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Chapter 6Integer and Goal
Programming Models
Jason C. H. Chen, Ph.D.Professor of MIS
School of Business AdministrationGonzaga UniversitySpokane, WA 99223
Dr. Chen, Decision Support Systems 2
Variations of Basic Linear Programming
• Integer Programming
• Goal Programming
• Nonlinear Programming (skip)
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
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
Dr. Chen, Decision Support Systems 5
Decision VariablesL = number of lamps to make
F = number of ceiling fans to make
Lamps(per lamp)
Fans(per fan)
Hours Available
Profit Contribution
$600 $700
Wiring Hours 2 hrs 3 hrs 12
Assembly Hours 6 hrs 5 hr 30
Dr. Chen, Decision Support Systems 6
LP Model Summary
Max 600 L + 700 F ($ of profit)
Subject to the constraints:
2L + 3F < 12 (wiring hours)
6L + 5F < 30 (assembly hours)
L, F > 0
Dr. Chen, Decision Support Systems 7
Graphical Solution
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
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
Dr. Chen, Decision Support Systems 10
Harrison Electric (General Integer)
L F
Lamp
s 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
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 $1000’s)
Dr. Chen, Decision Support Systems 12
Stock Data
Dr. Chen, Decision Support Systems 13
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
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
Dr. Chen, Decision Support Systems 15
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
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
Dr. Chen, Decision Support Systems 17
If British Petro is included (B=1), then
Trans-Texas Oil must also be included (T=1)
T=0 T=1
B=0 ok ok
B=1 not ok ok
allows the 3 acceptable combinations and prevents the unacceptable one
Combinationsof B and T
B < T
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 + 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
Dr. Chen, Decision Support Systems 19
Simkin and Steinberg (Binary)
T B D H L S C
Trans-Texas Oil
British
Petro
Dutch
Shell
Houston Oil
Lone Star Petr
o
San Diego Oil
Calif
Petro
Invest? (1 = Yes, 0 = No) 0 0 1 1 1 1 0
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 1 1 1 <= 1
British & Trans-Texas 1 0 <= 0
Texas companies 1 1 1 2 >= 2
California companies 1 1 1 = 1
LHS Sign RHS
Go to file 6-3.xls
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
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
Dr. Chen, Decision Support Systems 22
Data
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
Dr. Chen, Decision Support Systems 24
Wilson Doors (LP)
E I C
Exterior doors
Interior doors
Comm doors
Number of units 1400.00 800.00 0.00
Revenue $70 $110 $110 $186,000.00
Constraints:
Steel usage 4 3 7 8000.00 <= 9000
Forming time 2 4 3 6000.00 <= 6000
Assembly time 2 3 4 5200.00 <= 5200
LHS Sign RHS
LP Solution (File: 6-6.xls)
E: 1400, I=800, and C=0 with a total sales of $186,000
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
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
di+ = amount by which goal i is overachieved
di- = amount by which goal i is underachieved
Dr. Chen, Decision Support Systems 27
Goal ConstraintsGoal 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
Dr. Chen, Decision Support Systems 28
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
Dr. Chen, Decision Support Systems 29
Goals1. 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 + 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
Dr. Chen, Decision Support Systems 30
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
Dr. Chen, Decision Support Systems 31
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
+ > 0Go to file 6-6.xls
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 5dT- + dE
- + dI- + dC
-
Go to file 6-6.xls, sheet:6-6A
Dr. Chen, Decision Support Systems 33
Wilson Doors (Weighted GP #1)
E I C dT- dT
+ dE- dE
+ dI- dI
+ dC- dC
+
Exterior doors
Interior
doorsComm doors
Under
ach tota
l sale
s
Over
ach tota
l sale
s
Under
ach exter
doors
Over
ach exter
doors
Under
ach inte
r doors
Over ach inter
doors
Under ach
comm doors
Over
ach co
mm doors
Solution value1000.0
0800.0
0200.0
00.00
0.00
0.00
0.00
0.00
28000.00
13000.00
0.00
Goal weights 5 1 1 1 13000.00
Constraints: Achieved
Total sales goal 70 110 110 1 -1 180000.0
0 =18000
0180000.0
0
Exterior doors goal 70 1 -1 70000.00 = 70000 70000.00
Interior doors goal 110 1 -1 60000.00 = 60000 88000.00
Comm doors goal 110 1 -1 35000.00 = 35000 22000.00
Steel usage 4 3 7 7800.00 <= 9000
Forming time 2 4 3 5800.00 <= 6000
Assembly time 2 3 4 5200.00 <= 5200
LHSSign RHS
Go to file 6-6.xls, sheet:6-6A GP#1
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 2.5dT- + dE
- + dI- + dC
-
Go to file 6-6.xls, sheet:6-6B GP#2, 6-6B IP
Dr. Chen, Decision Support Systems 35
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 sale
s
Under
ach exte
r door
s
Over
ach exte
r door
s
Under
ach inter door
s
Over ach inter doors
Under
ach comm
doors
Over
ach comm
doors
Solution value 1000.00 642.42 318.18 4333.33 0.00 0.00 0.00 0.00 10666.67 0.00 0.0
Goal weights 2.5 1 1 1 10833.33
Constraints: Achieved
Total sales goal 70 110 110 1 -1 180000.00 = 180000 175666.67
Exterior doors goal 70 1 -1 70000.00 = 70000 70000.00
Interior doors goal 110 1 -1 60000.00 = 60000 70666.67
Comm doors goal 110 1 -1 35000.00 = 35000 35000.00
Steel usage 4 3 7 8154.55 <= 9000
Forming time 2 4 3 5524.24 <= 6000
Assembly time 2 3 4 5200.00 <= 5200
LHS Sign RHS
Go to file 6-6.xls, sheet:6-6B GP#2
Dr. Chen, Decision Support Systems 36
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 sale
s
Under
ach exte
r door
s
Over
ach exte
r door
s
Under
ach inter door
s
Over ach inter doors
Under ach
comm doors
Over
ach comm
doors
Solution value 1000.00 644.00 317.00 4290.00 0.00 0.00 0.00 0.00 10840.00 130.00 0.0
Goal weights 2.5 1 1 1 10855.00
Constraints: Achieved
Total sales goal 70 110 110 1 -1 180000.00 = 180000 175710.00
Exterior doors goal 70 1 -1 70000.00 = 70000 70000.00
Interior doors goal 110 1 -1 60000.00 = 60000 70840.00
Comm doors goal 110 1 -1 35000.00 = 35000 34870.00
Steel usage 4 3 7 8151.00 <= 9000
Forming time 2 4 3 5527.00 <= 6000
Assembly time 2 3 4 5200.00 <= 5200
LHS Sign RHS
Go to file 6-6.xls, sheet:6-6B IP
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 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)
Dr. Chen, Decision Support Systems 38
• Rank R1: Goal 1
• Rank R2: Goal 5
• Rank R3: Goals 2, 3, and 4
A series of LP models must be solved
1) Solve for the R1 goal while ignoring the other goals
Objective Function: Min dT-
Dr. Chen, Decision Support Systems 39
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
Dr. Chen, Decision Support Systems 40
Wilson Doors (Rank R1 Goals Only)
E I C dT- dT
+ dS- dE
- dE+ dI
- dI+ dC
- dC+
Exterior doors
Interior
doorsComm doors
Under
ach tota
l sale
s
Over
ach tota
l sale
s
Under ach steel
usage
Under
ach exter
doors
Over
ach exter
doors
Under
ach inte
r doors
Over ach inter doors
Under ach
comm doors
Over
ach co
mm doors
Solution value 1000.00800.0
0200.0
00.00
0.00 1200.00
0.00
0.00
0.00 28000.00 13000.00
0.00
Objective coeff 1 0.00
Constraints: Achieved
Total sales goal 70 110 110 1 -1 180000.00 = 180000 180000.00
Steel usage goal 4 3 7 1 9000.00 = 9000 7800.00
Exterior doors goal 70 1 -1 70000.00 = 70000 70000.00
Interior doors goal 110 1 -1 60000.00 = 60000 88000.00
Comm doors goal 110 1 -1 35000.00 = 35000 22000.00
Forming time 2 4 3 5800.00 <= 6000
Assembly time 2 3 4 5200.00 <= 5200
LHSSign RHS
Go to file 6-7A R1.xls
Dr. Chen, Decision Support Systems 41
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
Dr. Chen, Decision Support Systems 42
Wilson Doors (Rank R2 Goals Only)
E I C dT- dT
+ dS- dE
- dE+ dI
- dI+ dC
- dC+
Exterior doors
Interior doors
Comm
doors
Under
ach total sale
s
Over ach total sales
Under
ach stee
l usage
Under
ach exte
r door
s
Over ach exter doors
Under ach inter
doors
Over
ach inter door
s
Under ach comm doors
Over
ach comm
doors
Solution value 1900.00 466.67 0.00 0.00 4333.33 0.00 0.00 63000.00 8666.67 0.00 35000.00 0.00
Objective coeff 1 0.00
Constraints: Achieved
Total sales goal 70 110 110 1 -1 180000.00 = 180000 184333.33
Steel usage goal 4 3 7 1 9000.00 = 9000 9000.00
Exterior doors goal 70 1 -1 70000.00 = 70000 133000.00
Interior doors goal 110 1 -1 60000.00 = 60000 51333.33
Comm doors goal 110 1 -1 35000.00 = 35000 0.00
Forming time 2 4 3 5666.67 <= 6000
Assembly time 2 3 4 5200.00 <= 5200
LHS Sign RHS
Go to file 6-7B R2.xls
Dr. Chen, Decision Support Systems 43
Wilson Doors (Rank R2 Goals Only - IP)
E I C dT- dT
+ dS- dE
- dE+ dI
- dI+ dC
- dC+
Exterior doors
Interior doors
Comm doors
Under
ach total sale
s
Over ach total sales
Under
ach steel usag
e
Under
ach exte
r door
sOver ach
exter doorsUnder ach inter doors
Over ach inter door
s
Under ach comm doors
Over ach comm
doors
Solution value 1699.00 422.00 134.00 0.00 90.00 0.00 0.00 48930.00 13580.00 0.00 20260.00 0.00
Objective coeff 1 0.00
Constraints: Achieved
Total sales goal 70 110 110 1 -1 180000.00 = 180000 180090.00
Steel usage goal 4 3 7 1 9000.00 = 9000 9000.00
Exterior doors goal 70 1 -1 70000.00 = 70000 118930.00
Interior doors goal 110 1 -1 60000.00 = 60000 46420.00
Comm doors goal 110 1 -1 35000.00 = 35000 14740.00
Forming time 2 4 3 5488.00 <= 6000
Assembly time 2 3 4 5200.00 <= 5200
LHS Sign RHS
Go to file 6-7B R2 IP.xls
Dr. Chen, Decision Support Systems 44
Wilson Doors (Rank R3 Goals Only)
E I C dT- dT
+ dS- dE
- dE+ dI
- dI+ dC
- dC+
Exterior doors
Interior doors
Comm doors
Under
ach total sale
s
Over ach total sale
s
Under
ach steel usag
e
Under
ach exter door
sOver ach
exter doorsUnder ach inter doors
Over ach inter door
s
Under ach comm doors
Over ach comm
doors
Solution value 1694.74 421.05 136.84 0.00 0.00 0.00 0.00 48631.58 13684.21 0.00 19947.37 0.00
Objective coeff 1 1 1 33631.58
Constraints: Achieved
Total sales goal 70 110 110 1 -1 180000.00 = 180000 180000.00
Steel usage goal 4 3 7 1 9000.00 = 9000 9000.00
Exterior doors goal 70 1 -1 70000.00 = 70000 118631.58
Interior doors goal 110 1 -1 60000.00 = 60000 46315.79
Comm doors goal 110 1 -1 35000.00 = 35000 15052.63
Forming time 2 4 3 5484.21 <= 6000
Assembly time 2 3 4 5200.00 <= 5200
LHS Sign RHS
Go to file 6-7C R3.xls
Dr. Chen, Decision Support Systems 45
Go to file 6-7C R3 IP.xls
Wilson Doors (Rank R3 Goals Only - IP)
E I C dT- dT
+ dS- dE
- dE+ dI
- dI+ dC
- dC+
Exterior doors
Interior doors
Comm doors
Under
ach total sale
s
Over ach total sales
Under
ach steel usage
Under
ach exte
r door
s
Over ach exter doors
Under ach inter doors
Over
ach inter door
s
Under ach comm doors
Over
ach comm
doors
Solution value 1699.00 422.00 134.00 0.00 90.00 0.00 0.00 48930.00 13580.00 0.00 20260.00 0.00
Objective coeff 1 1 1 33840.00
Constraints: Achieved
Total sales goal 70 110 110 1 -1 180000.00 = 180000 180090.00
Steel usage goal 4 3 7 1 9000.00 = 9000 9000.00
Exterior doors goal 70 1 -1 70000.00 = 70000 118930.00
Interior doors goal 110 1 -1 60000.00 = 60000 46420.00
Comm doors goal 110 1 -1 35000.00 = 35000 14740.00
Forming time 2 4 3 5488.00 <= 6000
Assembly time 2 3 4 5200.00 <= 5200
LHS Sign RHS