1 Million+ Step-by-step solutions

f(x) = x12 – x2, x0 = [1 1]T. Sketch your path, predict the outcome of further steps.

Minimum cost Hard brick, Inc, has two kilns, Kiln I can produce 3000 grey bricks, 2000 red bricks, and 300 glazed bricks daily. For Kiln II the corresponding figures are 2000, 5000 and 1500. Daily operating costs of Kilns I and II are $400 and $ 600, respectively. Find the number of days of operation of each kiln so that the operation cost in filling an order of 18000 grey, 34000 red, and 9000 glazed bricks is minimized.

Nutrition Foods A and B have 600 and 500 calories, contain 15g and 30g of protein, and cost $1.80and $2.10 per unit, respectively, Find the minimum cost diet of at least 3900 calories containing at least 150g of protein.

Do Prob. 1 with the last two constraints interchanged.

Maximize z = 300x1 + 500x2 subject to 2x1 + 8x2 < 60, 2x1 + x2 < 30, 4x1 + 4x2 < 60.

Maximize the total output f = x1 + x2 + x3 (production figures of three different production processes subject to input constraints (limitation of machinetime)

Using an artificial variable, minimize f = 2x1 – x2 subject to x1 > 0, x2 > 0, x1 + x2 > 5 – x1 + x2 < 1, 5x1 + 4x1 < 40.

If one uses the method of artificial variables in a problem without solution, this non-existence will become apparent by the fact that one cannot get rid of the artificial variable. Illustrate this by trying to maximize f = 2x1 + x2 subject to x1 > 0, x2 > 0, 2x1 + x2 < 2, x1 + 2x2 > 6, x1 + x2 < 4.

In Prob. 8 start from x0 = [1.5 1] T, show that the next even-numbered approximations are x2 = kx0, x4 = k2x0, etc., where k = 0.04.

Apply the method of steepest descent to f(x) = 9x12 + x22 + 18x1 – 4x2, 5 steps, starting from x0 = [2 4]T.

Show that the gradients in Prob. 13 are orthogonal. Give a reason.

Maximize f = x1 + x2 subject to x1 + 2x2 < 10, 2x1 + x2 < 10, x2 < 4.

A factory produces two kinds of gaskets, G1, G2, with net profit of $60 and $30, respectively, Maximize the total daily profit subject to the constraints (xj = number of gaskets Gj produced per day) 40x1 + 40x2 < 1800 (Machine hours) 200x1 + 20x2 < 6300 (Labor)

Find the adjacency matrix of the graph in Fig 476.

The graph in Prob. 8, incidence Matrix of a Digraph; Matrix B = [bkj] with entries find the incidence matrixof;

Call the length of a shortest path s → v the distance of v from s. Show that if v has distance l, it has label λ(v) = l.

Show that the length of a shortest postman trail is the same for every starting vertex.

Uniqueness the path connecting any two vertices u and v in a tree is unique.

If a graph has no cycles, it must have at least 2 vertices of degree 1 (definition in Sec. 23.1)

A tree with n vertices has n – 1 edges (Proof by induction).

A graph with n vertices is a tree if and only if it has n – 1 edge and has no cycles.

Complexity show that Prim’s algorithm has complexity O(n2).

For a complete graph (or one that is almost complete), if our data is n n x n distance table (as in Prob. 12, Sec. 23.4) show that the present algorithm [which is O (n2)] cannot easily be replaced by an algorithm of order less than O(n2).

Show that in a network G with all cij = 1, the maximum flow equals the number of edge-disjoint paths s → t.

Show that in a network G with capacities all equal to 1, the capacity of a minimum cut set (S, T) equals the minimum number q of edges whose deletion destroys all directed paths s → t. (A directed path v → w is a path in which each edge has the direction in which it is traversed in going from v to w.)

Three factories 1, 2, 3 are each supplied underground by water, gas, and electricity, from poins A, B, C respectively. Show that this can be represented by K3,3 (the complete bipartite graph G = (S, T; E) with S and T consisting of three vertices each) and that eight of the nine supply lines (edges) can be laid out without crossing. Make it plausible that K3,3 is not planar by attempting to draw the ninth line without crossing the others.

Edge coloring the edge chromatic number Xe (G) of a graph G is the minimum number of colors needed for coloring the edges of G so that incident edges get different colors. Clearly, Xe (G) > max d(u), where d(u) is the degree of vertex u. If G = (S, T; E) is bipartite, the equality sign holds. Prove this for Kn,n.

Edge coloring the edge chromatic number Xe (G) of a graph G is the minimum number of colors needed for coloring the edges of G so that incident edges get different colors. Clearly, Xe (G) > max d(u), where d(u) is the degree of vertex u. If G = (S, T; E) is bipartite, the equality sign holds. Prove this for Kn,n. Discuss.

Which of the following mathematical relationships could be found in a linear programming model, and which could not? For the relationships that are unacceptable for linear programs, state why.

a. –1A + 2B ≤ 70

b. 2A – 2B = 50

c. 1A – 2B2 ≤ 10

d. 3 2 Ā + 2 B ≥ 15

e. 1A + 1B = 6

f. 2A + 5B + 1AB ≤ 25

Find the solutions that satisfy the following constraints:

a. 4A + 2B ≤ 16

b. 4A + 2B ≥ 16

c. 4A + 2B = 16

a. 4A + 2B ≤ 16

b. 4A + 2B ≥ 16

c. 4A + 2B = 16

Show a separate graph of the constraint lines and the solutions that satisfy each of the following constraints:

a. 3A + 2B ≤ 18

b. 12A + 8B ≥ 480

c. 5A + 10B = 200

a. 3A + 2B ≤ 18

b. 12A + 8B ≥ 480

c. 5A + 10B = 200

Show a separate graph of the constraint lines and the solutions that satisfy each of the following constraints:

a. 3A - 4B ≥ 60

b. -6A + 5B ≤ 60

c. 5A - 2B ≤ 0

a. 3A - 4B ≥ 60

b. -6A + 5B ≤ 60

c. 5A - 2B ≤ 0

Show a separate graph of the constraint lines and the solutions that satisfy each of the following constraints:

a. A ≥ 0.25 (A + B)

b. B ≤ 0.10 (A + B)

c. A ≤ 0.50 (A + B)

a. A ≥ 0.25 (A + B)

b. B ≤ 0.10 (A + B)

c. A ≤ 0.50 (A + B)

Three objective functions for linear programming problems are 7A + 10B, 6A + 4B, and -4A + 7B. Show the graph of each for objective function values equal to 420.

Identify the feasible region for the following set of constraints:

0.5A + 0.25B ≥ 30

1A + 5B ≥ 250

0.25A + 0.5B ≤ 50

A, B ≥ 0

0.5A + 0.25B ≥ 30

1A + 5B ≥ 250

0.25A + 0.5B ≤ 50

A, B ≥ 0

Identify the feasible region for the following set of constraints:

2A - 1B ≤ 0

-1A + 1.5B ≤ 200

A, B ≥ 0

2A - 1B ≤ 0

-1A + 1.5B ≤ 200

A, B ≥ 0

Identify the feasible region for the following set of constraints:

3A - 2B ≥ 0

2A - 1B ≤ 200

1A ≤ 150

A, B ≥ 0

3A - 2B ≥ 0

2A - 1B ≤ 200

1A ≤ 150

A, B ≥ 0

For the linear program

Max2A + 3B

s.t.

1A + 3B ≤ 6

5A + 3B ≤ 15

A, B ≥ 0

Find the optimal solution using the graphical solution procedure. What is the value of the objective function at the optimal solution?

Solve the following linear program using the graphical solution procedure:

Max 5A + 5B

s.t.

1A ≤ 100

1B ≤ 80

2A + 4B ≤ 400

A, B ≥ 0

Max 5A + 5B

s.t.

1A ≤ 100

1B ≤ 80

2A + 4B ≤ 400

A, B ≥ 0

Consider the following linear programming problem:

Max 3A + 3B

s.t.

2A + 4B ≤ 12

6A + 4B ≤ 24

A, B ≥ 0

a. Find the optimal solution using the graphical solution procedure.

b. If the objective function is changed to 2A + 6B, what will the optimal solution be?

c. How many extreme points are there? What are the values of A and B at each extreme point?

Max 3A + 3B

s.t.

2A + 4B ≤ 12

6A + 4B ≤ 24

A, B ≥ 0

a. Find the optimal solution using the graphical solution procedure.

b. If the objective function is changed to 2A + 6B, what will the optimal solution be?

c. How many extreme points are there? What are the values of A and B at each extreme point?

Consider the following linear program:

Max 1A + 2B

s.t.

1A ≤ 5

1B ≤ 4

2A + 2B = 12

A, B ≥ 0

a. Show the feasible region.

b. What are the extreme points of the feasible region?

c. Find the optimal solution using the graphical procedure.

Max 1A + 2B

s.t.

1A ≤ 5

1B ≤ 4

2A + 2B = 12

A, B ≥ 0

a. Show the feasible region.

b. What are the extreme points of the feasible region?

c. Find the optimal solution using the graphical procedure.

Par, Inc., is a small manufacturer of golf equipment and supplies. Parâ€™s distributor believes a market exists for both a medium-priced golf bag, referred to as a standard model, and a high-priced golf bag, referred to as a deluxe model. The distributor is so confident of the market that, if Par can make the bags at a competitive price, the distributor will purchase all the bags that Par can manufacture over the next three months. A careful analysis of the manufacturing requirements resulted in the following table, which shows the production time requirements for the four required manufacturing operations and the accounting departmentâ€™s estimate of the profit contribution per bag:

The director of manufacturing estimates that 630 hours of cutting and dyeing time, 600 hours of sewing time, 708 hours of finishing time, and 135 hours of inspection and packaging time will be available for the production of golf bags during the next three months.

a. If the company wants to maximize total profit contribution, how many bags of each model should it manufacture?

b. What profit contribution can Par earn on those production quantities?

c. How many hours of production time will be scheduled for each operation?

d. What is the slack time in eachoperation?

Suppose that Par’s management encounters the following situations:

a. The accounting department revises its estimate of the profit contribution for the deluxe bag to $18 per bag.

b. A new low-cost material is available for the standard bag, and the profit contribution per standard bag can be increased to $20 per bag. (Assume that the profit contribution of the deluxe bag is the original $9 value.)

c. New sewing equipment is available that would increase the sewing operation capacity to 750 hours. (Assume that 10A + 9B is the appropriate objective function.)

If each of these situations is encountered separately, what are the optimal solution and the total profit contribution?

a. The accounting department revises its estimate of the profit contribution for the deluxe bag to $18 per bag.

b. A new low-cost material is available for the standard bag, and the profit contribution per standard bag can be increased to $20 per bag. (Assume that the profit contribution of the deluxe bag is the original $9 value.)

c. New sewing equipment is available that would increase the sewing operation capacity to 750 hours. (Assume that 10A + 9B is the appropriate objective function.)

If each of these situations is encountered separately, what are the optimal solution and the total profit contribution?

Refer to the feasible region for Par, Inc., in Problem 14.

a. Develop an objective function that will make extreme point (0, 540) the optimal extreme point.

b. What is the optimal solution for the objective function you selected in part (a)?

c. What are the values of the slack variables associated with this solution?

Write the following linear program in standard form:

Max5A + 2B

s.t.

1A - 2B ≤ 420

2A + 3B ≤ 610

6A - 1B ≤ 125

A, B ≥ 0

For the linear program

Max4A + 1B

s.t.

10A + 2B ≤ 30

3A + 2B ≤ 12

2A + 2B ≤ 10

A, B ≥ 0

a. Write this problem in standard form.

b. Solve the problem using the graphical solution procedure.

c. What are the values of the three slack variables at the optimal solution?

Given the linear program

Max 3A + 4B

s.t

-1A + 2B ≤ 8

1A + 2B ≤ 12

2A + 1B ≤ 16

A, B ≥ 0

a. Write the problem in standard form.

b. Solve the problem using the graphical solution procedure.

c. What are the values of the three slack variables at the optimal solution?

Max 3A + 4B

s.t

-1A + 2B ≤ 8

1A + 2B ≤ 12

2A + 1B ≤ 16

A, B ≥ 0

a. Write the problem in standard form.

b. Solve the problem using the graphical solution procedure.

c. What are the values of the three slack variables at the optimal solution?

For the linear program

Max3A + 2B

s.t.

A + B ≥ 4

3A + 4B ≤ 24

A ≥ 2

A – B ≤ 0

A, B ≥ 0

a. Write the problem in standard form.

b. Solve the problem.

c. What are the values of the slack and surplus variables at the optimal solution?

Consider the following linear program:

Max 2A + 3B

s.t.

5A + 5B â‰¤ 400 Constraint1

-1A + 1B â‰¤ 10 Constraint 2

1A + 3B â‰¥ 90 Constraint 3

A, B â‰¥ 0

Figure shows a graph of the constraint lines.

FIGURE GRAPH OF THE CONSTRAINT LINES FOR EXERCISE 21

Max 2A + 3B

s.t.

5A + 5B â‰¤ 400 Constraint1

-1A + 1B â‰¤ 10 Constraint 2

1A + 3B â‰¥ 90 Constraint 3

A, B â‰¥ 0

Figure shows a graph of the constraint lines.

FIGURE GRAPH OF THE CONSTRAINT LINES FOR EXERCISE 21

a. Place a number (1, 2, or 3) next to each constraint line to identify which constraint it represents.

b. Shade in the feasible region on the graph.

c. Identify the optimal extreme point. What is the optimal solution?

d. Which constraints are binding? Explain.

e. How much slack or surplus is associated with the nonbindingconstraint?

Reiser Sports Products wants to determine the number of All-Pro (A) and College (C) footballs to produce in order to maximize profit over the next four-week planning horizon. Constraints affecting the production quantities are the production capacities in three departments: cutting and dyeing; sewing; and inspection and packaging. For the four-week planning period, 340 hours of cutting and dyeing time, 420 hours of sewing time, and 200 hours of inspection and packaging time are available. All-Pro footballs provide a profit of $5 per unit and College footballs provide a profit of $4 per unit. The linear programming model with production times expressed in minutes is as follows:

Max5A + 4C

s.t.

12A + 6C â‰¤ 20,400 Cutting and dyeing

9A + 15C â‰¤ 25,200 Sewing

6A + 6C â‰¤ 12,000 Inspection and packaging

A, C â‰¥ 0

A portion of the graphical solution to the Reiser problem is shown in Figure.

FIGURE PORTION OF THE GRAPHICAL SOLUTION FOR EXERCISE 22

a. Shade the feasible region for this problem.

b. Determine the coordinates of each extreme point and the corresponding profit. Which extreme point generates the highest profit?

c. Draw the profit line corresponding to a profit of $4000. Move the profit line as far from the origin as you can in order to determine which extreme point will provide the optimal solution. Compare your answer with the approach you used in part (b).

d. Which constraints are binding? Explain.

e. Suppose that the values of the objective function coefficients are $4 for each All-Pro model produced and $5 for each College model. Use the graphical solution procedure to determine the new optimal solution and the corresponding value ofprofit.

Embassy Motorcycles (EM) manufacturers two lightweight motorcycles designed for easy handling and safety. The EZ-Rider model has a new engine and a low profile that make it easy to balance. The Lady-Sport model is slightly larger, uses a more traditional engine, and is specifically designed to appeal to women riders. Embassy produces the engines for both models at its Des Moines, Iowa, plant. Each EZ-Rider engine requires 6 hours of manufacturing time and each Lady-Sport engine requires 3 hours of manufacturing time. The Des Moines plant has 2100 hours of engine manufacturing time available for the next production period. Embassy’s motorcycle frame supplier can supply as many EZ-Rider frames as needed. However, the Lady-Sport frame is more complex and the supplier can only provide up to 280 Lady-Sport frames for the next production period. Final assembly and testing requires 2 hours for each EZ-Rider model and 2.5 hours for each Lady-Sport model. A maximum of 1000 hours of assembly and testing time are available for the next production period. The company’s accounting department projects a profit contribution of $2400 for each EZ-Rider produced and $1800 for each Lady-Sport produced.

a. Formulate a linear programming model that can be used to determine the number of units of each model that should be produced in order to maximize the total contribution to profit.

b. Solve the problem graphically. What is the optimal solution?

c. Which constraints are binding?

Kelson Sporting Equipment, Inc., makes two different types of baseball gloves: a regular model and a catcherâ€™s model. The firm has 900 hours of production time available in its cutting and sewing department, 300 hours available in its finishing department, and 100 hours available in its packaging and shipping department. The production time requirements and the profit contribution per glove are given in the following table:

Assuming that the company is interested in maximizing the total profit contribution, answer the following:

a. What is the linear programming model for this problem?

b. Find the optimal solution using the graphical solution procedure. How many gloves of each model should Kelson manufacture?

c. What is the total profit contribution Kelson can earn with the given production quantities?

d. How many hours of production time will be scheduled in each department?

e. What is the slack time in eachdepartment?

George Johnson recently inherited a large sum of money; he wants to use a portion of this money to set up a trust fund for his two children. The trust fund has two investment options: (1) a bond fund and (2) a stock fund. The projected returns over the life of the investments are 6% for the bond fund and 10% for the stock fund. Whatever portion of the inheritance he finally decides to commit to the trust fund, he wants to invest at least 30% of that amount in the bond fund. In addition, he wants to select a mix that will enable him to obtain a total return of at least 7.5%.

a. Formulate a linear programming model that can be used to determine the percentage that should be allocated to each of the possible investment alternatives.

b. Solve the problem using the graphical solution procedure.

The Sea Wharf Restaurant would like to determine the best way to allocate a monthly advertising budget of $1000 between newspaper advertising and radio advertising. Management decided that at least 25% of the budget must be spent on each type of media, and that the amount of money spent on local newspaper advertising must be at least twice the amount spent on radio advertising. A marketing consultant developed an index that measures audience exposure per dollar of advertising on a scale from 0 to 100, with higher values implying greater audience exposure. If the value of the index for local newspaper advertising is 50 and the value of the index for spot radio advertising is 80, how should the restaurant allocate its advertising budget in order to maximize the value of total audience exposure?

a. Formulate a linear programming model that can be used to determine how the restaurant should allocate its advertising budget in order to maximize the value of total audience exposure.

b. Solve the problem using the graphical solution procedure.

a. Formulate a linear programming model that can be used to determine how the restaurant should allocate its advertising budget in order to maximize the value of total audience exposure.

b. Solve the problem using the graphical solution procedure.

Blair & Rosen, Inc. (B&R) is a brokerage firm that specializes in investment portfolios designed to meet the specific risk tolerances of its clients. A client who contacted B&R this past week has a maximum of $50,000 to invest. B&R’s investment advisor decides to recommend a portfolio consisting of two investment funds: an Internet fund and a Blue Chip fund. The Internet fund has a projected annual return of 12%, while the Blue Chip fund has a projected annual return of 9%. The investment advisor requires that at most $35,000 of the client’s funds should be invested in the Internet fund. B&R services include a risk rating for each investment alternative. The Internet fund, which is the more risky of the two investment alternatives, has a risk rating of 6 per thousand dollars invested. The Blue Chip fund has a risk rating of 4 per thousand dollars invested. For example, if $10,000 is invested in each of the two investment funds, B&R’s risk rating for the portfolio would be 6(10) + 4(10) = 100. Finally, B&R developed a questionnaire to measure each client’s risk tolerance. Based on the responses, each client is classified as a conservative, moderate, or aggressive investor. Suppose that the questionnaire results classified the current client as a moderate investor. B&R recommends that a client who is a moderate investor limit his or her portfolio to a maximum risk rating of 240.

a. What is the recommended investment portfolio for this client? What is the annual return for the portfolio?

b. Suppose that a second client with $50,000 to invest has been classified as an aggressive investor. B&R recommends that the maximum portfolio risk rating for an aggressive investor is 320. What is the recommended investment portfolio for this aggressive investor? Discuss what happens to the portfolio under the aggressive investor strategy.

c. Suppose that a third client with $50,000 to invest has been classified as a conservative investor. B&R recommends that the maximum portfolio risk rating for a conservative investor is 160. Develop the recommended investment portfolio for the conservative investor. Discuss the interpretation of the slack variable for the total investment fund constraint.

Tom’s, Inc., produces various Mexican food products and sells them to Western Foods, a chain of grocery stores located in Texas and New Mexico. Tom’s, Inc., makes two salsa products: Western Foods Salsa and Mexico City Salsa. Essentially, the two products have different blends of whole tomatoes, tomato sauce, and tomato paste. The Western Foods Salsa is a blend of 50% whole tomatoes, 30% tomato sauce, and 20% tomato paste. The Mexico City Salsa, which has a thicker and chunkier consistency, consists of 70% whole tomatoes, 10% tomato sauce, and 20% tomato paste. Each jar of salsa produced weighs 10 ounces. For the current production period, Tom’s, Inc., can purchase up to 280 pounds of whole tomatoes, 130 pounds of tomato sauce, and 100 pounds of tomato paste; the price per pound for these ingredients is $0.96, $0.64, and $0.56, respectively. The cost of the spices and the other ingredients is approximately $0.10 per jar. Tom’s, Inc., buys empty glass jars for $0.02 each, and labeling and filling costs are estimated to be $0.03 for each jar of salsa produced. Tom’s contract with Western Foods results in sales revenue of $1.64 for each jar of Western Foods Salsa and $1.93 for each jar of Mexico City Salsa.

a. Develop a linear programming model that will enable Tom’s to determine the mix of salsa products that will maximize the total profit contribution.

b. Find the optimal solution.

AutoIgnite produces electronic ignition systems for automobiles at a plant in Cleveland, Ohio. Each ignition system is assembled from two components produced at AutoIgnite’s plants in Buffalo, New York, and Dayton, Ohio. The Buffalo plant can produce 2000 units of component 1, 1000 units of component 2, or any combination of the two components each day. For instance, 60% of Buffalo’s production time could be used to produce component 1 and 40% of Buffalo’s production time could be used to produce component 2; in this case, the Buffalo plant would be able to produce 0.6(2000) = 1200 units of component 1 each day and 0.4(1000) = 400 units of component 2 each day. The Dayton plant can produce 600 units of component 1, 1400 units of component 2, or any combination of the two components each day. At the end of each day, the component production at Buffalo and Dayton is sent to Cleveland for assembly of the ignition systems on the following workday.

a. Formulate a linear programming model that can be used to develop a daily production schedule for the Buffalo and Dayton plants that will maximize daily production of ignition systems at Cleveland.

b. Find the optimal solution.

A financial advisor at Diehl Investments identified two companies that are likely candidates for a takeover in the near future. Eastern Cable is a leading manufacturer of flexible cable systems used in the construction industry, and ComSwitch is a new firm specializing in digital switching systems. Eastern Cable is currently trading for $40 per share, and ComSwitch is currently trading for $25 per share. If the takeovers occur, the financial advisor estimates that the price of Eastern Cable will go to $55 per share and ComSwitch will go to $43 per share. At this point in time, the financial advisor has identified Com- Switch as the higher risk alternative. Assume that a client indicated a willingness to invest a maximum of $50,000 in the two companies. The client wants to invest at least $15,000 in Eastern Cable and at least $10,000 in ComSwitch. Because of the higher risk associated with ComSwitch, the financial advisor has recommended that at most $25,000 should be invested in ComSwitch.

a. Formulate a linear programming model that can be used to determine the number of shares of Eastern Cable and the number of shares of ComSwitch that will meet the investment constraints and maximize the total return for the investment.

b. Graph the feasible region.

c. Determine the coordinates of each extreme point.

d. Find the optimal solution.

Consider the following linear program:

Min3A + 4B

s.t.

1A + 3B ≥ 6

1A + 1B ≥ 4

A, B ≥ 0

Identify the feasible region and find the optimal solution using the graphical solution procedure. What is the value of the objective function?

Identify the three extreme-point solutions for the M&D Chemicals problem Identify the value of the objective function and the values of the slack and surplus variables at each extreme point.

Consider the following linear programming problem:

MinA + 2B

s.t.

A + 4B ≤ 21

2A + B ≥ 7

3A + 1.5B ≤ 21

-2A + 6B ≥ 0

A, B ≥ 0

a. Find the optimal solution using the graphical solution procedure and the value of the objective function.

b. Determine the amount of slack or surplus for each constraint.

c. Suppose the objective function is changed to max 5A + 2B. Find the optimal solution and the value of the objective function.

Consider the following linear program:

Min 2A + 2B

s.t.

1A + 3B ≤ 12

3A + 1B ≥ 13

1A - 1B = 3

A, B ≥ 0

a. Show the feasible region.

b. What are the extreme points of the feasible region?

c. Find the optimal solution using the graphical solution procedure.

Min 2A + 2B

s.t.

1A + 3B ≤ 12

3A + 1B ≥ 13

1A - 1B = 3

A, B ≥ 0

a. Show the feasible region.

b. What are the extreme points of the feasible region?

c. Find the optimal solution using the graphical solution procedure.

For the linear program

Min6A + 4B

s.t.

2A + 1B ≥ 12

1A + 1B ≥ 10

1B ≤ 4

A, B ≥ 0

a. Write the problem in standard form.

b. Solve the problem using the graphical solution procedure.

c. What are the values of the slack and surplus variables?

As part of a quality improvement initiative, Consolidated Electronics employees complete a three-day training program on teaming and a two-day training program on problem solving. The manager of quality improvement has requested that at least 8 training programs on teaming and at least 10 training programs on problem solving be offered during the next six months. In addition, senior-level management has specified that at least 25 training programs must be offered during this period. Consolidated Electronics uses a consultant to teach the training programs. During the next quarter, the consultant has 84 days of training time available. Each training program on teaming costs $10,000 and each training program on problem solving costs $8000.

a. Formulate a linear programming model that can be used to determine the number of training programs on teaming and the number of training programs on problem solving that should be offered in order to minimize total cost.

b. Graph the feasible region.

c. Determine the coordinates of each extreme point.

d. Solve for the minimum cost solution.

The New England Cheese Company produces two cheese spreads by blending mild cheddar cheese with extra sharp cheddar cheese. The cheese spreads are packaged in 12-ounce containers, which are then sold to distributors throughout the Northeast. The Regular blend contains 80% mild cheddar and 20% extra sharp, and the Zesty blend contains 60% mild cheddar and 40% extra sharp. This year, a local dairy cooperative offered to provide up to 8100 pounds of mild cheddar cheese for $1.20 per pound and up to 3000 pounds of extra sharp cheddar cheese for $1.40 per pound. The cost to blend and package the cheese spreads, excluding the cost of the cheese, is $0.20 per container. If each container of Regular is sold for $1.95 and each container of Zesty is sold for $2.20, how many containers of Regular and Zesty should New England Cheese produce?

Applied Technology, Inc. (ATI) produces bicycle frames using two fiberglass materials that improve the strength-to-weight ratio of the frames. The cost of the standard grade material is $7.50 per yard and the cost of the professional grade material is $9.00 per yard. The standard and professional grade materials contain different amounts of fiberglass, carbon fiber, and Kevlar, as shown in the following table:

ATI signed a contract with a bicycle manufacturer to produce a new frame with a carbon fiber content of at least 20% and a Kevlar content of not greater than 10%. To meet the required weight specification, a total of 30 yards of material must be used for each frame.

a. Formulate a linear program to determine the number of yards of each grade of fiberglass material that ATI should use in each frame in order to minimize total cost. Define the decision variables and indicate the purpose of each constraint.

b. Use the graphical solution procedure to determine the feasible region. What are the coordinates of the extreme points?

c. Compute the total cost at each extreme point. What is the optimal solution?

d. The distributor of the fiberglass material is currently overstocked with the professional grade material. To reduce inventory, the distributor offered ATI the opportunity to purchase the professional grade for $8 per yard. Will the optimal solution change?

e. Suppose that the distributor further lowers the price of the professional grade material to $7.40 per yard. Will the optimal solution change? What effect would an even lower price for the professional grade material have on the optimal solution?Explain.

Innis Investments manages funds for a number of companies and wealthy clients. The investment strategy is tailored to each client’s needs. For a new client, Innis has been authorized to invest up to $1.2 million in two investment funds: a stock fund and a money market fund. Each unit of the stock fund costs $50 and provides an annual rate of return of 10%; each unit of the money market fund costs $100 and provides an annual rate of return of 4%. The client wants to minimize risk subject to the requirement that the annual income from the investment be at least $60,000. According to Innis’s risk measurement system, each unit invested in the stock fund has a risk index of 8, and each unit invested in the money market fund has a risk index of 3; the higher risk index associated with the stock fund simply indicates that it is the riskier investment. Innis’s client also specified that at least $300,000 be invested in the money market fund.

a. Determine how many units of each fund Innis should purchase for the client to minimize the total risk index for the portfolio.

b. How much annual income will this investment strategy generate?

c. Suppose the client desires to maximize annual return. How should the funds be invested?

a. Determine how many units of each fund Innis should purchase for the client to minimize the total risk index for the portfolio.

b. How much annual income will this investment strategy generate?

c. Suppose the client desires to maximize annual return. How should the funds be invested?

Photo Chemicals produces two types of photographic developing fluids. Both products cost Photo Chemicals $1 per gallon to produce. Based on an analysis of current inventory levels and outstanding orders for the next month, Photo Chemicals’ management specified that at least 30 gallons of product 1 and at least 20 gallons of product 2 must be produced during the next two weeks. Management also stated that an existing inventory of highly perishable raw material required in the production of both fluids must be used within the next two weeks. The current inventory of the perishable raw material is 80 pounds. Although more of this raw material can be ordered if necessary, any of the current inventories that is not used within the next two weeks will spoil—hence, the management requirement that at least 80 pounds be used in the next two weeks. Furthermore, it is known that product 1 requires 1 pound of this perishable raw material per gallon and product 2 requires 2 pounds of the raw material per gallon. Because Photo Chemicals’ objective is to keep its production costs at the minimum possible level, the firm’s management is looking for a minimum cost production plan that uses all the 80 pounds of perishable raw material and provides at least 30 gallons of product1 and at least 20 gallons of product 2. What is the minimum cost solution?

Southern Oil Company produces two grades of gasoline: regular and premium. The profit contributions are $0.30 per gallon for regular gasoline and $0.50 per gallon for premium gasoline. Each gallon of regular gasoline contains 0.3 gallons of grade A crude oil and each gallon of premium gasoline contains 0.6 gallons of grade A crude oil. For the next production period, Southern has 18,000 gallons of grade A crude oil available. The refinery used to produce the gasolines has a production capacity of 50,000 gallons for the next production period. Southern Oil’s distributors have indicated that demand for the premium gasoline for the next production period will be at most 20,000 gallons.

a. Formulate a linear programming model that can be used to determine the number of gallons of regular gasoline and the number of gallons of premium gasoline that should be produced in order to maximize total profit contribution.

b. What is the optimal solution?

c. What are the values and interpretations of the slack variables?

d. What are the binding constraints?

Does the following linear program involve infeasibility, unbounded, and/or alternative optimal solutions? Explain.

Max 4A + 8B

s.t.

2A + 2B ≤ 10

-1A + 1B ≥ 8

A, B ≥ 0

Max 4A + 8B

s.t.

2A + 2B ≤ 10

-1A + 1B ≥ 8

A, B ≥ 0

Does the following linear program involve infeasibility, unbounded, and/or alternative optimal solutions? Explain.

Max 1A + 1B

s.t.

8A + 6B ≥ 24

2B ≥ 4

A, B ≥ 0

Max 1A + 1B

s.t.

8A + 6B ≥ 24

2B ≥ 4

A, B ≥ 0

Consider the following linear program:

Max 1A + 1B

s.t.

5A + 3B ≤15

3A + 5B ≤ 15

A, B ≥ 0

a. What is the optimal solution for this problem?

b. Suppose that the objective function is changed to 1A + 2B. Find the new optimal solution.

Max 1A + 1B

s.t.

5A + 3B ≤15

3A + 5B ≤ 15

A, B ≥ 0

a. What is the optimal solution for this problem?

b. Suppose that the objective function is changed to 1A + 2B. Find the new optimal solution.

Consider the following linear program:

Max 1A - 2B

s.t.

-4A + 3B ≤ 3

1A - 1B ≤ 3

A, B ≥ 0

a. Graph the feasible region for the problem.

b. Is the feasible region unbounded? Explain.

c. Find the optimal solution.

d. Does an unbounded feasible region imply that the optimal solution to the linear program will be unbounded?

Max 1A - 2B

s.t.

-4A + 3B ≤ 3

1A - 1B ≤ 3

A, B ≥ 0

a. Graph the feasible region for the problem.

b. Is the feasible region unbounded? Explain.

c. Find the optimal solution.

d. Does an unbounded feasible region imply that the optimal solution to the linear program will be unbounded?

The manager of a small independent grocery store is trying to determine the best use of her shelf space for soft drinks. The store carries national and generic brands and currently has 200 square feet of shelf space available. The manager wants to allocate at least 60% of the space to the national brands and, regardless of the profitability, allocate at least 10% of the space to the generic brands. How many square feet of space should the manager allocate to the national brands and the generic brands under the following circumstances?

a. The national brands are more profitable than the generic brands.

b. Both brands are equally profitable.

c. The generic brand is more profitable than the national brand.

Discuss what happens to the M&D Chemicals problem (see Section 7.5) if the cost per gallon for product A is increased to $3.00 per gallon. What would you recommend? Explain.

For the M&D Chemicals problem in Section 7.5, discuss the effect of management’s requiring total production of 500 gallons for the two products. List two or three actions M&D should consider to correct the situation you encounter.

PharmaPlus operates a chain of 30 pharmacies. The pharmacies are staffed by licensed pharmacists and pharmacy technicians. The company currently employs 85 full-time equivalent pharmacists (combination of full time and part time) and 175 full-time equivalent technicians. Each spring management reviews current staffing levels and makes hiring plans for the year. A recent forecast of the prescription load for the next year shows that at least 250 full-time equivalent employees (pharmacists and technicians) will be required to staff the pharmacies. The personnel department expects 10 pharmacists and 30 technicians to leave over the next year. To accommodate the expected attrition and prepare for future growth, management stated that at least 15 new pharmacists must be hired. In addition, PharmaPlus’s new service quality guidelines specify no more than two technicians per licensed pharmacist. The average salary for licensed pharmacists is $40 per hour and the average salary for technicians is $10 per hour.

a. Determine a minimum-cost staffing plan for PharmaPlus. How many pharmacists and technicians are needed?

b. Given current staffing levels and expected attrition, how many new hires (if any) must be made to reach the level recommended in part (a)? What will be the impact on the payroll?

Expedition Outfitters manufactures a variety of specialty clothing for hiking, skiing, and mountain climbing. They have decided to begin production on two new parkas designed for use in extremely cold weather: the Mount Everest Parka and the Rocky Mountain Parka. Their manufacturing plant has 120 hours of cutting time and 120 hours of sewing time available for producing these two parkas. Each Mount Everest Parka requires 30 minutes of cutting time and 45 minutes of sewing time, and each Rocky Mountain Parka requires 20 minutes of cutting time and 15 minutes of sewing time. The labor and material cost is $150 for each Mount Everest Parka and $50 for each Rocky Mountain Parka, and the retail prices through the firm’s mail order catalog are $250 for the Mount Everest Parka and $200 for the Rocky Mountain Parka. Because management believes that the Mount Everest Parka is a unique coat that will enhance the image of the firm, they specified that at least 20% of the total production must consist of this model. Assuming that Expedition Outfitters can sell as many coats of each type as they can produce, how many units of each model should they manufacture to maximize the total profit contribution?

English Motors, Ltd. (EML), developed a new all-wheel-drive sports utility vehicle. As part of the marketing campaign, EML produced a video tape sales presentation to send to both owners of current EML four-wheel-drive vehicles as well as to owners of four-wheel-drive sports utility vehicles offered by competitors; EML refers to these two target markets as the current customer market and the new customer market. Individuals who receive the new promotion video will also receive a coupon for a test drive of the new EML model for one weekend. A key factor in the success of the new promotion is the response rate, the percentage of individuals who receive the new promotion and test drive the new model. EML estimates that the response rate for the current customer market is 25% and the response rate for the new customer market is 20%. For the customers who test drive the new model, the sales rate is the percentage of individuals that makes a purchase. Marketing research studies indicate that the sales rate is 12% for the current customer market and 20% for the new customer market. The cost for each promotion, excluding the test drive costs, is $4 for each promotion sent to the current customer market and $6 for each promotion sent to the new customer market. Management also specified that a minimum of 30,000 current customers should test drive the new model and a minimum of 10,000 new customers should test drive the new model. In addition, the numbers of current customers who test drive the new vehicle must be at least twice the number of new customers who test drive the new vehicle. If the marketing budget, excluding test drive costs, is $1.2 million, how many promotions should be sent to each group of customers in order to maximize total sales?

Creative Sports Design (CSD) manufactures a standard-size racket and an oversize racket. The firm’s rackets are extremely light due to the use of a magnesium-graphite alloy that was invented by the firm’s founder. Each standard-size racket uses 0.125 kilograms of the alloy and each oversize racket uses 0.4 kilograms; over the next two-week production period only 80 kilograms of the alloy are available. Each standard-size racket uses 10 minutes of manufacturing time and each oversize racket uses 12 minutes. The profit contributions are $10 for each standard-size racket and $15 for each oversize racket, and 40 hours of manufacturing time are available each week. Management specified that at least 20% of the total production must be the standard-size racket. How many rackets of each type should CSD manufacture over the next two weeks to maximize the total profit contribution? Assume that because of the unique nature of their products, CSD can sell as many rackets as they can produce.

Management of High Tech Services (HTS) would like to develop a model that will help allocate their technicians’ time between service calls to regular contract customers and new customers. A maximum of 80 hours of technician time is available over the two-week planning period. To satisfy cash flow requirements, at least $800 in revenue (per technician) must be generated during the two-week period. Technician time for regular customers generates $25 per hour. However, technician time for new customers only generates an average of $8 per hour because in many cases a new customer contact does not provide billable services. To ensure that new customer contacts are being maintained, the technician time spent on new customer contacts must be at least 60% of the time spent on regular customer contacts. Given these revenue and policy requirements, HTS would like to determine how to allocate technician time between regular customers and new customers so that the total number of customers contacted during the two-week period will be maximized. Technicians require an average of 50 minutes for each regular customer contact and 1 hour for each new customer contact.

a. Develop a linear programming model that will enable HTS to allocate technician time between regular customers and new customers.

b. Find the optimal solution.

Jackson Hole Manufacturing is a small manufacturer of plastic products used in the automotive and computer industries. One of its major contracts is with a large computer company and involves the production of plastic printer cases for the computer company’s portable printers. The printer cases are produced on two injection molding machines. The M-100 machine has a production capacity of 25 printer cases per hour, and the M-200 machine has a production capacity of 40 cases per hour. Both machines use the same chemical material to produce the printer cases; the M-100 uses 40 pounds of the raw material per hour and the M-200 uses 50 pounds per hour. The computer company asked Jackson Hole to produce as many of the cases during the upcoming week as possible; it will pay $18 for each case Jackson Hole can deliver. However, next week is a regularly scheduled vacation period for most of Jackson Hole’s production employees; during this time, annual maintenance is performed for all equipment in the plant. Because of the downtime for maintenance, the M-100 will be available for no more than 15 hours, and the M-200 will be available for no more than 10 hours. However, because of the high setup cost involved with both machines, management requires that, if production is scheduled on either machine, the machine must be operated for at least 5 hours. The supplier of the chemical material used in the production process informed Jackson Hole that a maximum of 1000 pounds of the chemical material will be available for next week’s production; the cost for this raw material is $6 per pound. In addition to the raw material cost, Jackson Hole estimates that the hourly cost of operating the M-100 and the M-200 are $50 and $75, respectively.

a. Formulate a linear programming model that can be used to maximize the contribution to profit.

b. Find the optimal solution.

Digital Imaging (DI) produces photo printers for both the professional and consumer markets. The DI consumer division recently introduced two photo printers that provide color prints rivaling those produced by a professional processing lab. The DI-910 model can produce a 4” × 6” borderless print in approximately 37 seconds. The more sophisticated and faster DI-950 can even produce a 13” × 19” borderless print. Financial projections show profit contributions of $42 for each DI-910 and $87 for each DI-950. The printers are assembled, tested, and packaged at DI’s plant located in New Bern, North Carolina. This plant is highly automated and uses two manufacturing lines to produce the printers. Line 1 performs the assembly operation with times of 3 minutes per DI-910 printer and 6 minutes per DI-950 printer. Line 2 performs both the testing and packaging operations. Times are 4 minutes per DI-910 printer and 2 minutes per DI-950 printer. The shorter time for the DI-950 printer is a result of its faster print speed. Both manufacturing lines are in operation one 8-hour shift per day.

Managerial Report

Perform an analysis for Digital Imaging in order to determine how many units of each printer to produce. Prepare a report to DI’s president presenting your findings and recommendations. Include (but do not limit your discussion to) a consideration of the following:

1. The recommended number of units of each printer to produce to maximize the total contribution to profit for an 8-hour shift. What reasons might management have for not implementing your recommendation?

2. Suppose that management also states that the number of DI-910 printers produced must be at least as great as the number of DI-950 units produced. Assuming that the objective is to maximize the total contribution to profit for an 8-hour shift, how many units of each printer should be produced?

3. Does the solution you developed in part (2) balance the total time spent on line 1 and the total time spent on line 2? Why might this balance or lack of it be a concern to management?

4. Management requested an expansion of the model in part (2) that would provide a better balance between the total time on line 1 and the total time on line 2. Management wants to limit the difference between the total time on line 1 and the total time on line 2 to 30 minutes or less. If the objective is still to maximize the total contribution to profit, how many units of each printer should be produced? What effect does this workload balancing have on total profit in part (2)?

5. Suppose that in part (1) management specified the objective of maximizing the total number of printers produced each shift rather than total profit contribution. With this objective, how many units of each printer should be produced per shift? What effect does this objective have on total profit and workload balancing?

For each solution that you develop, include a copy of your linear programming model and graphical solution in the appendix to your report.

Better Fitness, Inc. (BFI) manufactures exercise equipment at its plant in Freeport, Long Island. It recently designed two universal weight machines for the home exercise market. Both machines use BFI-patented technology that provides the user with an extremely wide range of motion capability for each type of exercise performed. Until now, such capabilities have been available only on expensive weight machines used primarily by physical therapists.

At a recent trade show, demonstrations of the machines resulted in significant dealer interest. In fact, the number of orders that BFI received at the trade show far exceeded its manufacturing capabilities for the current production period. As a result, management decided to begin production of the two machines. The two machines, which BFI named the Body- plus 100 and the BodyPlus 200, require different amounts of resources to produce.

Managerial Report

Analyze the production problem at Better Fitness, Inc., and prepare a report for BFI’s president presenting your findings and recommendations. Include (but do not limit your discussion to) a consideration of the following items:

1. The recommended number of BodyPlus 100 and BodyPlus 200 machines to produce

2. The effect on profits of the requirement that the number of units of the BodyPlus

200 produced must be at least 25% of the total production

3. Where efforts should be expended in order to increase contribution to profits Include a copy of your linear programming model and graphical solution in an appendix to your report.

At a recent trade show, demonstrations of the machines resulted in significant dealer interest. In fact, the number of orders that BFI received at the trade show far exceeded its manufacturing capabilities for the current production period. As a result, management decided to begin production of the two machines. The two machines, which BFI named the Body- plus 100 and the BodyPlus 200, require different amounts of resources to produce.

Managerial Report

Analyze the production problem at Better Fitness, Inc., and prepare a report for BFI’s president presenting your findings and recommendations. Include (but do not limit your discussion to) a consideration of the following items:

1. The recommended number of BodyPlus 100 and BodyPlus 200 machines to produce

2. The effect on profits of the requirement that the number of units of the BodyPlus

200 produced must be at least 25% of the total production

3. Where efforts should be expended in order to increase contribution to profits Include a copy of your linear programming model and graphical solution in an appendix to your report.

Hart Venture Capital (HVC) specializes in providing venture capital for software development and Internet applications. Currently HVC has two investment opportunities: (1) Security Systems, a firm that needs additional capital to develop an Internet security software package.

(2) Market Analysis, a market research company that needs additional capital to develop a software package for conducting customer satisfaction surveys.

In exchange for Security Systems stock, the firm asked HVC to provide $600,000 in year 1, $600,000 in year 2, and $250,000 in year 3 over the coming three-year period. In exchange for their stock, Market Analysis asked HVC to provide $500,000 in year 1, $350,000 in year 2, and $400,000 in year 3 over the same three-year period. HVC believes that both investment opportunities are worth pursuing. However, because of other investments, they are willing to commit at most $800,000 for both projects in the first year, at most $700,000 in the second year, and $500,000 in the third year.

Managerial Report

Perform an analysis of HVC’s investment problem and prepare a report that presents your findings and recommendations. Be sure to include information on the following:

1. The recommended percentage of each project that HVC should fund and the net present value of the total investment

2. A capital allocation plan for Security Systems and Market Analysis for the coming three-year period and the total HVC investment each year

3. The effect, if any, on the recommended percentage of each project that HVC should fund if HVC is willing to commit an additional $100,000 during the first year

4. A capital allocation plan if an additional $100,000 is made available

5. Your recommendation as to whether HVC should commit the additional $100,000 in the first year.

Provide model details and relevant computer output in a report appendix.

Consider the following linear program:

Max3A + 2B

s.t.

1A + 1B â‰¤ 10

3A + 1B â‰¤ 24

1A + 2B â‰¤ 16

A, B â‰¥ 0

a. Use the graphical solution procedure to find the optimal solution.

b. Assume that the objective function coefficient for A changes from 3 to 5. Does the optimal solution change? Use the graphical solution procedure to find the new optimal solution.

c. Assume that the objective function coefficient for A remains 3, but the objective function coefficient for B changes from 2 to 4. Does the optimal solution change? Use the graphical solution procedure to find the new optimal solution.

d. The Management Scientist computer solution for the linear program in part (a) provides the following objective coefficient range information:

Use this objective coefficient range information to answer parts (b) and(c).

Consider the linear program in Problem 1. The value of the optimal solution is 27. Suppose that the right-hand side for constraint 1 is increased from 10 to 11.

a. Use the graphical solution procedure to find the new optimal solution.

b. Use the solution to part (a) to determine the dual price for constraint 1.

c. The Management Scientist computer solution for the linear program in Problem 1 provides the following right-hand-side range information:

What does the right-hand-side range information for constraint 1 tell you about the dual price for constraint 1?

d. The dual price for constraint 2 is 0.5. Using this dual price and the right-hand-side range information in part (c), what conclusion can be drawn about the effect of changes to the right-hand side of constraint2?

Consider the following linear program:

Min8X + 12Y

s.t.

1X + 3Y â‰¥ 9

2X + 2Y â‰¥ 10

6X + 2Y â‰¥ 18

X, Y â‰¥ 0

a. Use the graphical solution procedure to find the optimal solution.

b. Assume that the objective function coefficient for X changes from 8 to 6. Does the optimal solution change? Use the graphical solution procedure to find the new optimal solution.

c. Assume that the objective function coefficient for X remains 8, but the objective function coefficient for Y changes from 12 to 6. Does the optimal solution change? Use the graphical solution procedure to find the new optimal solution.

d. The Management Scientist computer solution for the linear program in part (a) provides the following objective coefficient range information:

How would this objective coefficient range information help you answer parts (b) and (c) prior to resolving theproblem?

Consider the linear program in Problem 3. The value of the optimal solution is 48. Suppose that the right-hand side for constraint 1 is increased from 9 to 10.

a. Use the graphical solution procedure to find the new optimal solution.

b. Use the solution to part (a) to determine the dual price for constraint 1.

c. The Management Scientist computer solution for the linear program in Problem 3 provides the following right-hand-side range information:

What does the right-hand-side range information for constraint 1 tell you about the dual price for constraint 1?

d. The dual price for constraint 2 is - 3. Using this dual price and the right-hand-side range information in part (c), what conclusion can be drawn about the effect of changes to the right-hand side of constraint2?

Refer to the Kelson Sporting Equipment problem. Letting

R = number of regular gloves

C = number of catcherâ€™s mitts leads to the following formulation:

Max 5R + 8C

s.t.

R + 3/2 C â‰¤ 900 Cutting and sewing

1/2 R + 1/3 C â‰¤ 300 Finishing

1/8 R + 1/4 C â‰¤ 100 Packaging and shipping

R, C â‰¥ 0

The computer solution obtained using The Management Scientist is shown in Figure.

THE MANAGEMENT SCIENTIST SOLUTION FOR THE KELSON SPORTING

EQUIPMENT PROBLEM

R = number of regular gloves

C = number of catcherâ€™s mitts leads to the following formulation:

Max 5R + 8C

s.t.

R + 3/2 C â‰¤ 900 Cutting and sewing

1/2 R + 1/3 C â‰¤ 300 Finishing

1/8 R + 1/4 C â‰¤ 100 Packaging and shipping

R, C â‰¥ 0

The computer solution obtained using The Management Scientist is shown in Figure.

THE MANAGEMENT SCIENTIST SOLUTION FOR THE KELSON SPORTING

EQUIPMENT PROBLEM

a. What is the optimal solution, and what is the value of the total profit contribution?

b. Which constraints are binding?

c. What are the dual prices for the resources? Interpret each.

d. If overtime can be scheduled in one of the departments, where would you recommend doingso?

Refer to the computer solution of the Kelson Sporting Equipment problem in Figure.

THE MANAGEMENT SCIENTIST SOLUTION FOR THE KELSON SPORTING EQUIPMENT PROBLEM

a. Determine the objective coefficient ranges.

b. Interpret the ranges in part (a).

c. Interpret the right-hand-side ranges.

d. How much will the value of the optimal solution improve if 20 extra hours of packaging and shipping time are madeavailable?

Investment Advisors, Inc., is a brokerage firm that manages stock portfolios for a number of clients. A particular portfolio consists of U shares of U.S. Oil and H shares of Huber Steel. The annual return for U.S. Oil is $3 per share and the annual return for Huber Steel is $5 per share. U.S. Oil sells for $25 per share and Huber Steel sells for $50 per share. The portfolio has $80,000 to be invested. The portfolio risk index (0.50 per share of U.S. Oil and 0.25 per share for Huber Steel) has a maximum of 700. In addition, the portfolio is limited to a maximum of 1000 shares of U.S. Oil. The linear programming formulation that will maximize the total annual return of the portfolio is as follows:

Max3U + 5H Maximize total annual return

s.t.

25U + 50H â‰¤ 80,000 Funds available

0.50U + 0.25H â‰¤ 700 Risk maximum

1U â‰¤ 1000 U.S. Oil maximum

U, H â‰¥ 0

The computer solution of this problem is shown in Figure.

THE MANAGEMENT SCIENTIST SOLUTION FOR THE INVESTMENT

ADVISORS PROBLEM

a. What is the optimal solution, and what is the value of the total annual return?

b. Which constraints are binding? What is your interpretation of these constraints in terms of the problem?

c. What are the dual prices for the constraints? Interpret each.

d. Would it be beneficial to increase the maximum amount invested in U.S. Oil? Why or whynot?

Refer to Figure which shows the computer solution

THE MANAGEMENT SCIENTIST SOLUTION FOR THE INVESTMENT

ADVISORS PROBLEM

THE MANAGEMENT SCIENTIST SOLUTION FOR THE INVESTMENT

ADVISORS PROBLEM

a. How much would the return for U.S. Oil have to increase before it would be beneficial to increase the investment in this stock?

b. How much would the return for Huber Steel have to decrease before it would be beneficial to reduce the investment in this stock?

c. How much would the total annual return be reduced if the U.S. Oil maximum were reduced to 900shares?

Recall the Tomâ€™s, Inc., problem. Letting

W = jars of Western Foods Salsa

M = jars of Mexico City Salsa

Leads to the formulation:

Max1W + 1.25M

s.t.

5W + 7M â‰¤ 4480 Whole tomatoes

3W + 1M â‰¤ 2080 Tomato sauce

2W + 2M â‰¥ 1600

W, M â‰¥ 0 Tomato paste

The Management Scientist solution is shown in Figure.

THE MANAGEMENT SCIENTIST SOLUTION FOR THE TOMâ€™S, INC., PROBLEM

a. What is the optimal solution, and what are the optimal production quantities?

b. Specify the objective function ranges.

c. What are the dual prices for each constraint? Interpret each.

d. Identify each of the right-hand-sideranges.

Recall the Innis Investments problem. Letting

S = units purchased in the stock fund

M = units purchased in the money market fund

Leads to the following formulation:

Min8S + 3M

s.t.

50S + 100M â‰¤ 1,200,000Funds available

5S + 4M â‰¥ 60,000Annual income

M â‰¥ 3,000Units in money market

S, M â‰¥ 0

The computer solution is shown in Figure.

THE MANAGEMENT SCIENTIST SOLUTION FOR THE INNIS INVESTMENTS PROBLEM

a. What is the optimal solution, and what is the minimum total risk?

b. Specify the objective coefficient ranges.

c. How much annual income will be earned by the portfolio?

d. What is the rate of return for the portfolio?

e. What is the dual price for the funds available constraint?

f. What is the marginal rate of return on extra funds added to theportfolio?

Refer to Problem 10 and the computer solution shown in Refer Figure

THE MANAGEMENT SCIENTIST SOLUTION FOR THE INNIS INVESTMENTS PROBLEM

THE MANAGEMENT SCIENTIST SOLUTION FOR THE INNIS INVESTMENTS PROBLEM

a. Suppose the risk index for the stock fund (the value of CS) increases from its current value of 8 to 12. How does the optimal solution change, if at all?

b. Suppose the risk index for the money market fund (the value of CM) increases from its current value of 3 to 3.5. How does the optimal solution change, if at all?

c. Suppose CS increases to 12 and CM increases to 3.5. How does the optimal solution change, if atall?

Quality Air Conditioning manufactures three home air conditioners: an economy model, a standard model, and a deluxe model. The profits per unit are $63, $95, and $135, respectively. The production requirements per unit are as follows:

For the coming production period, the company has 200 fan motors, 320 cooling coils, and 2400 hours of manufacturing time available. How many economy models (E), standard models (S), and deluxe models (D) should the company produce in order to maximize profit? The linear programming model for the problem is as follows.

Max 63E + 95S + 135D

s.t.

1E + 1S + 1D â‰¤ 200 Fan motors

1E + 2S + 4D â‰¤ 320 Cooling coils

8E + 12S + 14D â‰¤ 2400 Manufacturing time

E, S, D â‰¥ 0

The computer solution using The Management Scientist is shown in Figure.

THE MANAGEMENT SCIENTIST SOLUTION FOR THE QUALITY AIR CONDITIONING PROBLEM

a. What is the optimal solution, and what is the value of the objective function?

b. Which constraints are binding?

c. Which constraint shows extra capacity? How much?

d. If the profit for the deluxe model were increased to $150 per unit, would the optimal solution change? Use the information in Figure 8.19 to answer thisquestion.

Refer to the computer solution of Problem 12 in Figure

THE MANAGEMENT SCIENTIST SOLUTION FOR THE QUALITY AIR CONDITIONING PROBLEM

a. Identify the range of optimality for each objective function coefficient.

b. Suppose the profit for the economy model is increased by $6 per unit, the profit for the standard model is decreased by $2 per unit, and the profit for the deluxe model is increased by $4 per unit. What will the new optimal solution be?

c. Identify the range of feasibility for the right-hand-side values.

d. If the number of fan motors available for production is increased by 100, will the dual price for that constraint change?Explain.

Digital Controls, Inc. (DCI) manufactures two models of a radar gun used by police to monitor the speed of automobiles. Model A has an accuracy of plus or minus 1 mile per hour, whereas the smaller model B has an accuracy of plus or minus 3 miles per hour. For the next week, the company has orders for 100 units of model A and 150 units of model B. Although DCI purchases all the electronic components used in both models, the plastic cases for both models are manufactured at a DCI plant in Newark, New Jersey. Each model A case requires 4 minutes of injection-molding time and 6 minutes of assembly time. Each model B case requires 3 minutes of injection-molding time and 8 minutes of assembly time. For next week the Newark plant has 600 minutes of injection-molding time available and 1080 minutes of assembly time available. The manufacturing cost is $10 per case for model A and $6 per case for model B. Depending upon demand and the time available at the Newark plant, DCI occasionally purchases cases for one or both models from an outside supplier in order to fill customer orders that could not be filled otherwise. The purchase cost is $14 for each model A case and $9 for each model B case. Management wants to develop a minimum cost plan that will determine how many cases of each model should be produced at the Newark plant and how many cases of each model should be purchased. The following decision variables were used to formulate a linear programming model for this problem:

AM = number of cases of model A manufactured

BM = number of cases of model B manufactured

AP = number of cases of model A purchased

BP = number of cases of model B purchased

The linear programming model that can be used to solve this problem is as follows:

Min10AM + 6AM + 14AP + 9BP

s.t.

1AM + + 1AP + = 100Demand for model A

1BM + 1BP = 150Demand for model B

4AM + 3BM â‰¤ 600Injection molding time

6BM + 8BM â‰¤ 1080Assembly time

AM, BM, AP, BP â‰¥ 0

The computer solution developed using The Management Scientist is shown in Figure.

THE MANAGEMENT SCIENTIST SOLUTION FOR THE DIGITAL CONTROLS, INC., PROBLEM

a. What is the optimal solution and what is the optimal value of the objective function?

b. Which constraints are binding?

c. What are the dual prices? Interpret each.

d. If you could change the right-hand side of one constraint by one unit, which one would you choose?Why?

Refer to the computer solution to Problem 14 in Figure

THE MANAGEMENT SCIENTIST SOLUTION FOR THE DIGITAL

CONTROLS, INC., PROBLEM

a. Interpret the ranges of optimality for the objective function coefficients.

b. Suppose that the manufacturing cost increases to $11.20 per case for model A. What is the new optimal solution?

c. Suppose that the manufacturing cost increases to $11.20 per case for model A and the manufacturing cost for model B decreases to $5 per unit. Would the optimal solution change? Use the 100 percent rule anddiscuss.

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