Problems

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1.

Following is a simplex tableau for a linear programming model.

c j	Basic Variables	Quantity	2	10	6
			x 1	x 2	x 3	s 1	s 2	s 3
2	x 1	10	1		2	1	1/2
10	x 2	40		1	2		1/2
	s 3	30			8	3	3/2	1
	z j	420	2	10	16	2	4
	c j z j				10	2	4

1. What is the solution given in this tableau?

2. Is the solution in this tableau optimal? Why?

3. What does x ₃ equal in this tableau? s ₂ ?

4. Write out the original objective function for the linear programming model, using only decision variables.

5. How many constraints are in the linear programming model?

6. 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.

c j	Basic Variables	Quantity	6	20	12
			x 1	x 2	x 3	s 1	s 2
6	x 1	20	1	1
12	x 3	10		1/3	1		1/6
	s 1	10		1/3		1	1/6
	z j	240	6	10	12		2
	z j c j			10			2

1. Is this a maximization or a minimization problem? Why?

2. What is the solution given in this tableau?

3. Is the solution given in this tableau optimal? Why?

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4. Write out the original objective function for the linear programming model using only decision variables.

5. How many constraints are in the linear programming model?

6. Were any of the constraints originally equations? Why?

7. What is the value of x ₂ in this tableau?

3.

The following is a simplex tableau for a linear programming problem.

c j	Basic Variables	Quantity	60	50	45	50
			x 1	x 2	x 3	x 4	s 1	s 2	s 3	s 4
	s 1	20		1			1
50	x 4	15				1		1
60	x 1	12	1	1/2					1/10
	s 4	45			8	6		6		1
	z j	1,470	60	30		50		50	6
	c j z j			20	45			50	6

1. Is this a maximization or a minimization problem?

2. What are the values of the decision variables in this tableau?

3. What are the values of the slack variables in this tableau?

4. What does the c _j z _j value of "20" in the " x ₂ " column mean?

5. 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.

c j	Basic Variables	Quantity	8	10	4				M	M
			x 1	x 2	x 3	s 1	s 2	s 3	A 1	A 2
M	A 1	30	2/3		1	1	1/6
10	x 2	10	1/3	1			1/6		1
M	A 3	100			1			1		1
	z j	130 M + 100	2 M /3 + 10/3	10	2 M	M	M /6 5/3	M	M	M
	z j c j		2 M /3 14/3		2 M 4	M	M /6 5/3	M

1. Is this a maximization or a minimization problem?

2. What is the value of x ₃ in this tableau?

3. What does the value " M /6 5/3" in the " s ₂ " column of the z _j c _j row mean?

4. What is the minimum number of additional simplex iterations that this problem must go through to determine a feasible optimal solution?

5. Is this solution optimal? Why? If the solution is not optimal, compute the optimal solution.

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5.

Given is the following simplex tableau for a linear programming problem.

c j	Basic Variables	Quantity	4	6			M
			x 1	x 2	s 1	s 2	A 1
M	A 1	2		1/2	1	1/2	1
4	x 1	6	1	1/2		1/2
	z j	24 + 2 M	4	M /2 + 2	M	M /2 2	M
	z j c j			M /2 4	M	M /2 2

1. Is this a maximization or a minimization problem? Why?

2. What are the values of the decision variables in this tableau?

3. Were any of the constraints in this problem originally equations? Why?

4. What is the value of s ₂ in this tableau?

5. 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.

c j	Basic Variables	Quantity	10	5			M
			x 1	x 2	s 1	s 2	A 2
10	x 1	5	1	1/2	1/2
M	A 2	4		1			1
	s 2	15		7/2	1/2	1
	z j	4 M + 50	10	M + 5	5		M
	c j z j			M	5

1. Is this a maximization or a minimization problem? Why?

2. What is the value of x ₂ in this tableau?

3. Does the fact that x ₁ has a c _j z _j value equal to "0" in this tableau mean that multiple optimal solutions exist? Why?

4. What does the c _j z _j value for the s ₁ column mean?

5. 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 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.

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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 ( x ₁ and x ₂ ) is

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.

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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 ( x ₁ ), telephone interviews at night ( x ₂ ), personal interviews during the day ( x ₃ ), and personal interviews at night ( x ₄ ) that should be conducted to minimize cost.

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 ( x ₁ and x ₂ ) that it needs to make to maximize profit.

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 ( x ₁ and x ₂ ) to maximize profits:

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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 ( x ₁ and x ₂ ) to maximize profits:

Solve this model using the simplex method.

19.

Solve the following linear programming model using the simplex method.

20.

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

c j	Basic Variables	Quantity	1	2	1
			x 1	x 2	x 3	s 1	s 2	s 3
2	x 2	10		1	1/4	1/4
	s 2	20			3/4	3/4	1	1/2
1	x 1	10	1		1	1/2		1/2
	z j	30	1	2	3/2			1/2
	c j z j				5/2			1/2

1. Is this a maximization or a minimization problem? Why?

2. What is the solution given in this tableau?

3. Write out the original objective function for the linear programming model, using only decision variables.

4. How many constraints are in the linear programming model?

5. Were any of the constraints originally equations? Why?

6. What does s ₁ equal in this tableau?

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7. This solution is optimal. Are there multiple optimal solutions? Why?

8. 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 ( x ₁ ), spruce ( x ₂ ), and ponderosa pines ( x ₃ ) to plant to minimize cost.

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 ( x ₁ ) and milk ( x ₂ ) to include in each jar of baby food to meet the requirements for protein and iron at the minimum cost.

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 cookieschocolate 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 pecan chip cookies ( x ₁ ), chocolate chip cookies ( x ₂ ), and pecan sandies ( x ₃ ) to make to maximize profit.

Solve this model using the simplex method.

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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 ( x ₁ ), bolts ( x ₂ ), and washers ( x ₃ ) to produce to maximize profit.

Solve this model using the simplex method.

26.

Solve the following linear programming model using the simplex method.

27.

Solve the following linear programming model using the simplex method.

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.

30.

Transform the following linear programming model into proper form for solution by the simplex method.

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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 ( x ₁ ) and product 2 ( x ₂ ) to produce to maximize profit.

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 ( x ₁ ) and jackets ( x ₂ ) to make each week to maximize profit.

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 ( x ₁ ) and oak ( x ₂ ) to produce to minimize cost.

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Solve this model using the simplex method.

34.

Solve the following linear programming model using the simplex method.

35.

Solve the following linear programming problem using the simplex method.

36.

Solve the following linear programming problem using the simplex method.

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 ( x ₁ ), peanuts ( x ₂ ), and cotton (x ₃ ) to plant to maximize profit.

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Solve this model using the simplex method.

38.

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

39.

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

40.

Solve the following linear programming model using the simplex method.

41.

Solve the following linear programming model using the simplex method.

42.

The Old English Metal Crafters Company makes brass trays and buckets. The number of trays ( x ₁ ) and buckets ( x ₂ ) that can be produced daily is constrained by the availability of brass and labor, as reflected in the following linear programming model.

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The final optimal simplex tableau for this model is as follows .

c j	Basic Variables	Quantity	6	10
			x 1	x 2	s 1	s 2
10	x 2	20		1	1/3	1/6
6	x 1	10	1		1/3	2/3
	z j	260	6	10	4/3	7/3
	c j z j				4/3	7/3

1. Formulate the dual of this model.

2. Define the dual variables and indicate their value.

3. Determine the optimal ranges for c ₁ and c ₂ .

4. Determine the feasible ranges for q ₁ (pounds of brass) and q ₂ (labor hours).

5. 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 chiliRazorback and Longhornfrom several ingredients, including chili beans and ground beef. The number of 100-gallon batches of Razorback chili ( x ₁ ) and Longhorn chili ( x ₂ ) that can be produced daily is constrained by the availability of chili beans and ground beef, as shown in the following linear programming model.

The final optimal simplex tableau for this model is as follows.

c j	Basic Variables	Quantity	200	300
			x 1	x 2	s 1	s 2
300	x 2	6		1	17/750	1/150
200	x 1	20	1		1/75	1/30
	z j	5,800	200	300	310/75	70/15
	c j z j				310/75	70/15

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2. Formulate the dual of this model and indicate what the dual variables equal.

3. What profit for Razorback chili will result in no Longhorn chili being produced? What will the new optimal solution values be?

4. 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.

5. Determine the optimal ranges for c ₁ and c ₂ .

6. Determine the feasible ranges for q ₁ (pounds of beans) and q ₂ (pounds of ground beef).

7. 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?

8. If the company could secure an additional 100 pounds of only one of the ingredients, beans or ground beef, which should it be?

9. 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 spreadersregular and cyclone. Each spreader must undergo two production processes. Letting x ₁ the number of regular spreaders produced and x ₂ the number of cyclone spreaders produced, the problem can be formulated as follows.

The final optimal simplex tableau for this problem is as follows.

	Basic Variables	Quantity	9	7
c j			x 1	x 2	s 1	s 2
9	x 1	4	1		1/10	1/20
7	x 2	3		1	1/20	3/20
	z j	57	9	7	11/20	12/20
	c j z j				11/20	12/20

1. Formulate the dual for this problem.

2. Define the dual variables and indicate their values.

3. Determine the optimal ranges for c ₁ and c ₂ .

4. Determine the feasible ranges for q ₁ and q ₂ (production hours for processes ₁ and 2, respectively).

5. 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?

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45.

The Stratford House Furniture Company makes two kinds of tablesend tables ( x ₁ ) and coffee tables ( x ₂ ). The manufacturer is restricted by material and labor constraints, as shown in the following linear programming formulation.

The final optimal simplex tableau for this problem is as follows.

	Basic Variables	Quantity	200	300
c j			x 1	x 2	s 1	s 2
300	x 2	30		1	1/3	2/9
200	x 1	15	1		1/3	5/9
	z j	12,000	200	300	100/3	400/9
	c j z j				100/3	400/9

1. Formulate the dual for this problem.

2. Define the dual variables and indicate their values.

3. What profit for coffee tables will result in no end tables being produced, and what will the new optimal solution values be?

4. What will be the effect on the optimal solution if the available wood is increased from 135 to 165 board feet?

5. Determine the optimal ranges for c ₁ and c ₂ .

6. Determine the feasible ranges for q ₁ (labor hours) and q ₂ (board feet of wood).

7. 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?

8. 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 ( x ₁ ) and vacuum cleaner motors ( x ₂ ) to produce has been formulated as follows.

The final optimal simplex tableau for this model is as follows.

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c j	Basic Variables	Quantity	70	80
			x 1	x 2	s 1	s 2	s 3
70	x 1	6	1		2/3		1/3
	s 2	1			1/3	1	1/3
80	x 2	7		1	1/3		2/3
	z j	980	70	80	20		30
	c j z j				20		30

1. Formulate the dual for this problem.

2. What do the dual variables equal, and what do they mean?

3. Determine the optimal ranges for c ₁ and c ₂ .

4. Determine the feasible ranges for q ₁ (production hours), q ₂ (pounds of steel), and q ₃ (feet of wire).

5. 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 ( x ₁ ) and the number of product 2 to product ( x ₂ ) is given as follows.

The final optimal simplex tableau for this model is as follows.

c j	Basic Variables	Quantity	9	12
			x 1	x 2	s 1	s 2	s 3	s 4	s 5
9	x 1	4	1		1/4	2/5
	s 5	1			1/4	1/5			1
	s 3	12			7/4	22/5	1
	s 4	3			1/4	2/5		1
12	x 2	6		1	1/4	1/5
	z j	108	9	12	3/4	6/5
	c j z j				3/4	6/5

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2. Formulate the dual for this problem.

3. What do the dual variables equal, and what does this dual solution mean?

4. Determine the optimal ranges for c ₁ and c ₂ .

5. Determine the range for q ₁ (process 1, labor hr).

6. 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( x ₁ ), product 2( x ₂ ), and product 3( x ₃ ) to produce.

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The final optimal simplex tableau for this model is as follows.

c j	Basic Variables	Quantity	40	35	45
			x 1	x 2	x 3	s 1	s 2	s 3	s 4
	s 1	10	1/2			1		1/2	4
	s 2	60	2				1	1	5
45	x 3	10	1/2		1			1/2	1
35	x 2	30	1/2	1				1/2	2
	z j	1,500	40	35	45			5	25
	c j z j							5	25

1. Formulate the dual for this problem.

2. What do the dual variables equal, and what do they mean?

3. How does the fact that this is a multiple optimum solution affect the interpretation of the dual solution values?

4. Determine the optimal range for c ₂ .

5. Determine the feasible range for q ₄ (square feet of storage space).

6. 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 ( x ₁ ) and ounces of potatoes ( x ₂ ) to put in a lunch.

The final optimal simplex tableau for this model is as follows.

c j	Basic Variables	Quantity	0.03	0.02
			x 1	x 2	s 1	s 2	s 3
0.02	x 2	3.6		1		0.20	0.80
0.03	x 1	1.6	1			0.13	0.20
	s 1	4.4			1	0.47	3.2
	z j	0.12	0.03	0.02			0.01
	z j c j						0.01

1. Formulate the dual for this problem. b. What do the dual variables equal, and what do they mean?

2. Determine the optimal ranges for c ₁ and c ₂ .

3. Determine the ranges for q ₁ , q ₂ , and q ₃ (milligrams of protein, iron, and carbohydrates, respectively).

4. 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 ( x ₁ ), a ham salad sandwich ( x ₂ ), and a pimento cheese sandwich ( x ₃ ). 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.

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The optimal simplex tableau follows.

c j	Basic Variables	Quantity	1.25	2.00	1.75
			x 1	x 2	x 3	s 1	s 2	s 3
	s 1	200	0.3			1	0.8	2
1.75	x 3	500	1		1		1	1
2.00	x 2	500		1				1
	z j	1,875	1.75	2.00	1.75		1.75	3.75
	c j z j		.5				1.75	3.75

1. Formulate the dual for this problem and define the dual variables.

2. Determine the optimal ranges for c ₁ , c ₂ , and c ₃ .

3. Determine the range for q ₃ (ham salad sandwiches).

4. 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,

and its optimal simplex tableau,

c j	Basic Variables	Quantity	3	5	2
			x 1	x 2	x 3	s 1	s 2	s 3
	s 2	15			4	3/2	1	1/2
3	x 1	5	1		2	1/2		1/2
5	x 2	30		1	1	1/2		1/2
	z j	165	3	5	11	4		1
	z j c j				13	4		1

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2. Find the optimal ranges for all c _j values.

3. Find the feasible ranges for all q _i values.

52.

The Sunshine Food Processing Company produces three canned fruit productsmixed fruit ( x ₁ ), fruit cocktail ( x ₂ ), and fruit delight ( x ₃ ). The main ingredients in each product are pears and peaches. Each product is produced in lots and must undergo three processesmixing, canning, and packaging. The resource requirements for each product and each process are shown in the following linear programming formulation.

The optimal simplex tableau is as follows.

c j	Basic Variables	Quantity	10	6	8
			x 1	x 2	x 3	s 1	s 2	s 3	s 4	s 5
10	x 1	8	1		8/15	1/15	1/30
6	x 2	16		1	8/15	1/30	1/15
	s 3	3			2/5		1/10	1
	s 4	36			1/15	1/30	1/30		1
	s 5	8			3/5	1/0				1
	z j	176	10	6	128/15	7/15	1/15
	c j z j				8/15	7/15	1/15

1. What is the maximum price the company would be willing to pay for additional pears? How much could be purchased at that price?

2. What is the marginal value of peaches? Over what range is this price valid?

3. 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?

4. 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?

5. 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 ( x ₁ ), Old English ( x ₂ ), and Colonial ( x ₃ ). Each sheet must be cut and pressed. The resource requirements are given in the following linear programming formulation.

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The optimal simplex tableau is as follows.

c j	Basic Variables	Quantity	4	10	8
			x 1	x 2	x 3	s 1	s 2	s 3	s 4
	s 1	80	7/3			1		4/3	2/3
	s 2	70	1/3				1	1/3	4/3
8	x 3	20	1/3		1			2/3	1/6
10	x 2	10	1/3	1				1/3	1/3
	z j	260	6	10	8			2	2
	c j z j		2					2	2

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

2. What is the marginal value of an additional hour of cutting? Over what range is this value valid?

3. Given a choice between securing more cutting hours or more pressing hours, which should management select? Why?

4. If the amount of spruce available to the firm were decreased from 160 to 100 pounds, would this reduction affect the solution?

5. What unit profit would have to be made from Western paneling before management would consider producing it?

6. 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 ( x ₁ ), product 2 ( x ₂ ), product 3 ( x ₃ ), and product 4 ( x ₄ ) to produce is as follows.

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[Page A-63]

The optimal simplex tableau is as follows.

c j	Basic Variables	Quantity	2	8	10	6
			x 1	x 2	x 3	x 4	s 1	s 2
6	x 4	80	1		2	1	2/3	1/3
8	x 2	40		1			1/3	2/3
	z j	800	6	8	12	6	4/3	10/3
	c j z j		4		2		4/3	10/3

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