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**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

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**Variations of Basic Linear Programming**

Integer Programming Goal Programming Nonlinear Programming (skip)

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**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

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**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

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**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

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**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

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Graphical Solution

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**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

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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

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**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

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**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)

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Stock Data

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**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

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**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

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**Objective Function (in $1000’s return)**

Max 50T + 80B + 90D + 120H + 110L S + 75C Subject to the constraints: Invest up to $3 Million 480T + 540B + 680D H + 700L + 510S + 900C < 3000

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**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

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**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

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**IP Model for Portfolio Selection**

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

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**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

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**Goal Programming Models**

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

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**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

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Data

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**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

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**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 800.00 0.00 Revenue $70 $110 $186,000.00 Constraints: Steel usage 4 3 7 <= 9000 Forming time 2 6000 Assembly time 5200 LHS Sign RHS E: 1400, I=800, and C=0 with a total sales of $186,000

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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 800.00 200.00 0.00 Goal weights 5 1 Constraints: Achieved Total sales goal 70 110 -1 = 180000 Exterior doors goal 70000 Interior doors goal 60000 Comm doors goal 35000 Steel usage 4 3 7 <= 9000 Forming time 2 6000 Assembly time 5200 LHS Sign RHS Go to file 6-6.xls, sheet:6-6A GP#1

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**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

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**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 642.42 318.18 0.00 0.0 Goal weights 2.5 1 Constraints: Achieved Total sales goal 70 110 -1 = 180000 Exterior doors goal 70000 Interior doors goal 60000 Comm doors goal 35000 Steel usage 4 3 7 <= 9000 Forming time 2 6000 Assembly time 5200 LHS Sign RHS Go to file 6-6.xls, sheet:6-6B GP#2

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**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 644.00 317.00 0.00 130.00 0.0 Goal weights 2.5 1 Constraints: Achieved Total sales goal 70 110 -1 = 180000 Exterior doors goal 70000 Interior doors goal 60000 Comm doors goal 35000 Steel usage 4 3 7 <= 9000 Forming time 2 6000 Assembly time 5200 LHS Sign RHS Go to file 6-6.xls, sheet:6-6B IP

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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)

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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-

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**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

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**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 800.00 200.00 0.00 Objective coeff 1 Constraints: Achieved Total sales goal 70 110 -1 = 180000 Steel usage goal 4 3 7 9000 Exterior doors goal 70000 Interior doors goal 60000 Comm doors goal 35000 Forming time 2 <= 6000 Assembly time 5200 LHS Sign RHS Go to file 6-7A R1.xls

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**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

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**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 466.67 0.00 Objective coeff 1 Constraints: Achieved Total sales goal 70 110 -1 = 180000 Steel usage goal 4 3 7 9000 Exterior doors goal 70000 Interior doors goal 60000 Comm doors goal 35000 Forming time 2 <= 6000 Assembly time 5200 LHS Sign RHS Go to file 6-7B R2.xls

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**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 422.00 134.00 0.00 90.00 Objective coeff 1 Constraints: Achieved Total sales goal 70 110 -1 = 180000 Steel usage goal 4 3 7 9000 Exterior doors goal 70000 Interior doors goal 60000 Comm doors goal 35000 Forming time 2 <= 6000 Assembly time 5200 LHS Sign RHS Go to file 6-7B R2 IP.xls

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**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 421.05 136.84 0.00 Objective coeff 1 Constraints: Achieved Total sales goal 70 110 -1 = 180000 Steel usage goal 4 3 7 9000 Exterior doors goal 70000 Interior doors goal 60000 Comm doors goal 35000 Forming time 2 <= 6000 Assembly time 5200 LHS Sign RHS Go to file 6-7C R3.xls

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**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 422.00 134.00 0.00 90.00 Objective coeff 1 Constraints: Achieved Total sales goal 70 110 -1 = 180000 Steel usage goal 4 3 7 9000 Exterior doors goal 70000 Interior doors goal 60000 Comm doors goal 35000 Forming time 2 <= 6000 Assembly time 5200 LHS Sign RHS Go to file 6-7C R3 IP.xls

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