A-42 Module A The Simplex Solution Methodcau.ac.kr/~orist/2005_1/files/module_a2.pdfA-44 Module A...

A-42 Module A The Simplex Solution Method

The value 360 can be eliminated, because q2 cannot exceed 240. Thus, the range overwhich 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) withoutchanging the optimal variable solution mix in the shadow price.

Sensitivity analysis of constraint quantity values can be used in conjunction with thedual solution to make decisions regarding model resources. Recall from our analysis of thedual solution of the Hickory Furniture Company example that

y1 � $20, marginal value of labory2 � $6.67, marginal value of woody3 � $0, marginal value of storage space

Because the resource with the greatest marginal value is labor, the manager might desireto secure some additional hours of labor. How many hours should the manager get? Giventhat the range for q1 is 32 � q1 � 48, the manager could secure up to an additional 8 hoursof labor (i.e., 48 total hours) before the solution basis changes and the shadow price alsochanges. If the manager did purchase 8 more hours, the solution values could be found byobserving the quantity values in Table A-37.

x2 � 8 � ��2x1 � 4 � ��2s3 � 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 � constraintsin determining the sensitivity ranges for qi values. To compute the qi sensitivity range, weobserved 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 addinga slack variable to form an equality (in addition to adding an artificial variable). Thus, fora constraint we must consider a �� change in qi in order to use the si (surplus) columnto perform sensitivity analysis. In that case sensitivity analysis would be performed exactly

�

The shadow prices are only validwith the sensitivity range for the

right-hand-side values.

Problems A-43

Problems

as shown in this example, except that the value of qi � � would be used instead of qi � �when computing the sensitivity range for qi.

1. Following is a simplex tableau for a linear programming model.

Basic10 2 6 0 0 0

cj Variables Quantity x1 x2 x3 s1 s2 s3

2 x1 10 0 1 �2 1 �1�2 010 x2 40 1 0 2 0 1�2 0

0 s3 30 0 0 8 �3 3�2 1

zj 420 10 2 16 2 4 0

cj � zj 0 0 �10 �2 �4 0

a. What is the solution given in this tableau?b. Is the solution in this tableau optimal? Why?c. What does x3 equal in this tableau? s2?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. 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.

Basic6 20 12 0 0

cj Variables Quantity x1 x2 x3 s1 s2

6 x1 20 1 1 0 0 012 x3 10 0 1�3 1 0 �1�60 s1 10 0 1�3 0 1 �1�6

zj 240 6 10 12 0 �2

zj � cj 0 �10 0 0 �2

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?


d. Write out the original objective function for the linear programming model using only decisionvariables.

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.

Basic60 50 45 50 0 0 0 0

cj Variables Quantity x1 x2 x3 x4 s1 s2 s3 s4

0 s1 20 0 1 0 0 1 0 0 050 x4 15 0 0 0 1 0 1 0 060 x1 12 1 1�2 0 0 0 0 1�10 0

0 s4 45 0 0 8 6 0 �6 0 1

zj 1,470 60 30 0 50 0 50 6 0

cj � zj 0 20 45 0 0 �50 �6 0

a. Is this a maximization or a minimization problem?b. What are the values of the decision variables in this tableau?c. What are the values of the slack variables in this tableau?d. What does the cj � zj value of “20” in the “x2” column mean?e. 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.

Basic8 10 4 0 0 0 M M

cj Variables Quantity x1 x2 x3 s1 s2 s3 A1 A2

M A1 30 2�3 0 1 �1 1�6 0 0 010 x2 10 1�3 1 0 0 �1�6 0 1 0M A3 100 0 0 1 0 0 �1 0 1

zj 130M � 100 2M�3 � 10�3 10 2M �M M�6 � 5�3 �M M M

zj � cj 2M�3 � 14�3 0 2M � 4 �M M�6 � 5�3 �M 0 0

a. Is this a maximization or a minimization problem?b. What is the value of x3 in this tableau?c. What does the value “M�6 � 5�3” in the “s2” column of the zj � cj row mean?d. 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.

Basic4 6 0 0 M

cj Variables Quantity x1 x2 s1 s2 A1

M A1 2 0 1�2 �1 1�2 14 x1 6 1 1�2 0 �1�2 0

zj 24 � 2M 4 M�2 � 2 �M M�2 � 2 M

zj � cj 0 M�2 � 4 �M M�2 � 2 0

a. Is this a maximization or a minimization problem? Why?b. What are the values of the decision variables in this tableau?c. Were any of the constraints in this problem originally equations? Why?d. What is the value of s2 in this tableau?e. 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.

Basic10 5 0 0 �M

cj Variables Quantity x1 x2 s1 s2 A2

10 x1 5 1 1�2 �1�2 0 0�M A2 4 0 1 0 0 1

0 s2 15 0 7�2 1�2 1 0

zj �4M � 50 10 �M � 5 �5 0 �M

cj � zj 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 solu-

tions 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 andrice it should include in each box of cereal to meet the minimum requirements of 48 milligrams ofvitamin A and 12 milligrams of vitamin B while minimizing cost. An ounce of oats contributes8 milligrams of vitamin A and 1 milligram of vitamin B, whereas an ounce of rice contributes


6 milligrams of vitamin A and 2 milligrams of vitamin B. An ounce of oats costs $0.05, and anounce of rice costs $0.03. Formulate a linear programming model for this problem and solve usingthe simplex method.

8. A company makes product 1 and product 2 from two resources. The linear programming modelfor 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 has100 available hours, and assembly line 2 has 42 available hours. Each product requires 10 hours ofprocessing time on line 1, while on line 2 product 1 requires 7 hours and product 2 requires3 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 andwood. The company has 80 hours of labor and 36 pounds of wood available each day. Demand forchairs is limited to 6 per day. Each chair requires 8 hours of labor and 2 pounds of wood to pro-duce, while a table requires 10 hours of labor and 6 pounds of wood. The profit derived from eachchair is $400 and from each table, $100. The company wants to determine the number of chairs andtables to produce each day to maximize profit. Formulate a linear programming model for thisproblem and solve using the simplex method.

11. The Crumb and Custard Bakery makes both coffee cakes and Danish in large pans. The main ingre-dients are flour and sugar. There are 25 pounds of flour and 16 pounds of sugar available and thedemand for coffee cakes is 8. Five pounds of flour and 2 pounds of sugar are required to make onepan of coffee cake, and 5 pounds of flour and 4 pounds of sugar are required to make one pan ofDanish. 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 thatprofit will be maximized. Formulate a linear programming model for this problem and solve usingthe simplex method.

12. The Kalo Fertilizer Company makes a fertilizer using two chemicals that provide nitrogen, phos-phate, and potassium. A pound of ingredient 1 contributes 10 ounces of nitrogen and 6 ounces ofphosphate, whereas a pound of ingredient 2 contributes 2 ounces of nitrogen, 6 ounces of phos-phate, and 1 ounce of potassium. Ingredient 1 costs $3 per pound, and ingredient 2 costs $5 perpound. The company wants to know how many pounds of each chemical ingredient to put into abag 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 thisproblem 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 123x1 � 6x2 125x1 � 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 12tons of high-grade ore, 8 tons of medium-grade ore, and 24 tons of low-grade ore. Each mine pro-duces a certain amount of each type of ore each hour it is in operation. Mine 1 produces 6 tons ofhigh-grade, 2 tons of medium-grade, and 4 tons of low-grade ore per hour. Mine 2 produces 2 tonsof high-grade, 2 tons of medium-grade, and 12 tons of low-grade ore per hour. It costs $200 perhour to mine each ton of ore from mine 1, and it costs $160 per hour to mine a ton of ore frommine 2. The company wants to determine the number of hours it needs to operate each mine sothat contractual obligations can be met at the lowest cost. Formulate a linear programming modelfor 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 houran interviewer works at each type of interview results in an average number of interviews. In orderto have a representative survey, the firm has determined that there must be at least 400 day inter-views, 100 personal interviews, and 1,200 interviews overall. The company has developed thefollowing linear programming model to determine the number of hours of telephone interviewsduring the day (x1), telephone interviews at night (x2), personal interviews during the day (x3), andpersonal 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


16. A jewelry store makes both necklaces and bracelets from gold and platinum. The store has devel-oped the following linear programming model for determining the number of necklaces andbracelets (x1 and x2) that it needs to make to maximize profit.


subject to

3x1 � 2x2 � 18 (gold, oz)2x1 � 4x2 � 20 (platinum, oz)

x2 � 4 (demand, bracelets)x1, x2 0


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 numberof baseballs and softballs to produce (x1 and x2) to maximize profits:



subject to

0.3x1 � 0.5x2 � 150 (leather, ft2)10x1 � 4x2 � 2,000 (yarn, yd)

x1, x2 0


18. A clothing shop makes suits and blazers. Three main resources are used: material, rack space, andlabor. The shop has developed this linear programming model for determining the number of suitsand blazers to make (x1 and x2) to maximize profits:


subject to

10x1 � 4x2 � 160 (material, yd2)x1 � x2 � 20 (rack space)

10x1 � 20x2 � (labor, hr)x1, x2 0


19. Solve the following linear programming model using the simplex method.

maximize Z � 100x1 � 20x2 � 60x3

subject to

3x1 � 5x2 � 602x1 � 2x2 � 2x3 � 100

x3 � 40x1, x2, x3 0

20. The following is a simplex tableau for a linear programming model.

Basic1 2 �1 0 0 0


2 x2 10 0 1 1�4 1�4 0 00 s2 20 0 0 �3�4 3�4 1 �1�21 x1 10 1 0 1 �1�2 0 1�2

zj 30 1 2 3�2 0 0 1�2

cj � zj 0 0 �5�2 0 0 �1�2

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 ponderosapines — to produce pulp for paper products and wood for lumber. The company wants to plantenough acres of each type of tree to produce at least 27 tons of pulp and 30 tons of lumber. Thecompany has developed the following linear programming model to determine the number ofacres 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


22. A baby products firm produces a strained baby food containing liver and milk, each of which con-tribute protein and iron to the baby food. Each jar of baby food must have 36 milligrams of proteinand 50 milligrams of iron. The company has developed the following linear programming model todetermine the number of ounces of liver (x1) and milk (x2) to include in each jar of baby food tomeet 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


23. Solve the linear programming model in problem 22 graphically, and identify the points on thegraph 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. Thefollowing linear programming model has been developed for determining the number of batchesof chocolate chip cookies (x1), pecan chip cookies (x2), and pecan sandies (x3) to make to maximizeprofit.

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



25. The Eastern Iron and Steel Company makes nails, bolts, and washers from leftover steel and coatsthem with zinc. The company has 24 tons of steel and 30 tons of zinc. The following linear pro-gramming 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



maximize Z � 100x1 � 75x2 � 90x3 � 95x4

subject to

3x1 � 2x2 � 404x3 � x4 � 25

200x1 � 250x3 � 2,000100x1 � 200x4 � 2,200

x1, x2, x3, x4 0


minimize Z � 20x1 � 16x2

subject to

3x1 � x2 6x1 � x2 4

2x1 � 6x2 12x1, x2 0

28. Solve the linear programming model in problem 27 graphically, and identify the points on thegraph that correspond to each simplex tableau.

29. Transform the following linear programming model into proper form for solution by the simplexmethod.

minimize Z � 8x1 � 2x2 � 7x3

subject to

2x1 � 6x2 � x3 � 303x2 � 4x3 60

4x1 � x2 � 2x3 � 50x1 � 2x2 20x1, x2, x3 0

30. Transform the following linear programming model into proper form for solution by the simplexmethod.

Problems A-51

minimize Z � 40x1 � 55x2 � 30x3

subject to

x1 � 2x2 � 3x3 � 602x1 � x2 � x3 � 40x1 � 3x2 � x3 50

5x2 � 3x3 100x1, x2, x3 0

31. A manufacturing firm produces two products using labor and material. The company has acontract to produce 5 of product 1 and 12 of product 2. The company has developed the followinglinear programming model to determine the number of units of product 1 (x1) and product 2 (x2)to produce to maximize 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


32. A custom tailor makes pants and jackets from imported Irish wool cloth. To get any cloth at all, thetailor must purchase at least 25 square feet each week. Each pair of pants and each jacket requires5 square feet of material. The tailor has 16 hours available each week to make pants and jackets. Thedemand for pants is never more than 5 pairs per week. The tailor has developed the following lin-ear programming model to determine the number of pants (x1) and jackets (x2) to make each weekto 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


33. A sawmill in Tennessee produces cherry and oak boards for a large furniture manufacturer. Eachmonth the sawmill must deliver at least 5 tons of wood to the manufacturer. It takes the sawmill3 days to produce a ton of cherry and 2 days to produce a ton of oak, and the sawmill can allocate18 days out of a month for this contract. The sawmill can get enough cherry to make 4 tons ofwood and enough oak to make 7 tons of wood. The sawmill owner has developed the following lin-ear programming model to determine the number of tons of cherry (x1) and oak (x2) to produce tominimize cost.


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



maximize Z � 10x1 � 5x2

subject to

2x1 � x2 10x2 � 4

x1 � 4x2 � 20x1, x2 0

35. Solve the following linear programming problem using the simplex method.

maximize Z � x1 � 2x2 � x3

subject to

4x2 � x3 � 40x1 � x2 � 20

2x1 � 4x2 � 3x3 � 60x1, x2, x3 0

36. Solve the following linear programming problem using the simplex method.

maximize Z � x1 � 2x2 � 2x3

subject to

x1 � x2 � 2x3 � 122x1 � x2 � 5x3 � 20

x1 � x2 � x3 8x1, 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 hasdeveloped the following linear programming model to determine the number of acres of corn (x1),peanuts (x2), and cotton (x3) to plant to maximize 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


38. Solve the following linear programming model (a) graphically and (b) using the simplex method.

maximize Z � 3x1 � 2x2

subject to

x1 � x2 � 1x1 � 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


maximize Z � 7x1 � 5x2 � 5x3

subject to

x1 � x2 � x3 � 252x1 � x2 � x3 � 40

x1 � x2 � 25x3 � 6

x1, x2, x3 0


minimize Z � 15x1 � 25x2

subject to

3x1 � 4x2 122x1 � x2 6

3x1 � 2x2 � 9x1, 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, asreflected in the following linear programming model.



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.

Basic6 10 0 0

cj Variables Quantity x1 x2 s1 s2

10 x2 20 0 1 1�3 �1�66 x1 10 1 0 �1�3 2�3

zj 260 6 10 4�3 7�3

cj � zj 0 0 �4�3 �7�3

a. Formulate the dual of this model.b. Define the dual variables and indicate their value.c. Determine the optimal ranges for c1 and c2.d. Determine the feasible ranges for q1 (pounds of brass) and q2 (labor hours).e. 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 — fromseveral ingredients, including chili beans and ground beef. The number of 100-gallon batches ofRazorback chili (x1) and Longhorn chili (x2) that can be produced daily is constrained by the avail-ability of chili beans and ground beef, as shown in the following linear programming model.


subject to

10x1 � 50x2 � 500 (chili beans, lb)34x1 � 20x2 � 800 (ground beef, lb)

x1, x2 0


Basic200 300 0 0


300 x2 6 0 1 17�750 �1�150200 x1 20 1 0 �1�75 1�30

zj 5,800 200 300 310�75 70�15

cj � zj 0 0 �310�75 �70�15

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. Eachspreader must undergo two production processes. Letting x1 � the number of regular spreadersproduced and x2 � the number of cyclone spreaders produced, the problem can be formulated asfollows.


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.

Basic9 7 0 0


9 x1 4 1 0 1�10 �1�207 x2 3 0 1 �1�20 3�20

zj 57 9 7 11�20 12�20

cj � zj 0 0 �11�20 �12�20

a. Formulate the dual for this problem.b. Define the dual variables and indicate their values.c. Determine the optimal ranges for c1 and c2.d. Determine the feasible ranges for q1 and q2 (production hours for processes 1 and 2, respect-

ively).e. What is the maximum price the Agrimaster Company would be willing to pay for addi-

tional hours of process 1 production time, and how many hours could be purchased atthat price?


45. The Stratford House Furniture Company makes two kinds of tables — end tables (x1) and coffeetables (x2). The manufacturer is restricted by material and labor constraints, as shown in thefollowing linear programming formulation.


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.

Basic200 300 0 0


300 x2 30 0 1 1�3 �2�9200 x1 15 1 0 �1�3 5�9

zj 12,000 200 300 100�3 400�9

cj � zj 0 0 �100�3 �400�9

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. Thefirm has resource constraints for production time, steel, and wire. The linear programming modelfor determining the number of washing machine motors (x1) and vacuum cleaner motors (x2) toproduce has been formulated as follows.


subject to

2x1 � x2 � 19 (production, hr)x1 � x2 � 14 (steel, lb)

x1 � 2x2 � 20 (wire, ft)x1, x2 0


Problems A-57

Basic70 80 0 0 0

cj Variables Quantity x1 x2 s1 s2 s3

70 x1 6 1 0 2�3 0 �1�30 s2 1 0 0 �1�3 1 �1�3

80 x2 7 0 1 �1�3 0 2�3

zj 980 70 80 20 0 30

cj � zj 0 0 �20 0 �30

a. Formulate the dual for this problem.b. What do the dual variables equal, and what do they mean?c. Determine the optimal ranges for c1 and c2.d. 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 theoptimal solution?

47. A manufacturer produces products 1 and 2, for which profits are $9 and $12, respectively. Eachproduct must undergo two production processes that have labor constraints. There are alsomaterial constraints and storage limitations. The linear programming model for determiningthe number of product 1 to produce (x1) and the number of product 2 to product (x2) is given asfollows.


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


Basic9 12 0 0 0 0 0

cj Variables Quantity x1 x2 s1 s2 s3 s4 s5

9 x1 4 1 0 �1�4 2�5 0 0 00 s5 1 0 0 �1�4 1�5 0 0 10 s3 12 0 0 7�4 �22�5 1 0 00 s4 3 0 0 1�4 �2�5 0 1 0

12 x2 6 0 1 1�4 �1�5 0 0 0

zj 108 9 12 3�4 6�5 0 0 0

cj � zj 0 0 �3�4 �6�5 0 0 0


a. Formulate the dual for this problem.b. What do the dual variables equal, and what does this dual solution mean?c. Determine the optimal ranges for c1 and c2.d. Determine the range for q1 (process 1, labor hr).e. 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 throughthree 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 prod-uct 1(x1), product 2(x2), and product 3(x3) to produce.


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


Basic40 35 45 0 0 0 0

cj Variables Quantity x1 x2 x3 s1 s2 s3 s4

0 s1 10 �1�2 0 0 1 0 1�2 �40 s2 60 2 0 0 0 1 1 �5

45 x3 10 1�2 0 1 0 0 1�2 �135 x2 30 1�2 1 0 0 0 �1�2 2

zj 1,500 40 35 45 0 0 5 25

cj � zj 0 0 0 0 0 �5 �25

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 stor-

age 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 certainminimum dietary requirements. The two staples in the meal are meat and potatoes, which provideprotein, iron, and carbohydrates. The following linear programming model has been formulated todetermine 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


Basic0.03 0.02 0 0 0

cj Variables Quantity x1 x2 s1 s2 s3

0.02 x2 3.6 0 1 0 0.20 �0.800.03 x1 1.6 1 0 0 �0.13 0.200 s1 4.4 0 0 1 0.47 �3.2

zj 0.12 0.03 0.02 0 0 �0.01

zj � cj 0 0 0 0 �0.01

a. Formulate the dual for this problem.b. What do the dual variables equal, and what do they mean?c. Determine the optimal ranges for c1 and c2.d. Determine the ranges for q1, q2, and q3 (milligrams of protein, iron, and carbohydrates, respec-

tively).e. What would it be worth for the school dietitian to be able to reduce the requirement for carbo-

hydrates, and what is the smallest number of milligrams of carbohydrates that would berequired at that value?

50. The Overnight Food Processing Company prepares sandwiches (among other processed fooditems) for vending machines, markets, and business canteens around the city. The sandwiches aremade at night and delivered early the following morning. Any sandwiches not purchased duringthe previous day are thrown away. Three kinds of sandwiches are made each night, a basic cheesesandwich (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 tomake 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 saladsandwiches 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 com-pany has formulated the following linear programming model in order to determine how many ofeach 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


The optimal simplex tableau follows.

Basic 1.25 2.00 1.75 0 0 0


0 s1 200 �0.3 0 0 1 �0.8 �21.75 x3 500 1 0 1 0 1 12.00 x2 500 0 1 0 0 0 1

zj 1,875 1.75 2.00 1.75 0 1.75 3.75

cj � zj �.5 0 0 0 �1.75 �3.75

a. Formulate the dual for this problem and define the dual variables.b. Determine the optimal ranges for c1, c2, and c3.c. Determine the range for q3 (ham salad sandwiches).d. Overnight Foods is considering advertising its cheese sandwiches to increase demand. The com-

pany estimates that spending $100 on some leaflets that would be packaged with all other sand-wiches would increase the demand for both kinds of cheese sandwiches by 200. Should it makethis expenditure?

51. Given the linear programming model,

minimize Z � 3x1 � 5x2 � 2x3

subject to

x1 � x2 � 3x3 35x1 � 2x2 50

�x1 � x2 25x1, x2, x3 0

and its optimal simplex tableau,

Basic3 5 2 0 0 0


0 s2 15 0 0 �4 �3�2 1 �1�23 x1 5 1 0 �2 �1�2 0 1�25 x2 30 0 1 �1 �1�2 0 �1�2

zj 165 3 5 �11 �4 0 �1

zj � cj 0 0 �13 �4 0 �1

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 andpeaches. 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 thefollowing linear programming formulation.


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.

Basic10 6 8 0 0 0 0 0

cj Variables Quantity x1 x2 x3 s1 s2 s3 s4 s5

10 x1 8 1 0 8�15 1�15 1�30 0 0 06 x2 16 0 1 8�15 �1�30 1�15 0 0 00 s3 3 0 0 2�5 0 �1�10 1 0 00 s4 36 0 0 �1�15 �1�30 �1�30 0 1 00 s5 8 0 0 �3�5 �1�0 0 0 0 1

zj 176 10 6 128�15 7�15 1�15 0 0 0

cj � zj 0 0 �8�15 �7�15 �1�15 0 0 0

a. What is the maximum price the company would be willing to pay for additional pears? Howmuch 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. Thethree types of paneling are Western (x1), Old English (x2), and Colonial (x3). Each sheet must be cutand pressed. The resource requirements are given in the following linear programming formulation.



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


Basic4 10 8 0 0 0 0

cj Variables Quantity x1 x2 x3 s1 s2 s3 s4

0 s1 80 7�3 0 0 1 0 �4�3 �2�30 s2 70 �1�3 0 0 0 1 1�3 �4�38 x3 20 1�3 0 1 0 0 2�3 �1�6

10 x2 10 1�3 1 0 0 0 �1�3 1�3

zj 260 6 10 8 0 0 2 2

cj � zj �2 0 0 0 0 �2 �2

a. What is the marginal value of an additional pound of spruce? Over what range is this valuevalid?

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 con-

sider 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 pro-cessing. 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


Basic2 8 10 6 0 0

cj Variables Quantity x1 x2 x3 x4 s1 s2

6 x4 80 1 0 2 1 2�3 �1�38 x2 40 0 1 0 0 �1�3 2�3

zj 800 6 8 12 6 4�3 10�3

cj�zj �4 0 �2 0 �4�3 �10�3

What is the marginal value of an additional pound of material? Over what range is this value valid?

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