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an advertising budget of $18,200 is
z 1,052,000 Maximum weekly audience
and this occurs when x1 4, x2 10, and x3 14. The results are summarized as follows.
Number of
Media Advertisements Cost Audience
Television 4 $8000 400,000
Newspaper 10 $6000 400,000
Radio 14 $4200 252,000
Total 28 $18,200 1,052,000
SECTION 9.3 Q' EXERCISES
In Exercises 1 4, write the simplex tableau for the linear program- 7. (Maximize) 8. (Maximize)
ming problem. You do not need to solve the problem. (In each case Objective function: Objective function:
the objective function is to be maximized.)
z x1 x2 z x1 x2
1. Objective function: 2. Objective function:
Constraints: Constraints:
z x1 2x2 z x1 3x2 x1 x2 3x3 d" 5 x1 x2 e" 4
Constraints: Constraints: 2x1 2x3 e" 1 2x1 x2 e" 6
2x1 x2 d" 8 x1 x2 d" 4 x2 x3 d" 0 x1, x2 e" 0
x1 x2 d" 5 x1 x2 d" 1 x1, x2, x3 e" 0
x1, x2 e" 0 x1, x2 e" 0
In Exercises 9 20, use the simplex method to solve the linear pro-
3. Objective function: 4. Objective function: gramming problem. (In each case the objective function is to be
maximized.)
z 2x1 3x2 4x3 z 6x1 9x2
9. Objective function: 10. Objective function:
Constraints: Constraints:
z x1 2x2 z x1 x2
x1 2x2 d" 12 2x1 3x2 d" 6
Constraints: Constraints:
x1 x3 d" 8 x1 x2 d" 20
x1 4x2 d" 8 x1 2x2 d" 6
x1, x2, x3 e" 0 x1, x2 e" 0
x1 x2 d" 12 3x1 2x2 d" 12
In Exercises 5 8, explain why the linear programming problem is
x1, x2 e" 0 x1, x2 e" 0
not in standard form.
11. Objective function: 12. Objective function:
5. (Minimize) 6. (Maximize)
z 5x1 2x2 8x3 z x1 x2 2x3
Objective function: Objective function:
Constraints: Constraints:
z x1 x2 z x1 x2
2x1 4x2 x3 d" 42 2x1 2x2 d" 8
Constraints: Constraints:
2x1 3x2 x3 d" 42 x3 d" 5
x1 2x2 d" 4 x1 2x2 d" 6
6x1 x2 3x3 d" 42 x1, x2, x3 e" 0
x1, x2, e" 0 2x1 x2 d" 1
x1, x2, x3 e" 0
x1, x2 e" 0
544 CHAPTER 9 LINEAR PROGRAMMING
13. Objective function: 14. Objective function: 22. A fruit grower has 150 acres of land available to raise two
crops, A and B. It takes one day to trim an acre of crop A and
z 4x1 5x2 z x1 2x2
two days to trim an acre of crop B, and there are 240 days per
Constraints: Constraints:
year available for trimming. It takes 0.3 day to pick an acre of
x1 x2 d" 10 x1 3x2 d" 15
crop A and 0.1 day to pick an acre of crop B, and there are 30
3x1 7x2 d" 42 2x1 x2 d" 12
days per year available for picking. Find the number of acres
of each fruit that should be planted to maximize profit, assum-
x1, x2 e" 0 x1, x2 e" 0
ing that the profit is $140 per acre for crop A and $235 per
15. Objective function: 16. Objective function:
acre for B. (See Exercise 22 in Section 9.2.)
z 3x1 4x2 x3 7x4 z x1
23. A grower has 50 acres of land for which she plans to raise
Constraints: Constraints: three crops. It costs $200 to produce an acre of carrots and
8x1 3x2 4x3 x4 d" 7 3x1 2x2 d" 60 the profit is $60 per acre. It costs $80 to produce an acre of
celery and the profit is $20 per acre. Finally, it costs $140 to
2x1 6x2 x3 5x4 d" 3 x1 2x2 d" 28
produce an acre of lettuce and the profit is $30 per acre. Use
x1 4x2 5x3 2x4 d" 8 x1 4x2 d" 48
the simplex method to find the number of acres of each crop
x1, x2, x3, x4 e" 0 x1, x2 e" 0
she should plant in order to maximize her profit. Assume that
her cost cannot exceed $10,000.
17. Objective function: 18. Objective function:
24. A fruit juice company makes two special drinks by blending
z x1 x2 x3 z 2x1 x2 3x3
apple and pineapple juices. The first drink uses 30% apple
Constraints: Constraints:
juice and 70% pineapple juice, while the second drink uses
2x1 x2 3x3 d" 40 x1 x2 x3 d" 59
60% apple juice and 40% pineapple juice. There are 1000
x1 x3 d" 25 2x1 3x3 d" 75 liters of apple juice and 1500 liters of pineapple juice avail-
able. If the profit for the first drink is $.60 per liter and that
2x2 3x3 d" 32 x2 6x3 d" 54
for the second drink is $.50, use the simplex method to find
x1, x2, x3 e" 0 x1, x2, x3 e" 0
the number of liters of each drink that should be produced in
order to maximize the profit.
19. Objective function:
25. A manufacturer produces three models of bicycles. The time
z x1 2x2 x4
(in hours) required for assembling, painting, and packaging
Constraints:
each model is as follows.
x1 2x2 3x3 d" 24
3x2 7x3 x4 d" 42
Model A Model B Model C
x1, x2, x3, x4 e" 0
Assembling 2 2.5 3
20. Objective function:
Painting 1.5 2 1
z x1 2x2 x3 x4
Packaging 1 0.75 1.25
Constraints:
x1 x2 3x3 4x4 d" 60
The total time available for assembling, painting, and packag-
ing is 4006 hours, 2495 hours, and 1500 hours, respectively.
x2 2x3 5x4 d" 50
The profit per unit for each model is $45 (Model A), $50
2x1 3x2 6x4 d" 72
(Model B), and $55 (Model C). How many of each type
x1, x2, x3, x4 e" 0
should be produced to obtain a maximum profit?
21. A merchant plans to sell two models of home computers at 26. Suppose in Exercise 25 the total time available for assem-
costs of $250 and $400, respectively. The $250 model yields a bling, painting, and packaging is 4000 hours, 2500 hours, and
profit of $45 and the $400 model yields a profit of $50. The 1500 hours, respectively, and that the profit per unit is $48
merchant estimates that the total monthly demand will not (Model A), $50 (Model B), and $52 (Model C). How many of
exceed 250 units. Find the number of units of each model that each type should be produced to obtain a maximum profit?
should be stocked in order to maximize profit. Assume that
the merchant does not want to invest more than $70,000 in
computer inventory. (See Exercise 21 in Section 9.2.)
SECTION 9.3 EXERCISES 545
27. A company has budgeted a maximum of $600,000 for adver- In the simplex method, it may happen that in selecting the depart-
tising a certain product nationally. Each minute of television ing variable all the calculated ratios are negative. This indicates an
time costs $60,000 and each one-page newspaper ad costs unbounded solution. Demonstrate this in Exercises 33 and 34.
$15,000. Each television ad is expected to be viewed by 15
33. (Maximize) 34. (Maximize)
million viewers, and each newspaper ad is expected to be seen
Objective function: Objective function:
by 3 million readers. The company s market research
department advises the company to use at most 90% of the
z x1 2x2 z x1 3x2
advertising budget on television ads. How should the
Constraints: Constraints:
advertising budget be allocated to maximize the total audi-
x1 3x2 d" 1 x1 x2 d" 20
ence?
x1 2x2 d" 4 2x1 x2 d" 50
28. Rework Exercise 27 assuming that each one-page newspaper
ad costs $30,000. x1, x2 e" 0 x1, x2 e" 0
29. An investor has up to $250,000 to invest in three types of in-
If the simplex method terminates and one or more variables not in
vestments. Type A pays 8% annually and has a risk factor of
the final basis have bottom-row entries of zero, bringing these
0. Type B pays 10% annually and has a risk factor of 0.06.
variables into the basis will determine other optimal solutions.
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