


1
Winston Chapter 9.3, Page 514, Number 2 (Integer Programming)
Problem Statement: Use branch-and-bound to solve the IP: The Dorian Auto example
from Section 3.2, Page 61: Dorian Auto manufactures luxury cars and trucks. The
company believes that its most likely customers are high-income women and men. To
reach these groups, Dorian Auto has embarked on an ambitious TV advertising campaign
and has decided to purchase 1-minute commercial spots on two types of programs:
comedy shows and football games. Each comedy commercial is seen by 7 million highincome women and 12 million high-income men. A 1-minute comedy ad costs $50,000,
and a 1-minute football ad costs $100,000. Dorian would like the commercials to be seen
by at least 28 million high-income women and 24 million high-income men. Use linear
integer programming to determine how Dorian Auto can meet its advertising
requirements at minimum cost.
Let x1 = number of 1-minute comedy ads purchased (in thousands of dollars).
Let x2 = number of 1-minute football ads purchased (in thousands of dollars).
Min z = 50x1 + 100x2
s.t.
7x1 + 2x2 ≥ 28
2x1 + 12x2 ≥ 24
x1, x2 ≥ 0
(HIW)
(HIM)
Questions:
A. Solve using the Integer Programming option of Quant. Print input and output of
Quant.
B. Solve using the Linear Programming option of Quant and creating the Branch-andBound tree. Print input and output of Quant.
C. Comment on these solutions.



2
A. The problem is solved using the Integer Programming option of Quant.
Quant Input:
Input Data of The Problem 051402A
Min +50.0000X1
Subject to
(1) +7.00000X1
(2) +2.00000X1
+100.000X2
+2.00000X2
+12.0000X2
Integrality and Bounds
Var. no.
1
2
Name
X1
X2
Page: 1
> +28.0000
> +24.0000
Page: 1
Integrality(C/I/B)
<I>
<I>
Lower bound
Upper bound
<+0
<+0
<+1.0E+20>
<+1.0E+20>
>
>
Quant Output:
|--------------------------------------------------------------------------|
|
Summarized Results for 051402A
Page : 1
|
|--------------------------------------------------------------------------|
| Variables |
|Obj. Fnctn.| Variables |
|Obj. Fnctn. |
| No. Names | Solution |Coefficient| No. Names | Solution |Coefficient |
|-------------+----------+-----------+-------------+----------+------------|
| 1
X1
|+4.0000000|+50.000000 | 2
X2
|+2.0000000|+100.00000 |
|--------------------------------------------------------------------------|
|
Minimized OBJ = 400 Iteration = 5 Elapsed CPU second = 0
|
| Branch selection: Newest problem Integer tolerance: .01 Max. #node: 3 |
|--------------------------------------------------------------------------|
The results indicate that four 1-minute comedy ads should be purchased and two 1minute football ads should be purchased. The total cost for this ad campaign will cost
$400,000.



3
B. Solve using the Linear Programming option of Quant and creating the Branch-andBound tree. Each circle within the branch-and-bound tree is assigned a letter that
corresponds with the appropriate Quant input and output.
Quant Input & Output, P1 (B):
Input Data of The Problem 051402B
Min +50.0000X1
Subject to
(1) +7.00000X1
(2) +2.00000X1
Page: 1
+100.000X2
+2.00000X2
+12.0000X2
> +28.0000
> +24.0000
|------------------------------------------------------------------------------|
|
Summarized Report for 051402B
Page : 1
|
|------------------------------------------------------------------------------|
|
|
|
|Opportunity| Objective | Minimum | Maximum |
|Number | Variable | Solution |
Cost
|Coefficient|Obj. Coeff.|Obj. Coeff.|
|-------+----------+-----------+-----------+-----------+-----------+-----------|
| 1
| X1
| +3.5999999| 0
| +50.000000| +16.666668| +350.00000|
| 2
| X2
| +1.4000000| 0
| +100.00000| +14.285713| +300.00000|
|------------------------------------------------------------------------------|
|
Minimized OBJ = 320 Iteration = 2 Elapsed CPU second = 0
|
|------------------------------------------------------------------------------|
|------------------------------------------------------------------------------|
|
Summarized Report for 051402B
Page : 2
|
|------------------------------------------------------------------------------|
|
|
|
|
Shadow | Slack or | Minimum | Maximum |
|Constr.| Status |
RHS
|
Price
| Surplus |
RHS
|
RHS
|
|-------+---------+------------+-----------+-----------+-----------+-----------|
| 1
| Tight | >+28.000000| +5.0000000| 0
| +4.0000000| +84.000000|
| 2
| Tight | >+24.000000| +7.5000000| 0
| +8.0000000| +168.00000|
|------------------------------------------------------------------------------|
|
Minimized OBJ = 320 Iteration = 2 Elapsed CPU second = 0
|
|------------------------------------------------------------------------------|



Quant Input & Output, P2 (C):
Input Data of The Problem 051402C
Min +50.0000X1
Subject to
(1) +7.00000X1
(2) +2.00000X1
(3) +1.00000X1
Page: 1
+100.000X2
+2.00000X2
+12.0000X2
+0
X2
> +28.0000
> +24.0000
< +3.00000
|------------------------------------------------------------------------------|
|
Summarized Report for 051402C
Page : 1
|
|------------------------------------------------------------------------------|
|
|
|
|Opportunity| Objective | Minimum | Maximum |
|Number | Variable | Solution |
Cost
|Coefficient|Obj. Coeff.|Obj. Coeff.|
|-------+----------+-----------+-----------+-----------+-----------+-----------|
| 1
| X1
| +3.0000000| 0
| +50.000000| - Infinity| +350.00000|
| 2
| X2
| +3.5000000| 0
| +100.00000| +14.285713| + Infinity|
|------------------------------------------------------------------------------|
|
Minimized OBJ = 500 Iteration = 3 Elapsed CPU second = 0
|
|------------------------------------------------------------------------------|
|------------------------------------------------------------------------------|
|
Summarized Report for 051402C
Page : 2
|
|------------------------------------------------------------------------------|
|
|
|
|
Shadow | Slack or | Minimum | Maximum |
|Constr.| Status |
RHS
|
Price
| Surplus |
RHS
|
RHS
|
|-------+---------+------------+-----------+-----------+-----------+-----------|
| 1
| Tight | >+28.000000| +50.000000| 0
| +24.000000| + Infinity|
| 2
| Loose | >+24.000000| 0
| +24.000000| - Infinity| +48.000000|
| 3
| Tight | <+3.0000000| -300.00000| 0
| 0
| +3.5999999|
|------------------------------------------------------------------------------|
|
Minimized OBJ = 500 Iteration = 3 Elapsed CPU second = 0
|
|------------------------------------------------------------------------------|
4



Quant Input & Output, P3 (D):
Input Data of The Problem 051402d
Min +50.0000X1
Subject to
(1) +7.00000X1
(2) +2.00000X1
(3) +1.00000X1
Page: 1
+100.000X2
+2.00000X2
+12.0000X2
+0
X2
> +28.0000
> +24.0000
> +4.00000
|------------------------------------------------------------------------------|
|
Summarized Report for 051402d
Page : 1
|
|------------------------------------------------------------------------------|
|
|
|
|Opportunity| Objective | Minimum | Maximum |
|Number | Variable | Solution |
Cost
|Coefficient|Obj. Coeff.|Obj. Coeff.|
|-------+----------+-----------+-----------+-----------+-----------+-----------|
| 1
| X1
| +4.0000000| 0
| +50.000000| +16.666668| + Infinity|
| 2
| X2
| +1.3333334| 0
| +100.00000| 0
| +300.00000|
|------------------------------------------------------------------------------|
|
Minimized OBJ = 333.3333 Iteration = 3 Elapsed CPU second = 0
|
|------------------------------------------------------------------------------|
|------------------------------------------------------------------------------|
|
Summarized Report for 051402d
Page : 2
|
|------------------------------------------------------------------------------|
|
|
|
|
Shadow | Slack or | Minimum | Maximum |
|Constr.| Status |
RHS
|
Price
| Surplus |
RHS
|
RHS
|
|-------+---------+------------+-----------+-----------+-----------+-----------|
| 1
| Loose | >+28.000000| 0
| +2.6666667| - Infinity| +30.666666|
| 2
| Tight | >+24.000000| +8.3333330| 0
| +8.0000000| + Infinity|
| 3
| Tight | >+4.0000000| +33.333332| 0
| +3.5999999| +12.000000|
|------------------------------------------------------------------------------|
|
Minimized OBJ = 333.3333 Iteration = 3 Elapsed CPU second = 0
|
|------------------------------------------------------------------------------|
5



Quant Input & Output, P4 (E):
Input Data of The Problem 051402E
Min +50.0000X1
Subject to
(1) +7.00000X1
(2) +2.00000X1
(3) +1.00000X1
(4) +0
X1
Page: 1
+100.000X2
+2.00000X2
+12.0000X2
+0
X2
+1.00000X2
>
>
<
<
+28.0000
+24.0000
+3.00000
+3.00000
NO FEASIBLE SOLUTION.
Quant Input & Output, P5 (F):
Input Data of The Problem 051402F
Min +50.0000X1
Subject to
(1) +7.00000X1
(2) +2.00000X1
(3) +1.00000X1
(4) +0
X1
Page: 1
+100.000X2
+2.00000X2
+12.0000X2
+0
X2
+1.00000X2
>
>
<
>
+28.0000
+24.0000
+3.00000
+4.00000
|------------------------------------------------------------------------------|
|
Summarized Report for 051402F
Page : 1
|
|------------------------------------------------------------------------------|
|
|
|
|Opportunity| Objective | Minimum | Maximum |
|Number | Variable | Solution |
Cost
|Coefficient|Obj. Coeff.|Obj. Coeff.|
|-------+----------+-----------+-----------+-----------+-----------+-----------|
| 1
| X1
| +2.8571429| 0
| +50.000000| +.00000381| +350.00000|
| 2
| X2
| +4.0000000| 0
| +100.00000| +14.285713| + Infinity|
|------------------------------------------------------------------------------|
|
Minimized OBJ = 542.8571 Iteration = 4 Elapsed CPU second = 0
|
|------------------------------------------------------------------------------|
|------------------------------------------------------------------------------|
|
Summarized Report for 051402F
Page : 2
|
|------------------------------------------------------------------------------|
|
|
|
|
Shadow | Slack or | Minimum | Maximum |
|Constr.| Status |
RHS
|
Price
| Surplus |
RHS
|
RHS
|
|-------+---------+------------+-----------+-----------+-----------+-----------|
| 1
| Tight | >+28.000000| +7.1428571| 0
| +8.0000000| +29.000000|
| 2
| Loose | >+24.000000| 0
| +29.714285| - Infinity| +53.714287|
| 3
| Loose | <+3.0000000| 0
| +.14285715| +2.8571429| + Infinity|
| 4
| Tight | >+4.0000000| +85.714287| 0
| +3.5000000| +14.000000|
|------------------------------------------------------------------------------|
|
Minimized OBJ = 542.8571 Iteration = 4 Elapsed CPU second = 0
|
|------------------------------------------------------------------------------|
6



Quant Input & Output, P6 (G):
Input Data of The Problem 051402G
Min +50.0000X1
Subject to
(1) +7.00000X1
(2) +2.00000X1
(3) +1.00000X1
(4) +0
X1
Page: 1
+100.000X2
+2.00000X2
+12.0000X2
+0
X2
+1.00000X2
>
>
>
<
+28.0000
+24.0000
+4.00000
+1.00000
|------------------------------------------------------------------------------|
|
Summarized Report for 051402G
Page : 1
|
|------------------------------------------------------------------------------|
|
|
|
|Opportunity| Objective | Minimum | Maximum |
|Number | Variable | Solution |
Cost
|Coefficient|Obj. Coeff.|Obj. Coeff.|
|-------+----------+-----------+-----------+-----------+-----------+-----------|
| 1
| X1
| +6.0000000| 0
| +50.000000| +16.666668| + Infinity|
| 2
| X2
| +1.0000000| 0
| +100.00000| - Infinity| +300.00000|
|------------------------------------------------------------------------------|
|
Minimized OBJ = 400 Iteration = 4 Elapsed CPU second = 0
|
|------------------------------------------------------------------------------|
|------------------------------------------------------------------------------|
|
Summarized Report for 051402G
Page : 2
|
|------------------------------------------------------------------------------|
|
|
|
|
Shadow | Slack or | Minimum | Maximum |
|Constr.| Status |
RHS
|
Price
| Surplus |
RHS
|
RHS
|
|-------+---------+------------+-----------+-----------+-----------+-----------|
| 1
| Loose | >+28.000000| 0
| +16.000000| - Infinity| +44.000000|
| 2
| Tight | >+24.000000| +25.000000| 0
| +20.000000| + Infinity|
| 3
| Loose | >+4.0000000| 0
| +2.0000000| - Infinity| +6.0000000|
| 4
| Tight | <+1.0000000| -200.00000| 0
| 0
| +1.3333334|
|------------------------------------------------------------------------------|
|
Minimized OBJ = 400 Iteration = 4 Elapsed CPU second = 0
|
|------------------------------------------------------------------------------|
7



Quant Input & Output, P7 (H):
Input Data of The Problem 051402H
Min +50.0000X1
Subject to
(1) +7.00000X1
(2) +2.00000X1
(3) +1.00000X1
(4) +0
X1
Page: 1
+100.000X2
+2.00000X2
+12.0000X2
+0
X2
+1.00000X2
>
>
>
>
+28.0000
+24.0000
+4.00000
+2.00000
|------------------------------------------------------------------------------|
|
Summarized Report for 051402H
Page : 1
|
|------------------------------------------------------------------------------|
|
|
|
|Opportunity| Objective | Minimum | Maximum |
|Number | Variable | Solution |
Cost
|Coefficient|Obj. Coeff.|Obj. Coeff.|
|-------+----------+-----------+-----------+-----------+-----------+-----------|
| 1
| X1
| +4.0000000| 0
| +50.000000| 0
| + Infinity|
| 2
| X2
| +2.0000000| 0
| +100.00000| 0
| + Infinity|
|------------------------------------------------------------------------------|
|
Minimized OBJ = 400 Iteration = 4 Elapsed CPU second = 5.993652E-02
|
|------------------------------------------------------------------------------|
|------------------------------------------------------------------------------|
|
Summarized Report for 051402H
Page : 2
|
|------------------------------------------------------------------------------|
|
|
|
|
Shadow | Slack or | Minimum | Maximum |
|Constr.| Status |
RHS
|
Price
| Surplus |
RHS
|
RHS
|
|-------+---------+------------+-----------+-----------+-----------+-----------|
| 1
| Loose | >+28.000000| 0
| +4.0000000| - Infinity| +32.000000|
| 2
| Loose | >+24.000000| 0
| +8.0000000| - Infinity| +32.000000|
| 3
| Tight | >+4.0000000| +50.000000| 0
| +3.4285715| + Infinity|
| 4
| Tight | >+2.0000000| +100.00000| 0
| +1.3333333| + Infinity|
|------------------------------------------------------------------------------|
|
Minimized OBJ = 400 Iteration = 4 Elapsed CPU second = 5.993652E-02
|
|------------------------------------------------------------------------------|
8



Quant Input & Output, P8 (I):
Input Data of The Problem 051402I
Min +50.0000X1
Subject to
(1) +7.00000X1
(2) +2.00000X1
(3) +1.00000X1
(4) +0
X1
(5) +1.00000X1
Page: 1
+100.000X2
+2.00000X2
+12.0000X2
+0
X2
+1.00000X2
+0
X2
>
>
<
>
<
+28.0000
+24.0000
+3.00000
+4.00000
+2.00000
|------------------------------------------------------------------------------|
|
Summarized Report for 051402I
Page : 1
|
|------------------------------------------------------------------------------|
|
|
|
|Opportunity| Objective | Minimum | Maximum |
|Number | Variable | Solution |
Cost
|Coefficient|Obj. Coeff.|Obj. Coeff.|
|-------+----------+-----------+-----------+-----------+-----------+-----------|
| 1
| X1
| +2.0000000| 0
| +50.000000| - Infinity| +350.00000|
| 2
| X2
| +7.0000000| 0
| +100.00000| +14.285713| + Infinity|
|------------------------------------------------------------------------------|
|
Minimized OBJ = 800 Iteration = 4 Elapsed CPU second = 0
|
|------------------------------------------------------------------------------|
|------------------------------------------------------------------------------|
|
Summarized Report for 051402I
Page : 2
|
|------------------------------------------------------------------------------|
|
|
|
|
Shadow | Slack or | Minimum | Maximum |
|Constr.| Status |
RHS
|
Price
| Surplus |
RHS
|
RHS
|
|-------+---------+------------+-----------+-----------+-----------+-----------|
| 1
| Tight | >+28.000000| +50.000000| 0
| +22.000000| + Infinity|
| 2
| Loose | >+24.000000| 0
| +64.000000| - Infinity| +88.000000|
| 3
| Loose | <+3.0000000| 0
| +1.0000000| +2.0000000| + Infinity|
| 4
| Loose | >+4.0000000| 0
| +3.0000000| - Infinity| +7.0000000|
| 5
| Tight | <+2.0000000| -300.00000| 0
| 0
| +2.8571429|
|------------------------------------------------------------------------------|
|
Minimized OBJ = 800 Iteration = 4 Elapsed CPU second = 0
|
|------------------------------------------------------------------------------|
9



Quant Input & Output, P9 (J):
Input Data of The Problem 051402J
Min +50.0000X1
Subject to
(1) +7.00000X1
(2) +2.00000X1
(3) +1.00000X1
(4) +0
X1
(5) +1.00000X1
Page: 1
+100.000X2
+2.00000X2
+12.0000X2
+0
X2
+1.00000X2
+0
X2
>
>
<
>
>
+28.0000
+24.0000
+3.00000
+4.00000
+3.00000
|------------------------------------------------------------------------------|
|
Summarized Report for 051402J
Page : 1
|
|------------------------------------------------------------------------------|
|
|
|
|Opportunity| Objective | Minimum | Maximum |
|Number | Variable | Solution |
Cost
|Coefficient|Obj. Coeff.|Obj. Coeff.|
|-------+----------+-----------+-----------+-----------+-----------+-----------|
| 1
| X1
| +3.0000000| 0
| +50.000000| - Infinity| + Infinity|
| 2
| X2
| +4.0000000| 0
| +100.00000| 0
| + Infinity|
|------------------------------------------------------------------------------|
|
Minimized OBJ = 550 Iteration = 5 Elapsed CPU second = 0
|
|------------------------------------------------------------------------------|
|------------------------------------------------------------------------------|
|
Summarized Report for 051402J
Page : 2
|
|------------------------------------------------------------------------------|
|
|
|
|
Shadow | Slack or | Minimum | Maximum |
|Constr.| Status |
RHS
|
Price
| Surplus |
RHS
|
RHS
|
|-------+---------+------------+-----------+-----------+-----------+-----------|
| 1
| Loose | >+28.000000| 0
| +1.0000000| - Infinity| +29.000000|
| 2
| Loose | >+24.000000| 0
| +30.000000| - Infinity| +54.000000|
| 3
| Tight | <+3.0000000| +50.000000| 0
| +2.8571429| +3.0000000|
| 4
| Tight | >+4.0000000| +100.00000| 0
| +3.5000000| + Infinity|
| 5
| Tight | >+3.0000000| 0
| 0
| +3.0000000| + Infinity|
|------------------------------------------------------------------------------|
|
Minimized OBJ = 550 Iteration = 5 Elapsed CPU second = 0
|
|------------------------------------------------------------------------------|
10


11

Branch-and-Bound Integer Programming Tree:
P1 (B)
Z = 320.00
LB = 320.00
x1 = 3.59
x2 = 1.40
X1 ≤ 3
X1 ≥ 4
P2 (C)
Z = 500.00
X1 = 3.00
X2 = 3.50
X2 ≤ 3
P3 (D)
Z = 333.33
X1 = 4.00
X2 = 1.33
X2 ≥ 4
P4 (E)
No
Feasible
Solution
X2 ≤ 1
P5 (F)
Z = 542.86
X1 = 2.86
X2 = 4.00
X1 ≤ 2
P8 (I)
Z = 800
X1 = 2.00
X2 = 7.00
P6 (G)
Z = 400
X1 = 6.00
X2 = 1.00
X2 ≥ 2
P7 (H)
Z = 400
X1 = 4.00
X2 = 2.00
X1 ≥ 3
P9 (J)
Z = 550
X1 = 3.00
X2 = 4.00
C. Comments on the solutions:
Candidate solutions to this problem are located within the darkened circles. The final
solutions to this problem are located in circles G and H above. These are optimum
solutions because both coefficients in each circle are integers and the objective function
is least compared to the other circles. As well, circles G and H are optimal because, due
to the nature of a minimization tree, the objective function only increases as the tree
grows. This means that there will be no possibility of a more optimal solution further
down the tree.



12
The linear programming results of this problem indicate that Dorian should buy 3.6
comedy commercials and 1.4 football commercials for a minimum cost of $320,000.
Since Dorian can’t buy a fraction of a commercial, the divisibility assumption is violated
and integer programming is recommended to solve this problem. The solution from
integer programming suggests that Dorian buy 6 comedy commercials and 1 football
commercial or 4 comedy commercials and 2 football commercials for a minimum cost of
$400,000. This is 25% higher than the cost obtained from the LP solution.