B R2 IP.xls Wilson Doors (Rank R2 Goals Only - IP) E I C   E I C dT- dT+ dS- dE- dE+ dI- dI+ dC- dC+ Exterior doors Interior doors Comm doors Under ach total sales Over ach total sales Under ach steel usage Under ach exter doors Over ach exter doors Under ach inter doors Over ach inter doors Under ach comm doors Over ach comm doors Solution value 1699.00 422.00 134.00 0.00 90.00 48930.00 13580.00 20260.00 Objective coeff 1 Constraints: Achieved Total sales goal 70 110 -1 180000.00 = 180000 180090.00 Steel usage goal 4 3 7 9000.00 9000 Exterior doors goal 70000.00 70000 118930.00 Interior doors goal 60000.00 60000 46420.00 Comm doors goal 35000.00 35000 14740.00 Forming time 2 5488.00 <= 6000 Assembly time 5200.00 5200 LHS Sign RHS Go to file 6-7B R2 IP.xls

Go to file 6-7C R3.xls Wilson Doors (Rank R3 Goals Only) E I C dT- dT+   E I C dT- dT+ dS- dE- dE+ dI- dI+ dC- dC+ Exterior doors Interior doors Comm doors Under ach total sales Over ach total sales Under ach steel usage Under ach exter doors Over ach exter doors Under ach inter doors Over ach inter doors Under ach comm doors Over ach comm doors Solution value 1694.74 421.05 136.84 0.00 48631.58 13684.21 19947.37 Objective coeff 1 33631.58 Constraints: Achieved Total sales goal 70 110 -1 180000.00 = 180000 Steel usage goal 4 3 7 9000.00 9000 Exterior doors goal 70000.00 70000 118631.58 Interior doors goal 60000.00 60000 46315.79 Comm doors goal 35000.00 35000 15052.63 Forming time 2 5484.21 <= 6000 Assembly time 5200.00 5200 LHS Sign RHS Go to file 6-7C R3.xls

Go to file 6-7C R3 IP.xls Wilson Doors (Rank R3 Goals Only - IP) E I C   E I C dT- dT+ dS- dE- dE+ dI- dI+ dC- dC+ Exterior doors Interior doors Comm doors Under ach total sales Over ach total sales Under ach steel usage Under ach exter doors Over ach exter doors Under ach inter doors Over ach inter doors Under ach comm doors Over ach comm doors Solution value 1699.00 422.00 134.00 0.00 90.00 48930.00 13580.00 20260.00 Objective coeff 1 33840.00 Constraints: Achieved Total sales goal 70 110 -1 180000.00 = 180000 180090.00 Steel usage goal 4 3 7 9000.00 9000 Exterior doors goal 70000.00 70000 118930.00 Interior doors goal 60000.00 60000 46420.00 Comm doors goal 35000.00 35000 14740.00 Forming time 2 5488.00 <= 6000 Assembly time 5200.00 5200 LHS Sign RHS Go to file 6-7C R3 IP.xls