A-42
Module A The Simplex Solution Method
The value 360 can be eliminated, because q2 cannot exceed 240. Thus, the range over
which the basic solution variables will remain the same is
180 q2 240
The range for q3 is
192 q3 The upper limit of means that q3 can increase indefinitely (without limit) without
changing the optimal variable solution mix in the shadow price.
Sensitivity analysis of constraint quantity values can be used in conjunction with the
dual solution to make decisions regarding model resources. Recall from our analysis of the
dual solution of the Hickory Furniture Company example that
y1 $20, marginal value of labor
y2 $6.67, marginal value of wood
y3 $0, marginal value of storage space
The shadow prices are only valid
with the sensitivity range for the
right-hand-side values.
Because the resource with the greatest marginal value is labor, the manager might desire
to secure some additional hours of labor. How many hours should the manager get? Given
that the range for q1 is 32 q1 48, the manager could secure up to an additional 8 hours
of labor (i.e., 48 total hours) before the solution basis changes and the shadow price also
changes. If the manager did purchase 8 more hours, the solution values could be found by
observing the quantity values in Table A-37.
x2 8 2
x1 4 2
s3 48 6
Since 8,
x2 8 (8)/2
12
x1 4 (8)/2
0
s3 48 6(8)
96
Total profit will be increased by $20 for each extra hour of labor.
Z $2,240 20
2,240 20(8)
2,240 160
$2,400
In this example for the Hickory Furniture Company, we considered only constraints
in determining the sensitivity ranges for qi values. To compute the qi sensitivity range, we
observed the slack column, si, since a change in qi was reflected in the si column.
However, recall that with a constraint we subtract a surplus variable rather than adding
a slack variable to form an equality (in addition to adding an artificial variable). Thus, for
a constraint we must consider a change in qi in order to use the si (surplus) column
to perform sensitivity analysis. In that case sensitivity analysis would be performed exactly
Problems
A-43
as shown in this example, except that the value of qi would be used instead of qi when computing the sensitivity range for qi.
Problems
1. Following is a simplex tableau for a linear programming model.
cj
2
10
0
10
2
6
0
0
0
Quantity
x1
x2
x3
s1
s2
s3
x1
x2
s3
10
40
30
0
1
0
1
0
0
2
2
8
1
0
3
12
12
32
0
0
1
zj
420
10
2
16
2
4
0
0
0
10
2
4
0
Basic
Variables
cj zj
a.
b.
c.
d.
What is the solution given in this tableau?
Is the solution in this tableau optimal? Why?
What does x3 equal in this tableau? s2?
Write out the original objective function for the linear programming model, using only decision
variables.
e. How many constraints are in the linear programming model?
f. Explain briefly why it would have been difficult to solve this problem graphically.
2. The following is a simplex tableau for a linear programming model.
cj
6
12
0
6
20
12
0
0
Quantity
x1
x2
x3
s1
s2
x1
x3
s1
20
10
10
1
0
0
1
13
13
0
1
0
0
0
1
0
16
16
zj
240
6
10
12
0
2
0
10
0
0
2
Basic
Variables
zj cj
a. Is this a maximization or a minimization problem? Why?
b. What is the solution given in this tableau?
c. Is the solution given in this tableau optimal? Why?
A-44
Module A The Simplex Solution Method
d. Write out the original objective function for the linear programming model using only decision
variables.
e. How many constraints are in the linear programming model?
f. Were any of the constraints originally equations? Why?
g. What is the value of x2 in this tableau?
3. The following is a simplex tableau for a linear programming problem.
Basic
Variables
cj
0
50
60
0
Quantity
50
45
50
0
0
0
0
x1
x2
x3
x4
s1
s2
s3
s4
s1
x4
x1
s4
20
15
12
45
0
0
1
0
1
0
12
0
0
0
0
8
0
1
0
6
1
0
0
0
0
1
0
6
0
0
110
0
0
0
0
1
zj
1,470
60
30
0
50
0
50
6
0
0
20
45
0
0
50
6
0
cj zj
a.
b.
c.
d.
e.
60
Is this a maximization or a minimization problem?
What are the values of the decision variables in this tableau?
What are the values of the slack variables in this tableau?
What does the cj zj value of “20” in the “x2” column mean?
Is this solution optimal? Why? If the solution is not optimal, determine the optimal solution.
4. The following is a simplex tableau for a linear programming problem.
cj
M
10
M
Basic
Variables
A1
x2
A3
zj
zj cj
a.
b.
c.
d.
Quantity
8
10
4
0
0
0
M
M
x1
x2
x3
s1
s2
s3
A1
A2
30
10
100
23
13
0
0
1
0
1
0
1
1
0
0
16
16
0
0
0
1
0
1
0
0
0
1
130M 100
2M3 103
10
2M
M
M6 53
M
M
M
2M3 143
0
2M 4
M
M6 53
M
0
0
Is this a maximization or a minimization problem?
What is the value of x3 in this tableau?
What does the value “M6 53” in the “s2” column of the zj cj row mean?
What is the minimum number of additional simplex iterations that this problem must go
through to determine a feasible optimal solution?
e. Is this solution optimal? Why? If the solution is not optimal, compute the optimal solution.
Problems
A-45
5. Given is the following simplex tableau for a linear programming problem.
Basic
Variables
cj
M
4
A1
x1
zj
4
6
0
0
M
Quantity
x1
x2
s1
s2
A1
2
6
0
1
12
12
1
0
12
12
1
0
24 2M
4
M2 2
M
M2 2
M
0
M2 4
M
M2 2
0
zj cj
a.
b.
c.
d.
e.
Is this a maximization or a minimization problem? Why?
What are the values of the decision variables in this tableau?
Were any of the constraints in this problem originally equations? Why?
What is the value of s2 in this tableau?
Is this solution optimal? Why? If the solution is not optimal, complete the next iteration
(tableau) and indicate if it is optimal.
6. Following is a simplex tableau for a linear programming problem.
cj
10
M
0
Basic
Variables
x1
A2
s2
zj
cj zj
10
5
0
0
M
Quantity
x1
x2
s1
s2
A2
5
4
15
1
0
0
12
1
72
12
0
12
0
0
1
0
1
0
4M 50
10
M 5
5
0
M
0
M
5
0
0
a. Is this a maximization or a minimization problem? Why?
b. What is the value of x2 in this tableau?
c. Does the fact that x1 has a cj zj value equal to “0” in this tableau mean that multiple optimal
solutions exist? Why?
d. What does the cj zj value for the s1 column mean?
e. Is this solution optimal? Why? If not, solve this problem and indicate if multiple optimal solutions exist.
7. The Munchies Cereal Company makes a cereal from several ingredients. Two of the ingredients,
oats and rice, provide vitamins A and B. The company wants to know how many ounces of oats and
rice it should include in each box of cereal to meet the minimum requirements of 48 milligrams of
vitamin A and 12 milligrams of vitamin B while minimizing cost. An ounce of oats contributes
8 milligrams of vitamin A and 1 milligram of vitamin B, whereas an ounce of rice contributes
A-46
Module A The Simplex Solution Method
6 milligrams of vitamin A and 2 milligrams of vitamin B. An ounce of oats costs $0.05, and an
ounce of rice costs $0.03. Formulate a linear programming model for this problem and solve using
the simplex method.
8. A company makes product 1 and product 2 from two resources. The linear programming model
for determining the amounts of product 1 and 2 to produce (x1 and x2) is
maximize Z 8x1 2x2 (profit, $)
subject to
4x1 5x2 20 (resource 1, lb)
2x1 6x2 18 (resource 2, lb)
x1, x2 0
Solve this model using the simplex method.
9. A company produces two products that are processed on two assembly lines. Assembly line 1 has
100 available hours, and assembly line 2 has 42 available hours. Each product requires 10 hours of
processing time on line 1, while on line 2 product 1 requires 7 hours and product 2 requires
3 hours. The profit for product 1 is $6 per unit, and the profit for product 2 is $4 per unit.
Formulate a linear programming model for this problem and solve using the simplex method.
10. The Pinewood Furniture Company produces chairs and tables from two resources — labor and
wood. The company has 80 hours of labor and 36 pounds of wood available each day. Demand for
chairs is limited to 6 per day. Each chair requires 8 hours of labor and 2 pounds of wood to produce, while a table requires 10 hours of labor and 6 pounds of wood. The profit derived from each
chair is $400 and from each table, $100. The company wants to determine the number of chairs and
tables to produce each day to maximize profit. Formulate a linear programming model for this
problem and solve using the simplex method.
11. The Crumb and Custard Bakery makes both coffee cakes and Danish in large pans. The main ingredients are flour and sugar. There are 25 pounds of flour and 16 pounds of sugar available and the
demand for coffee cakes is 8. Five pounds of flour and 2 pounds of sugar are required to make one
pan of coffee cake, and 5 pounds of flour and 4 pounds of sugar are required to make one pan of
Danish. One pan of coffee cake has a profit of $1, and one pan of Danish has a profit of $5.
Determine the number of pans of cake and Danish that the bakery must produce each day so that
profit will be maximized. Formulate a linear programming model for this problem and solve using
the simplex method.
12. The Kalo Fertilizer Company makes a fertilizer using two chemicals that provide nitrogen, phosphate, and potassium. A pound of ingredient 1 contributes 10 ounces of nitrogen and 6 ounces of
phosphate, whereas a pound of ingredient 2 contributes 2 ounces of nitrogen, 6 ounces of phosphate, and 1 ounce of potassium. Ingredient 1 costs $3 per pound, and ingredient 2 costs $5 per
pound. The company wants to know how many pounds of each chemical ingredient to put into a
bag of fertilizer to meet minimum requirements of 20 ounces of nitrogen, 36 ounces of phosphate,
and 2 ounces of potassium while minimizing cost. Formulate a linear programming model for this
problem and solve using the simplex method.
13. Solve the following model using the simplex method.
minimize Z 0.06x1 0.10x2
subject to
Problems
A-47
4x1 3x2 12
3x1 6x2 12
5x1 2x2 10
x1, x2 0
14. The Copperfield Mining Company owns two mines, both of which produce three grades of ore —
high, medium, and low. The company has a contract to supply a smelting company with at least 12
tons of high-grade ore, 8 tons of medium-grade ore, and 24 tons of low-grade ore. Each mine produces a certain amount of each type of ore each hour it is in operation. Mine 1 produces 6 tons of
high-grade, 2 tons of medium-grade, and 4 tons of low-grade ore per hour. Mine 2 produces 2 tons
of high-grade, 2 tons of medium-grade, and 12 tons of low-grade ore per hour. It costs $200 per
hour to mine each ton of ore from mine 1, and it costs $160 per hour to mine a ton of ore from
mine 2. The company wants to determine the number of hours it needs to operate each mine so
that contractual obligations can be met at the lowest cost. Formulate a linear programming model
for this problem and solve using the simplex method.
15. A marketing firm has contracted to do a survey on a political issue for a Spokane television station.
The firm conducts interviews during the day and at night, by telephone and in person. Each hour
an interviewer works at each type of interview results in an average number of interviews. In order
to have a representative survey, the firm has determined that there must be at least 400 day interviews, 100 personal interviews, and 1,200 interviews overall. The company has developed the
following linear programming model to determine the number of hours of telephone interviews
during the day (x1), telephone interviews at night (x2), personal interviews during the day (x3), and
personal interviews at night (x4) that should be conducted to minimize cost.
minimize Z 2x1 3x2 5x3 7x4 (cost, $)
subject to
10x1 4x3 400 (day interviews)
4x3 5x4 100 (personal interviews)
x1 x2 x3 x4 1,200 (total interviews)
x1, x2, x3, x4 0
Solve this model using the simplex method.
16. A jewelry store makes both necklaces and bracelets from gold and platinum. The store has developed the following linear programming model for determining the number of necklaces and
bracelets (x1 and x2) that it needs to make to maximize profit.
maximize Z 300x1 400x2 (profit, $)
subject to
3x1 2x2 18 (gold, oz)
2x1 4x2 20 (platinum, oz)
x2 4 (demand, bracelets)
x1, x2 0
Solve this model using the simplex method.
17. A sporting goods company makes baseballs and softballs on a daily basis from leather and yarn.
The company has developed the following linear programming model for determining the number
of baseballs and softballs to produce (x1 and x2) to maximize profits:
A-48
Module A The Simplex Solution Method
maximize Z 5x1 4x2 (profit, $)
subject to
0.3x1 0.5x2 150 (leather, ft2)
10x1 4x2 2,000 (yarn, yd)
x1, x2 0
Solve this model using the simplex method.
18. A clothing shop makes suits and blazers. Three main resources are used: material, rack space, and
labor. The shop has developed this linear programming model for determining the number of suits
and blazers to make (x1 and x2) to maximize profits:
maximize Z 100x1 150x2 (profit, $)
subject to
10x1 4x2 160 (material, yd2)
x1 x2 20 (rack space)
10x1 20x2 (labor, hr)
x1, x2 0
Solve this model using the simplex method.
19. Solve the following linear programming model using the simplex method.
maximize Z 100x1 20x2 60x3
subject to
3x1 5x2 60
2x1 2x2 2x3 100
x3 40
x1, x2, x3 0
20. The following is a simplex tableau for a linear programming model.
cj
2
0
1
1
2
1
0
0
0
Quantity
x1
x2
x3
s1
s2
s3
x2
s2
x1
10
20
10
0
0
1
1
0
0
14
34
1
14
34
12
0
1
0
0
12
12
zj
30
1
2
32
0
0
12
0
0
52
0
0
12
Basic
Variables
cj zj
a. Is this a maximization or a minimization problem? Why?
b. What is the solution given in this tableau?
c. Write out the original objective function for the linear programming model, using only decision
variables.
d. How many constraints are in the linear programming model?
e. Were any of the constraints originally equations? Why?
f. What does s1 equal in this tableau?
Problems
A-49
g. This solution is optimal. Are there multiple optimal solutions? Why?
h. If there are multiple optimal solutions, identify the alternate solutions.
21. A wood products firm in Oregon plants three types of trees — white pines, spruce, and ponderosa
pines — to produce pulp for paper products and wood for lumber. The company wants to plant
enough acres of each type of tree to produce at least 27 tons of pulp and 30 tons of lumber. The
company has developed the following linear programming model to determine the number of
acres of white pines (x1), spruce (x2), and ponderosa pines (x3) to plant to minimize cost.
minimize Z 120x1 40x2 240x3 (cost, $)
subject to
4x1 x2 3x3 27 (pulp, tons)
2x1 6x2 3x3 30 (lumber, tons)
x1, x2, x3 0
Solve this model using the simplex method.
22. A baby products firm produces a strained baby food containing liver and milk, each of which contribute protein and iron to the baby food. Each jar of baby food must have 36 milligrams of protein
and 50 milligrams of iron. The company has developed the following linear programming model to
determine the number of ounces of liver (x1) and milk (x2) to include in each jar of baby food to
meet the requirements for protein and iron at the minimum cost.
minimize Z 0.05x1 0.10x2 (cost, $)
subject to
6x1 2x2 36 (protein, mg)
5x1 5x2 50 (iron, mg)
x1, x2 0
Solve this model using the simplex method.
23. Solve the linear programming model in problem 22 graphically, and identify the points on the
graph that correspond to each simplex tableau.
24. The Cookie Monster Store at South Acres Mall makes three types of cookies — chocolate chip,
pecan chip, and pecan sandies. Three primary ingredients are chocolate chips, pecans, and sugar.
The store has 120 pounds of chocolate chips, 40 pounds of pecans, and 300 pounds of sugar. The
following linear programming model has been developed for determining the number of batches
of chocolate chip cookies (x1), pecan chip cookies (x2), and pecan sandies (x3) to make to maximize
profit.
maximize Z 10x1 12x2 7x3 (profit, $)
subject to
20x1 15x2 10x3 300 (sugar, lb)
10x1 5x2 120 (chocolate chips, lb)
x1 2x3 40 (pecans, lb)
x1, x2, x3 0
Solve this model using the simplex method.
A-50
Module A The Simplex Solution Method
25. The Eastern Iron and Steel Company makes nails, bolts, and washers from leftover steel and coats
them with zinc. The company has 24 tons of steel and 30 tons of zinc. The following linear programming model has been developed for determining the number of batches of nails (x1), bolts
(x2), and washers (x3) to produce to maximize profit.
maximize Z 6x1 2x2 12x3 (profit, $1,000s)
subject to
4x1 x2 3x3 24 (steel, tons)
2x1 6x2 3x3 30 (zinc, tons)
x1, x2, x3 0
Solve this model using the simplex method.
26. Solve the following linear programming model using the simplex method.
maximize Z 100x1 75x2 90x3 95x4
subject to
3x1 2x2 40
4x3 x4 25
200x1 250x3 2,000
100x1 200x4 2,200
x1, x2, x3, x4 0
27. Solve the following linear programming model using the simplex method.
minimize Z 20x1 16x2
subject to
3x1 x2 6
x1 x2 4
2x1 6x2 12
x1, x2 0
28. Solve the linear programming model in problem 27 graphically, and identify the points on the
graph that correspond to each simplex tableau.
29. Transform the following linear programming model into proper form for solution by the simplex
method.
minimize Z 8x1 2x2 7x3
subject to
2x1 6x2 x3 30
3x2 4x3 60
4x1 x2 2x3 50
x1 2x2 20
x1, x2, x3 0
30. Transform the following linear programming model into proper form for solution by the simplex
method.
Problems
A-51
minimize Z 40x1 55x2 30x3
subject to
x1 2x2 3x3 60
2x1 x2 x3 40
x1 3x2 x3 50
5x2 3x3 100
x1, x2, x3 0
31. A manufacturing firm produces two products using labor and material. The company has a
contract to produce 5 of product 1 and 12 of product 2. The company has developed the following
linear programming model to determine the number of units of product 1 (x1) and product 2 (x2)
to produce to maximize profit.
maximize Z 40x1 60x2 (profit, $)
subject to
x1 2x2 30 (material, lb)
4x1 4x2 72 (labor, hr)
x1 5 (contract, product 1)
x2 12 (contract, product 2)
x1, x2 0
Solve this model using the simplex method.
32. A custom tailor makes pants and jackets from imported Irish wool cloth. To get any cloth at all, the
tailor must purchase at least 25 square feet each week. Each pair of pants and each jacket requires
5 square feet of material. The tailor has 16 hours available each week to make pants and jackets. The
demand for pants is never more than 5 pairs per week. The tailor has developed the following linear programming model to determine the number of pants (x1) and jackets (x2) to make each week
to maximize profit.
maximize Z x1 5x2 (profit, $100s)
subject to
5x1 5x2 25 (wool, ft2)
2x1 4x2 16 (labor, hr)
x1 5 (demand, pants)
x1, x2 0
Solve this model using the simplex method.
33. A sawmill in Tennessee produces cherry and oak boards for a large furniture manufacturer. Each
month the sawmill must deliver at least 5 tons of wood to the manufacturer. It takes the sawmill
3 days to produce a ton of cherry and 2 days to produce a ton of oak, and the sawmill can allocate
18 days out of a month for this contract. The sawmill can get enough cherry to make 4 tons of
wood and enough oak to make 7 tons of wood. The sawmill owner has developed the following linear programming model to determine the number of tons of cherry (x1) and oak (x2) to produce to
minimize cost.
A-52
Module A The Simplex Solution Method
minimize Z 3x1 6x2 (cost, $)
subject to
3x1 2x2 18 (production time, days)
x1 x2 5 (contract, tons)
x1 4 (cherry, tons)
x2 7 (oak, tons)
x1, x2 0
Solve this model using the simplex method.
34. Solve the following linear programming model using the simplex method.
maximize Z 10x1 5x2
subject to
2x1 x2 10
x2 4
x1 4x2 20
x1, x2 0
35. Solve the following linear programming problem using the simplex method.
maximize Z x1 2x2 x3
subject to
4x2 x3 40
x1 x2 20
2x1 4x2 3x3 60
x1, x2, x3 0
36. Solve the following linear programming problem using the simplex method.
maximize Z x1 2x2 2x3
subject to
x1 x2 2x3 12
2x1 x2 5x3 20
x1 x2 x3 8
x1, x2, x3 0
37. A farmer has a 40-acre farm in Georgia. The farmer is trying to determine how many acres of corn,
peanuts, and cotton to plant. Each crop requires labor, fertilizer, and insecticide. The farmer has
developed the following linear programming model to determine the number of acres of corn (x1),
peanuts (x2), and cotton (x3) to plant to maximize profit.
maximize Z 400x1 350x2 450x3 (profit, $)
subject to
2x1 3x2 2x3 120 (labor, hr)
4x1 3x2 x3 160 (fertilizer, tons)
Problems
A-53
3x1 2x2 4x3 100 (insecticide, tons)
x1 x2 x3 40 (acres)
x1, x2, x3 0
Solve this model using the simplex method.
38. Solve the following linear programming model (a) graphically and (b) using the simplex method.
maximize Z 3x1 2x2
subject to
x1 x2 1
x 1 x2 2
x1, x2 0
39. Solve the following linear programming model (a) graphically and (b) using the simplex method.
maximize Z x1 x2
subject to
x1 x2 1
x1 2x2 4
x1, x2 0
40. Solve the following linear programming model using the simplex method.
maximize Z 7x1 5x2 5x3
subject to
x1 x2 x3 25
2x1 x2 x3 40
x1 x2 25
x3 6
x1, x2, x3 0
41. Solve the following linear programming model using the simplex method.
minimize Z 15x1 25x2
subject to
3x1 4x2 12
2x1 x2 6
3x1 2x2 9
x1, x2 0
42. The Old English Metal Crafters Company makes brass trays and buckets. The number of trays (x1)
and buckets (x2) that can be produced daily is constrained by the availability of brass and labor, as
reflected in the following linear programming model.
A-54
Module A The Simplex Solution Method
maximize Z 6x1 10x2 (profit, $)
subject to
x1 4x2 90 (brass, lb)
2x1 2x2 60 (labor, hr)
x1, x2 0
The final optimal simplex tableau for this model is as follows.
6
10
0
0
Quantity
x1
x2
s1
s2
x2
x1
20
10
0
1
1
0
13
13
16
23
zj
260
6
10
43
73
0
0
43
73
Basic
Variables
cj
10
6
cj zj
a.
b.
c.
d.
e.
Formulate the dual of this model.
Define the dual variables and indicate their value.
Determine the optimal ranges for c1 and c2.
Determine the feasible ranges for q1 (pounds of brass) and q2 (labor hours).
What is the maximum price the company would be willing to pay for additional labor hours,
and how many hours could be purchased at that price?
43. The Southwest Foods Company produces two brands of chili — Razorback and Longhorn — from
several ingredients, including chili beans and ground beef. The number of 100-gallon batches of
Razorback chili (x1) and Longhorn chili (x2) that can be produced daily is constrained by the availability of chili beans and ground beef, as shown in the following linear programming model.
maximize Z 200x1 300x2 (profit, $)
subject to
10x1 50x2 500 (chili beans, lb)
34x1 20x2 800 (ground beef, lb)
x1, x2 0
The final optimal simplex tableau for this model is as follows.
cj
300
200
Basic
Variables
Quantity
200
300
0
0
x1
x2
s1
s2
x2
x1
6
20
0
1
1
0
17750
175
1150
130
zj
5,800
200
300
31075
7015
0
0
31075
7015
cj zj
Problems
A-55
a. Formulate the dual of this model and indicate what the dual variables equal.
b. What profit for Razorback chili will result in no Longhorn chili being produced? What will the
new optimal solution values be?
c. Determine what the effect will be of changing the amount of beans in Razorback chili from
10 pounds per batch to 15 pounds per batch.
d. Determine the optimal ranges for c1 and c2.
e. Determine the feasible ranges for q1 (pounds of beans) and q2 (pounds of ground beef).
f. What is the maximum price the company would be willing to pay for additional pounds of chili
beans, and how many pounds could be purchased at that price?
g. If the company could secure an additional 100 pounds of only one of the ingredients, beans or
ground beef, which should it be?
h. If the company changed the selling price of Longhorn chili so that the profit was $400 instead of
$300, would the optimal solution be affected?
44. The Agrimaster Company produces two kinds of fertilizer spreaders — regular and cyclone. Each
spreader must undergo two production processes. Letting x1 the number of regular spreaders
produced and x2 the number of cyclone spreaders produced, the problem can be formulated as
follows.
maximize Z 9x1 7x2 (profit, $)
subject to
12x1 4x2 60 (process 1, production hr)
4x1 8x2 40 (process 2, production hr)
x1, x2 0
The final optimal simplex tableau for this problem is as follows.
cj
9
7
9
7
0
0
Quantity
x1
x2
s1
s2
x1
x2
4
3
1
0
0
1
110
120
120
320
zj
57
9
7
1120
1220
0
0
1120
1220
Basic
Variables
cj zj
a.
b.
c.
d.
Formulate the dual for this problem.
Define the dual variables and indicate their values.
Determine the optimal ranges for c1 and c2.
Determine the feasible ranges for q1 and q2 (production hours for processes 1 and 2, respectively).
e. What is the maximum price the Agrimaster Company would be willing to pay for additional hours of process 1 production time, and how many hours could be purchased at
that price?
A-56
Module A The Simplex Solution Method
45. The Stratford House Furniture Company makes two kinds of tables — end tables (x1) and coffee
tables (x2). The manufacturer is restricted by material and labor constraints, as shown in the
following linear programming formulation.
maximize Z 200x1 300x2 (profit, $)
subject to
2x1 5x2 180 (labor, hr)
3x1 3x2 135 (wood, bd ft)
x1, x2 0
The final optimal simplex tableau for this problem is as follows.
cj
300
200
Basic
Variables
Quantity
200
300
0
0
x1
x2
s1
s2
x2
x1
30
15
0
1
1
0
13
13
29
59
zj
12,000
200
300
1003
4009
0
0
1003
4009
cj zj
a. Formulate the dual for this problem.
b. Define the dual variables and indicate their values.
c. What profit for coffee tables will result in no end tables being produced, and what will the new
optimal solution values be?
d. What will be the effect on the optimal solution if the available wood is increased from 135 to
165 board feet?
e. Determine the optimal ranges for c1 and c2.
f. Determine the feasible ranges for q1 (labor hours) and q2 (board feet of wood).
g. What is the maximum price the Stratford House Furniture Company would be willing to pay
for additional wood, and how many board feet of wood could be purchased at that price?
h. If the furniture company wanted to secure additional units of only one of the resources, labor or
wood, which should it be?
46. A manufacturing firm produces electric motors for washing machines and vacuum cleaners. The
firm has resource constraints for production time, steel, and wire. The linear programming model
for determining the number of washing machine motors (x1) and vacuum cleaner motors (x2) to
produce has been formulated as follows.
maximize Z 70x1 80x2 (profit, $)
subject to
2x1 x2 19 (production, hr)
x1 x2 14 (steel, lb)
x1 2x2 20 (wire, ft)
x1, x2 0
The final optimal simplex tableau for this model is as follows.
Problems
cj
70
0
80
Basic
Variables
Quantity
70
80
0
0
0
x1
x2
s1
s2
s3
x1
s2
x2
6
1
7
1
0
0
0
0
1
23
13
13
0
1
0
13
13
23
zj
980
70
80
20
0
30
0
0
20
0
30
cj zj
A-57
a.
b.
c.
d.
Formulate the dual for this problem.
What do the dual variables equal, and what do they mean?
Determine the optimal ranges for c1 and c2.
Determine the feasible ranges for q1 (production hours), q2 (pounds of steel), and q3 (feet of
wire).
e. Managers at the firm have determined that the firm can purchase a new production machine
that will increase available production time from 19 to 25 hours. Would this change affect the
optimal solution?
47. A manufacturer produces products 1 and 2, for which profits are $9 and $12, respectively. Each
product must undergo two production processes that have labor constraints. There are also
material constraints and storage limitations. The linear programming model for determining
the number of product 1 to produce (x1) and the number of product 2 to product (x2) is given as
follows.
maximize Z 9x1 12x2 (profit, $)
subject to
4x1 8x2 64 (process 1, labor hr)
5x1 5x2 50 (process 2, labor hr)
15x1 8x2 120 (material, lb)
x1 7 (storage space, ft2)
x2 7 (storage space, ft2)
x1, x2 0
The final optimal simplex tableau for this model is as follows.
cj
9
0
0
0
12
Basic
Variables
Quantity
9
12
0
0
0
0
0
x1
x2
s1
s2
s3
s4
s5
x1
s5
s3
s4
x2
4
1
12
3
6
1
0
0
0
0
0
0
0
0
1
14
14
74
14
14
25
15
225
25
15
0
0
1
0
0
0
0
0
1
0
0
1
0
0
0
zj
108
9
12
34
65
0
0
0
0
0
34
65
0
0
0
cj zj
A-58
Module A The Simplex Solution Method
a.
b.
c.
d.
e.
Formulate the dual for this problem.
What do the dual variables equal, and what does this dual solution mean?
Determine the optimal ranges for c1 and c2.
Determine the range for q1 (process 1, labor hr).
Due to a problem with a supplier, only 100 pounds of material will be available for production
instead of 120 pounds. Will this affect the optimal solution mix?
48. A manufacturer produces products 1, 2, and 3 daily. The three products are processed through
three production operations that have time constraints, and the finished products are then stored.
The following linear programming model has been formulated to determine the number of product 1(x1), product 2(x2), and product 3(x3) to produce.
maximize Z 40x1 35x2 45x3 (profit, $)
subject to
2x1 3x2 2x3 120 (operation 1, hr)
4x1 3x2 x3 160 (operation 2, hr)
3x1 2x2 4x3 100 (operation 3, hr)
x1 x2 x3 40 (storage, ft2)
x1, x2, x3 0
The final optimal simplex tableau for this model is as follows.
cj
0
0
45
35
Basic
Variables
Quantity
40
35
45
0
0
0
0
x1
x2
x3
s1
s2
s3
s4
s1
s2
x3
x2
10
60
10
30
12
2
12
12
0
0
0
1
0
0
1
0
1
0
0
0
0
1
0
0
12
1
12
12
4
5
1
2
zj
1,500
40
35
45
0
0
5
25
0
0
0
0
0
5
25
cj zj
a. Formulate the dual for this problem.
b. What do the dual variables equal, and what do they mean?
c. How does the fact that this is a multiple optimum solution affect the interpretation of the dual
solution values?
d. Determine the optimal range for c2.
e. Determine the feasible range for q4 (square feet of storage space).
f. What is the maximum price the manufacturer would be willing to pay to lease additional storage space, and how many additional square feet could be leased at that price?
49. A school dietitian is attempting to plan a lunch menu that will minimize cost and meet certain
minimum dietary requirements. The two staples in the meal are meat and potatoes, which provide
protein, iron, and carbohydrates. The following linear programming model has been formulated to
determine how many ounces of meat (x1) and ounces of potatoes (x2) to put in a lunch.
minimize Z 0.03x1 0.02x2 (cost, $)
Problems
A-59
subject to
4x1 5x2 20 (protein, mg)
12x1 3x2 30 (iron, mg)
3x1 2x2 12 (carbohydrates, mg)
x1, x2 0
The final optimal simplex tableau for this model is as follows.
cj
0.02
0.03
0
Basic
Variables
Quantity
0.03
0.02
0
0
0
x1
x2
s1
s2
s3
x2
x1
s1
3.6
1.6
4.4
0
1
0
1
0
0
0
0
1
0.20
0.13
0.47
0.80
0.20
3.2
zj
0.12
0.03
0.02
0
0
0.01
0
0
0
0
0.01
zj cj
a.
b.
c.
d.
Formulate the dual for this problem.
What do the dual variables equal, and what do they mean?
Determine the optimal ranges for c1 and c2.
Determine the ranges for q1, q2, and q3 (milligrams of protein, iron, and carbohydrates, respectively).
e. What would it be worth for the school dietitian to be able to reduce the requirement for carbohydrates, and what is the smallest number of milligrams of carbohydrates that would be
required at that value?
50. The Overnight Food Processing Company prepares sandwiches (among other processed food
items) for vending machines, markets, and business canteens around the city. The sandwiches are
made at night and delivered early the following morning. Any sandwiches not purchased during
the previous day are thrown away. Three kinds of sandwiches are made each night, a basic cheese
sandwich (x1), a ham salad sandwich (x2), and a pimento cheese sandwich (x3). The profits are
$1.25, $2.00, and $1.75, respectively. It takes 0.5 minutes to make a cheese sandwich, 1.2 minutes to
make a ham salad sandwich, and 0.8 minutes to make a pimento cheese sandwich. The company,
has 20 hours of labor available to produce the sandwiches each night. The demand for ham salad
sandwiches is at least as great as the demand for the two types of cheese sandwiches combined.
However, the company has only enough ham salad to produce 500 sandwiches per night. The company has formulated the following linear programming model in order to determine how many of
each type of sandwich to make to maximize profit.
maximize Z $1.25x1 2.00x2 1.75x3
subject to
0.5x1 1.2x2 0.8x3 1,200 (production time, min)
x1 x3 x2 (demand for ham salad sandwiches)
x2 500 (ham salad sandwich limit)
x1, x2, x3 0
A-60
Module A The Simplex Solution Method
The optimal simplex tableau follows.
Basic
Variables
cj
0
1.75
2.00
1.25
2.00
1.75
0
0
0
x1
x2
x3
s1
s2
s3
0.3
1
0
0
0
1
0
1
0
1
0
0
0.8
1
0
2.00
1.75
0
1.75
3.75
0
0
0
1.75
3.75
Quantity
s1
x3
x2
200
500
500
zj
1,875
1.75
cj zj
a.
b.
c.
d.
.5
2
1
1
Formulate the dual for this problem and define the dual variables.
Determine the optimal ranges for c1, c2, and c3.
Determine the range for q3 (ham salad sandwiches).
Overnight Foods is considering advertising its cheese sandwiches to increase demand. The company estimates that spending $100 on some leaflets that would be packaged with all other sandwiches would increase the demand for both kinds of cheese sandwiches by 200. Should it make
this expenditure?
51. Given the linear programming model,
minimize Z 3x1 5x2 2x3
subject to
x1 x2 3x3 35
x1 2x2 50
x1 x2 25
x1, x2, x3 0
and its optimal simplex tableau,
cj
0
3
5
3
5
2
0
0
0
Quantity
x1
x2
x3
s1
s2
s3
s2
x1
x2
15
5
30
0
1
0
0
0
1
4
2
1
32
12
12
1
0
0
12
12
12
zj
165
3
5
11
4
0
1
0
0
13
4
0
1
Basic
Variables
zj cj
Problems
A-61
a. Find the optimal ranges for all cj values.
b. Find the feasible ranges for all qi values.
52. The Sunshine Food Processing Company produces three canned fruit products — mixed fruit (x1),
fruit cocktail (x2), and fruit delight (x3). The main ingredients in each product are pears and
peaches. Each product is produced in lots and must undergo three processes — mixing, canning,
and packaging. The resource requirements for each product and each process are shown in the
following linear programming formulation.
maximize Z 10x1 6x2 8x3 (profit, $)
subject to
20x1 10x2 16x3 320 (pears, lb)
10x1 20x2 16x3 400 (peaches, lb)
x1 2x2 2x3 43 (mixing, hr)
x1 x2 x3 60 (canning, hr)
2x1 x2 x3 40 (packaging, hr)
x1, x2, x3 0
The optimal simplex tableau is as follows.
cj
10
6
0
0
0
Basic
Variables
Quantity
10
6
8
0
0
0
0
0
x1
x2
x3
s1
s2
s3
s4
s5
x1
x2
s3
s4
s5
8
16
3
36
8
1
0
0
0
0
0
1
0
0
0
815
815
25
115
35
115
130
0
130
10
130
115
110
130
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
zj
176
10
6
12815
715
115
0
0
0
0
0
815
715
115
0
0
0
cj zj
a. What is the maximum price the company would be willing to pay for additional pears? How
much could be purchased at that price?
b. What is the marginal value of peaches? Over what range is this price valid?
c. The company can purchase a new mixing machine that would increase the available mixing
time from 43 to 60 hours. Would this affect the optimal solution?
d. The company can also purchase a new packaging machine that would increase the available
packaging time from 40 to 50 hours. Would this affect the optimal solution?
e. If the manager were to attempt to secure additional units of only one of the resources, which
should it be?
53. The Evergreen Products Firm produces three types of pressed paneling from pine and spruce. The
three types of paneling are Western (x1), Old English (x2), and Colonial (x3). Each sheet must be cut
and pressed. The resource requirements are given in the following linear programming formulation.
A-62
Module A The Simplex Solution Method
maximize Z 4x1 10x2 8x3 (profit, $)
subject to
5x1 4x2 4x3 200 (pine, lb)
2x1 5x2 2x3 160 (spruce, lb)
x1 x2 2x3 50 (cutting, hr)
2x1 4x2 2x3 80 (pressing, hr)
x1, x2, x3 0
The optimal simplex tableau is as follows.
cj
0
0
8
10
4
10
8
0
0
0
0
Quantity
x1
x2
x3
s1
s2
s3
s4
s1
s2
x3
x2
80
70
20
10
73
13
13
13
0
0
0
1
0
0
1
0
1
0
0
0
0
1
0
0
43
13
23
13
23
43
16
13
zj
260
6
10
8
0
0
2
2
2
0
0
0
0
2
2
Basic
Variables
cj zj
a. What is the marginal value of an additional pound of spruce? Over what range is this value
valid?
b. What is the marginal value of an additional hour of cutting? Over what range is this value valid?
c. Given a choice between securing more cutting hours or more pressing hours, which should
management select? Why?
d. If the amount of spruce available to the firm were decreased from 160 to 100 pounds, would this
reduction affect the solution?
e. What unit profit would have to be made from Western paneling before management would consider producing it?
f. Management is considering changing the profit of Colonial paneling from $8 to $13. Would this
change affect the solution?
54. A manufacturing firm produces four products. Each product requires material and machine processing. The linear programming model formulated to determine the number of product 1 (x1),
product 2 (x2), product 3 (x3), and product 4 (x4) to produce is as follows.
maximize Z 2x1 8x2 10x3 6x4 (profit, $)
subject to
2x1 x2 4x3 2x4 200 (material, lb)
x1 2x2 2x3 x4 160 (machine processing, hr)
x1, x2, x3, x4 0
Problems
A-63
The optimal simplex tableau is as follows.
cj
6
8
Basic
Variables
Quantity
2
8
10
6
0
0
x1
x2
x3
x4
s1
s2
x4
x2
80
40
1
0
0
1
2
0
1
0
23
13
13
23
zj
800
6
8
12
6
43
103
4
0
2
0
43
103
cjzj
What is the marginal value of an additional pound of material? Over what range is this value valid?