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```					       Chapter 6
Integer and Goal
Programming Models
Jason C. H. Chen, Ph.D.
Professor of MIS
Gonzaga University
Spokane, WA 99223
chen@jepson.gonzaga.edu
Variations of Basic
Linear Programming
• Integer Programming

• Goal Programming

• Nonlinear Programming (skip)

Dr. Chen, Decision Support Systems                         2
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                3
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                    4
Decision Variables
L = number of lamps to make
F = number of ceiling fans to make

Lamps        Fans
(per lamp)   (per fan)
Profit                                                         Hours
\$600        \$700
Contribution                                                  Available
Wiring Hours                           2 hrs       3 hrs         12
Assembly Hours                         6 hrs        5 hr         30

Dr. Chen, Decision Support Systems                                            5
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                                     6
Graphical Solution

Dr. Chen, Decision Support Systems                        7
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                  8
Using Solver for IP
• IP models are formulated in Excel in the
same way as LP models

• The additional integer restriction is entered
int - Means general integer variables
bin - Means binary variables
Go to file 6-1.xls

Dr. Chen, Decision Support Systems                            9
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                                                 10
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                     11
Stock Data

Dr. Chen, Decision Support Systems                12
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                        13
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                  14
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            15
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            16
If British Petro is included (B=1), then
Trans-Texas Oil must also be included (T=1)
T=0    T=1
Combinations            B=0   ok     ok
of B and T
B=1 not ok   ok

B<T
allows the 3 acceptable combinations and
prevents the unacceptable one

Dr. Chen, Decision Support Systems                      17
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                                 18
Simkin and Steinberg (Binary)

T      B       D       H       L     S      C
Tran
s-    Briti   Dut             Lone   San   Cali
Tex     sh      ch             Star   Die    f
as    Petr    She    Houst    Petr   go    Petr
Oil    o        ll   on Oil     o    Oil    o
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                                                                           19
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                   20
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              21
Data

Dr. Chen, Decision Support Systems          22
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                        23
LP Solution (File: 6-6.xls)

Wilson Doors (LP)

E          I        C

Exterior   Interior   Comm
doors      doors     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

E: 1400, I=800, and C=0 with a total sales of \$186,000
Dr. Chen, Decision Support Systems                                                              24
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                25
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                                26
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

Dr. Chen, Decision Support Systems                      27
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                  28
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 + 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                          29
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                        30
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
Dr. Chen, Decision Support Systems                                           31
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                                     32
Wilson Doors (Weighted GP #1)

E           I      C       dT -   dT +   dE-   dE+   d I-     d I+       dC-     d C+

Und    Ove    Und   Ov    Und                         Ove
er      r     er    er    er                           r
ach    ach    ach   ach   ach                         ach
tota   tota   ext   ext   inte              Under      co
Interio              l      l     er    er     r    Over ach    ach      mm
Exterior      r      Comm     sale   sale   doo   doo   doo      inter    comm      doo
doors     doors     doors      s      s     rs    rs    rs     doors     doors      rs

1000.0     800.0     200.0    0.0    0.0    0.0   0.0   0.0    28000.0    13000.0   0.0
Solution value          0          0         0       0      0      0     0     0        0          0       0

Goal weights                                         5             1           1                  1              13000.00

Constraints:                                                                                                                              Achieved

180000.0         18000   180000.0
Total sales goal        70        110       110      1      -1                                                          0   =         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

Sig
LHS        n    RHS
Go to file 6-6.xls, sheet:6-6A GP#1

Dr. Chen, Decision Support Systems                                                                                                            33
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                                                  34
Wilson Doors (Weighted GP #2)

E           I        C         dT-      dT+     dE-    dE+     dI-      dI+      dC-    dC+

Und    Ove                       Und    Ove
Ove      er     r     Und                 er      r
r     ach    ach      er               ach    ach
ach     exte   exte    ach               com    com
Under      total     r     r     inter   Over ach    m      m
Exterior   Interior   Comm     ach total   sale    door   door   door      inter    door   doo
doors      doors     doors     sales        s      s      s       s      doors      s      rs

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                                                                                                                      35
Wilson Doors (Weighted GP #2 - IP)

E           I        C         dT-      dT+     dE-    dE+     dI-      dI+       dC-     dC+

Und    Ove                                Ove
Ove      er      r    Und                           r
r     ach    ach      er                        ach
ach     exte   exte    ach               Under    com
Under      total     r      r    inter   Over ach    ach      m
Exterior   Interior   Comm     ach total   sale    door   door   door      inter    comm     doo
doors      doors     doors     sales        s      s      s       s      doors     doors     rs

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                                                                                                                      36
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                  37
•          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                 38
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                                           39
Wilson Doors (Rank R1 Goals Only)

E          I       C      dT-    dT+      dS-     dE-   dE+   dI-      dI+        dC-     dC+

Un     Ov              Un    Ov     Un                         Ov
der     er              der    er   der                          er
ach    ach              ach   ach   ach                         ach
tota   tota    Under    ext   ext   inte              Under      co
Interio             l      l       ach     er    er     r    Over ach    ach      mm
Exterior      r      Comm     sal    sal     steel   doo   doo   doo      inter    comm      doo
doors     doors     doors    es     es     usage     rs    rs    rs     doors     doors      rs

1000.0     800.0     200.0   0.0    0.0     1200.0   0.0   0.0   0.0    28000.0    13000.0   0.0
Solution value          0          0         0      0      0        0       0     0     0        0          0       0

Objective coeff                                     1                                                                         0.00

Constraints:                                                                                                                                       Achieved

18000
Total sales goal        70        110       110     1     -1                                                             180000.00   =         0   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

Sig
LHS        n    RHS

Go to file 6-7A R1.xls
Dr. Chen, Decision Support Systems                                                                                                                      40
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                        41
Wilson Doors (Rank R2 Goals Only)

E          I        C     dT-       dT+      dS-    dE-      dE+         dI-       dI+       dC-      dC+

Und    Und                                               Ove
Und                 er     er                           Ove                   r
er                ach    ach                             r                 ach
Com    ach                stee   exte                           ach                com
m     total   Over ach     l      r    Over ach    Under      inter   Under ach    m
Exterior   Interior   door   sale      total    usa    door    exter     ach inter   door     comm       door
doors      doors      s       s      sales      ge     s      doors      doors        s       doors      s

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                                                                                                                                    42
Wilson Doors (Rank R2 Goals Only - IP)

E          I         C      dT-      dT+       dS-    dE-      dE+          dI-        dI+       dC-      dC+

Und                                                 Ove
Und               Und      er                             Ove                   r
er                 er    ach                               r                 ach
ach     Over       ach    exte                             ach                com
total    ach      steel     r    Over ach                 inter   Under ach    m
Exterior   Interior   Comm     sale    total     usag    door    exter     Under ach     door     comm       door
doors      doors     doors      s     sales       e      s      doors     inter doors     s       doors      s

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                                                                                                                                 43
Wilson Doors (Rank R3 Goals Only)

E          I         C      dT-     dT+      dS-    dE-      dE+          dI-        dI+       dC-      dC+

Und
Und             Und      er                                                 Over
er     Over     er     ach                             Over                ach
ach     ach     ach     exte                             ach                com
total   total   steel     r    Over ach                 inter   Under ach    m
Exterior   Interior   Comm     sale    sale    usag    door    exter     Under ach     door     comm       door
doors      doors     doors      s       s       e      s      doors     inter doors     s       doors      s

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                                                                                                                               44
Wilson Doors (Rank R3 Goals Only - IP)

E          I         C      dT-      dT+    dS-    dE-      dE+          dI-        dI+       dC-      dC+

Und    Und                                                 Ove
Und              er     er                             Ove                   r
er             ach    ach                               r                 ach
ach     Over    stee   exte                             ach                com
total    ach      l      r    Over ach                 inter   Under ach    m
Exterior   Interior   Comm     sale    total   usa    door    exter     Under ach     door     comm       door
doors      doors     doors      s     sales    ge     s      doors     inter doors     s       doors      s

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

Go to file 6-7C R3 IP.xls

Dr. Chen, Decision Support Systems                                                                                                                                    45

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