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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.

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Presentation on theme: "Chapter 6 Integer and Goal Programming Models Jason C. H. Chen, Ph.D. Professor of MIS School of Business Administration Gonzaga University Spokane, WA."— Presentation transcript:

1 Chapter 6 Integer and Goal Programming Models Jason C. H. Chen, Ph.D. Professor of MIS School of Business Administration Gonzaga University Spokane, WA

2 Dr. Chen, Decision Support Systems 2 Variations of Basic Linear Programming Integer Programming Goal Programming Nonlinear Programming (skip)

3 Dr. Chen, Decision Support Systems 3 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

4 Dr. Chen, Decision Support Systems 4 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

5 Dr. Chen, Decision Support Systems 5 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 Hours2 hrs3 hrs12 Assembly Hours6 hrs5 hr30

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

7 Dr. Chen, Decision Support Systems 7 Graphical Solution

8 Dr. Chen, Decision Support Systems 8 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

9 Dr. Chen, Decision Support Systems 9 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

10 Dr. Chen, Decision Support Systems 10 Harrison Electric (General Integer) LF Lamp sFans Number of units Profit$600$700$3, Constraints: Wiring hours <=12 Assembly hours <=30 LHSSignRHS Go to file 6-1.xls

11 Dr. Chen, Decision Support Systems 11 Binary Integer Example: Portfolio Selection Choosing stocks to include in portfolio Decision: Which of 7 stocks to include? Objective: Maximize expected annual return (in $1000s)

12 Dr. Chen, Decision Support Systems 12 Stock Data

13 Dr. Chen, Decision Support Systems 13 Decision Variables Use the first letter of each stocks name Example for Trans-Texas Oil: T= 1 if Trans-Texas Oil is included T= 0 if not included

14 Dr. Chen, Decision Support Systems 14 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

15 Dr. Chen, Decision Support Systems 15 Objective Function(in $1000s return) Max 50T + 80B + 90D + 120H + 110L + 40S + 75C Subject to the constraints: Invest up to $3 Million 480T + 540B + 680D H + 700L + 510S + 900C < 3000

16 Dr. Chen, Decision Support Systems 16 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

17 Dr. Chen, Decision Support Systems 17 If British Petro is included (B=1), then Trans-Texas Oil must also be included (T=1) T=0T=1 B=0ok B=1not okok allows the 3 acceptable combinations and prevents the unacceptable one Combinations of B and T B < T

18 Dr. Chen, Decision Support Systems 18 IP Model for Portfolio Selection Max $50T + $80B + $90D + $120H + $110L + $40S + $75C Subject to the constraints: 480T + 540B + 680D H + 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

19 Dr. Chen, Decision Support Systems 19 Simkin and Steinberg (Binary) TBDHLSC Tran s- Tex as Oil Briti sh Petr o Dut ch She ll Houst on Oil Lone Star Petr o San Die go Oil Cali f Petr o Invest? (1 = Yes, 0 = No) Exp annual return ('000)$50$80$90$120$110$40$75$360 Constraints: Investment limit <=3000 Foreign companies111<=1 British & Trans-Texas10<=0 Texas companies1112>=2 California companies 111=1 LHSSignRHS Go to file 6-3.xls

20 Dr. Chen, Decision Support Systems 20 Goal Programming Models Permit multiple objectives Try to satisfy goals rather than optimize Objective is to minimize underachievement of goals

21 Dr. Chen, Decision Support Systems 21 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

22 Dr. Chen, Decision Support Systems 22 Data

23 Dr. Chen, Decision Support Systems 23 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

24 Dr. Chen, Decision Support Systems 24 Wilson Doors (LP) EIC Exterior doors Interior doors Comm doors Number of units Revenue$70$110 $186, Constraints: Steel usage <=9000 Forming time <=6000 Assembly time <=5200 LHSSignRHS LP Solution (File: 6-6.xls) E: 1400, I=800, and C=0 with a total sales of $186,000

25 Dr. Chen, Decision Support Systems 25 Goals 1.Total sales at least $180,000 2.Exterior door sales at least $70,000 3.Interior door sales at lest $60,000 4.Commercial door sales at least $35,000

26 Dr. Chen, Decision Support Systems 26 Regular Decision Variables E = number of exterior doors made I = number of interior doors made C = number of commercial doors made Deviation Variables d i + = amount by which goal i is overachieved d i - = amount by which goal i is underachieved

27 Dr. Chen, Decision Support Systems 27 Goal Constraints Goal 1: Total sales at least $180,000 70E I + 110C + d T - - d T + = 180,000 Goal 2: Exterior door sales at least $70,000 70E + d E - - d E + = 70,000 Note: Each highlighted deviation variable measures goal underachievement

28 Dr. Chen, Decision Support Systems 28 Goal 3: Interior door sales at least $60, I + d I - - d I + = 60,000 Goal 4: Commercial door sales at least $35, C + d C - - d C + = 35,000

29 Dr. Chen, Decision Support Systems 29 Goals 1.Total sales at least $180,000 2.Exterior door sales at least $70,000 3.Interior door sales at lest $60,000 4.Commercial door sales at least $35,000 Goal 1: 70E + 110I + 110C + d T - - d T + = 180,000 Goal 2: 70E + d E - - d E + = 70,000 Goal 3: 110 I + d I - - d I + = 60,000 Goal 4: 110C + d C - - d C + = 35,000

30 Dr. Chen, Decision Support Systems 30 Objective Function Minimize total goal underachievement Min d T - + d E - + d I - + d C - Subject to the constraints: The 4 goal constraints The regular constraints (3 limited resources) nonnegativity

31 Dr. Chen, Decision Support Systems 31 Objective Function Minimize d T - + d E - + d I - + d C - Subject to the constraints: 70E + 110I + 110C + d T - - d T + = 180,000 (total sales goal) 70E + d E - - d E + = 70,000 (exterior door sales goal) 110 I + d I - - d I + = 60,000 (interior door sales goal) 110C + d C - - d C + = 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, d T -, d T +, d E -, d E +, d I -, d I +, d C -, d C + > 0 Go to file 6-6.xls

32 Dr. Chen, Decision Support Systems 32 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 5d T - + d E - + d I - + d C - Go to file 6-6.xls, sheet:6-6A

33 Dr. Chen, Decision Support Systems 33 Wilson Doors (Weighted GP #1) EICdT-dT- dT+dT+ dE-dE- dE+dE+ dI-dI- dI+dI+ dC-dC- dC+dC+ Exterior doors Interio r doors Comm doors Und er ach tota l sale s Ove r ach tota l sale s Und er ach ext er doo rs Ov er ach ext er doo rs Und er ach inte r doo rs Over ach inter doors Under ach comm doors Ove r ach co mm doo rs Solution value Goal weights Constraints: Achieved Total sales goal = Exterior doors goal = Interior doors goal = Comm doors goal = Steel usage <=9000 Forming time <=6000 Assembly time <=5200 LHS Sig nRHS Go to file 6-6.xls, sheet:6-6A GP#1

34 Dr. Chen, Decision Support Systems 34 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 2.5d T - + d E - + d I - + d C - Go to file 6-6.xls, sheet:6-6B GP#2, 6-6B IP

35 Dr. Chen, Decision Support Systems 35 Wilson Doors (Weighted GP #2) EICdT-dT- dT+dT+ dE-dE- dE+dE+ dI-dI- dI+dI+ dC-dC- dC+dC+ Exterior doors Interior doors Comm doors Under ach total sales Ove r ach total sale s Und er ach exte r door s Ove r ach exte r door s Und er ach inter door s Over ach inter doors Und er ach com m door s Ove r ach com m doo rs Solution value Goal weights Constraints: Achieved Total sales goal = Exterior doors goal = Interior doors goal = Comm doors goal = Steel usage <=9000 Forming time <=6000 Assembly time <=5200 LHSSignRHS Go to file 6-6.xls, sheet:6-6B GP#2

36 Dr. Chen, Decision Support Systems 36 Wilson Doors (Weighted GP #2 - IP) EICdT-dT- dT+dT+ dE-dE- dE+dE+ dI-dI- dI+dI+ dC-dC- dC+dC+ Exterior doors Interior doors Comm doors Under ach total sales Ove r ach total sale s Und er ach exte r door s Ove r ach exte r door s Und er ach inter door s Over ach inter doors Under ach comm doors Ove r ach com m doo rs Solution value Goal weights Constraints: Achieved Total sales goal = Exterior doors goal = Interior doors goal = Comm doors goal = Steel usage <=9000 Forming time <=6000 Assembly time <=5200 LHSSignRHS Go to file 6-6.xls, sheet:6-6B IP

37 Dr. Chen, Decision Support Systems 37 Ranked Goals Lower ranked goals are considered only if all higher ranked goals are achieved Suppose they added a 5 th goal Goal 5: Steel usage as close to 9000 lb as possible 4E + 3 I + 7C + d S - = 9000 (lbs steel) (no d S + is needed because we cannot exceed 9000 pounds)

38 Dr. Chen, Decision Support Systems 38 Rank R 1 : Goal 1 Rank R 2 : Goal 5 Rank R 3 : Goals 2, 3, and 4 A series of LP models must be solved 1)Solve for the R 1 goal while ignoring the other goals Objective Function: Min d T -

39 Dr. Chen, Decision Support Systems 39 Objective Function Objective Function: Min d T - Subject to the constraints: 70E + 110I + 110C + d T - - d T + = 180,000 (total sales goal) 4E + 3I + 7C + d S - = 9000 (steel usage goal) 70E + d E - - d E + = 70,000 (exterior door sales goal) 110 I + d I - - d I + = 60,000 (interior door sales goal) 110C + d C - - d C + = 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, d T -, d T +, d E -, d E +, d I -, d I +, d C -, d C + > 0 Go to file 6-7.xls

40 Dr. Chen, Decision Support Systems 40 Wilson Doors (Rank R 1 Goals Only) EICdT-dT- dT+dT+ dS-dS- dE-dE- dE+dE+ dI-dI- dI+dI+ dC-dC- dC+dC+ Exterior doors Interio r doors Comm doors Un der ach tota l sal es Ov er ach tota l sal es Under ach steel usage Un der ach ext er doo rs Ov er ach ext er doo rs Un der ach inte r doo rs Over ach inter doors Under ach comm doors Ov er ach co mm doo rs Solution value Objective coeff Constraints: Achieved Total sales goal = Steel usage goal = Exterior doors goal = Interior doors goal = Comm doors goal = Forming time <=6000 Assembly time <=5200 LHS Sig nRHS Go to file 6-7A R1.xls

41 Dr. Chen, Decision Support Systems 41 2) If the R 1 goal can be achieved (d T - = 0), then this is added as a constraint and we attempt to satisfy the R 2 goal (Goal 5) Objective Function: Min d S - 3) If the R 2 goal can be achieved (d S - = 0), then this is added as a constraint and we solve for the R 3 goals (Goals 2, 3, and 4) Objective Function: Min d E - + d I - + d C - Go to file 6-7.xls

42 Dr. Chen, Decision Support Systems 42 Wilson Doors (Rank R 2 Goals Only) EICdT-dT- dT+dT+ dS-dS- dE-dE- dE+dE+ dI-dI- dI+dI+ dC-dC- dC+dC+ Exterior doors Interior doors Com m door s Und er ach total sale s Over ach total sales Und er ach stee l usa ge Und er ach exte r door s Over ach exter doors Under ach inter doors Ove r ach inter door s Under ach comm doors Ove r ach com m door s Solution value Objective coeff Constraints: Achieved Total sales goal = Steel usage goal = Exterior doors goal = Interior doors goal = Comm doors goal = Forming time <=6000 Assembly time <=5200 LHSSignRHS Go to file 6-7B R2.xls

43 Dr. Chen, Decision Support Systems 43 Wilson Doors (Rank R 2 Goals Only - IP) EICdT-dT- dT+dT+ dS-dS- dE-dE- dE+dE+ dI-dI- dI+dI+ dC-dC- dC+dC+ Exterior doors Interior doors Comm doors Und er ach total sale s Over ach total sales Und er ach steel usag e Und er ach exte r door s Over ach exter doors Under ach inter doors Ove r ach inter door s Under ach comm doors Ove r ach com m door s Solution value Objective coeff Constraints: Achieved Total sales goal = Steel usage goal = Exterior doors goal = Interior doors goal = Comm doors goal = Forming time <=6000 Assembly time <=5200 LHSSignRHS Go to file 6-7B R2 IP.xls

44 Dr. Chen, Decision Support Systems 44 Wilson Doors (Rank R 3 Goals Only) EICdT-dT- dT+dT+ dS-dS- dE-dE- dE+dE+ dI-dI- dI+dI+ dC-dC- dC+dC+ Exterior doors Interior doors Comm doors Und er ach total sale s Over ach total sale s Und er ach steel usag e Und er ach exte r door s Over ach exter doors Under ach inter doors Over ach inter door s Under ach comm doors Over ach com m door s Solution value Objective coeff Constraints: Achieved Total sales goal = Steel usage goal = Exterior doors goal = Interior doors goal = Comm doors goal = Forming time <=6000 Assembly time <=5200 LHSSignRHS Go to file 6-7C R3.xls

45 Dr. Chen, Decision Support Systems 45 Go to file 6-7C R3 IP.xls Wilson Doors (Rank R 3 Goals Only - IP) EICdT-dT- dT+dT+ dS-dS- dE-dE- dE+dE+ dI-dI- dI+dI+ dC-dC- dC+dC+ Exterior doors Interior doors Comm doors Und er ach total sale s Over ach total sales Und er ach stee l usa ge Und er ach exte r door s Over ach exter doors Under ach inter doors Ove r ach inter door s Under ach comm doors Ove r ach com m door s Solution value Objective coeff Constraints: Achieved Total sales goal = Steel usage goal = Exterior doors goal = Interior doors goal = Comm doors goal = Forming time <=6000 Assembly time <=5200 LHSSignRHS


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