# What is the optimal value of the objective function

What is the optimal value of the objective function

__Please be advised that you are required to submit linear programs (with sensitivity reports where necessary) and provide direct answers to each part of each question.__

1. You are the recently hired Chief Operations Officer at ABC Inc, a regional firm which produces specialized circuit boards used in the production of various makes and models of automobiles. The company currently owns three production plants, one much newer than the other two. Once the circuit boards are produced, the firm ships them to one of four warehouses throughout the state where they are placed in inventory until ordered by the end-user. Each circuit board sells for $150.

Being the good COO that you are, you are always looking for ways to minimize costs and increase profitability. You suspect that there is much room for improvement in this regard. Specifically, you have reason to believe that production and stocking levels of each plant and warehouse could be modified to yield the desired results.

The details of this managerial challenge are as follows:

Production Costs – Plant A — $100 per circuit board

Plant B — $120 per circuit board

Plant C — $90 per circuit board

Plant Capacity – Plant A – 1,200 circuit boards

Plant B – 1,200 circuit boards

Plant C – 500 circuit boards

Demand Forecast – Warehouse #1 – 700 circuit boards

Warehouse #2 – 400 circuit boards

Warehouse #3 – 600 circuit boards

Warehouse #4 – 500 circuit boards

Shipping Costs

Warehouse #1 | Warehouse #2 | Warehouse #3 | Warehouse #4 | |

Plant A | $10 | $15 | $40 | $30 |

Plant B | $50 | $20 | $25 | $20 |

Plant C | $25 | $45 | $30 | $22 |

Create a linear program that shows exactly how many circuit boards should be produced at each plant and then shipped to each warehouse in order to maximize resulting profits. Your completed model should show the optimal values of all 12 decision variables AS WELL AS the optimal value of the objective function.

2. Given the following LP model,

Minimize (costs) Z = 4X1 + 8X2

Subject to Fiber 5X1 + 8X2 __>__ 40

Protein 6X1 + 4X2 __>__ 24

X1, X2 __>__ 0

a.) What is the optimal value of the objective function?

b.) What are the optimal values of the two decision variables?

c.) Find the range of optimality for each objective function coefficient.

d.) How would a decrease of $1 in the X1 coefficient of the objective function affect the optimal values of the decision variables?

e.) How would a decrease of $1 in the X1 coefficient of the objective function affect the optimal value of the objective function?

f.) What is the dual value (AKA –“shadow price”) for the RHS of the protein constraint?

g.) What is the range of feasibility of the dual value for the RHS of the protein constraint?

h.) What impact on total cost would a decrease of 2 units in the RHS of the protein constraint have? Please explain the rationale for your answer.

3. The manager of FYZ Incorporated has been presented with the following LP model:

Minimize (costs) Z = 30A + 45B

Subject to 5A + 2B __>__ 100

4A + 8B __>__ 240

B __>__ 20

A and B __>__ 0

She would like your assistance with several questions below.

a.) What is the proper name of the last constraint shown in the model?

b.) What is the optimal value of the objective function?

c.) What are the optimal values of the two decision variables?

d.) If the cost of B could be reduced to $42 per unit, how many units of B would be optimal?

e.) If the cost of B could be reduced to $42 per unit, what would the minimum total cost be?

f.) What is the dual value (AKA–“shadow price”) for the RHS value of the first constraint?

g.) What is the range of feasibility for the RHS value of the first constraint?

h.) By what amount would the total cost change, and in what direction, if the RHS value of the first constraint was changed to 110? Please explain the rationale for your answer.

Case Problem 1 SOLUTIONS PLUS

Please keep your response between 2 and 3 pages (double-spaced) in length….not including the associated linear program and sensitivity report you will have to submit in an Excel format. The score will be primarily dependent on the content and accuracy of your responses to the first three sets of issues mentioned in the “Managerial Report” on Page 306 of your text. However, part of the evaluative process, will involve the overall professionalism of the write-up itself. I expect the work to be free of typographical errors and inclusive of the kind of responses you would be expected to provide to a reporting supervisor.

Solutions Plus is an industrial chemicals company that produces specialized cleaning ﬂuids and solvents for a wide variety of applications. Solutions Plus just received an invitation to submit a bid to supply Great North American railroad with a cleaning ﬂuid for locomotives. Great North American needs the cleaning ﬂuid at 11 locations (railway stations); it provided the following information to Solutions Plus regarding the number of gallons of cleaning ﬂuid required at each location (see Table 6.8):

Solutions Plus can produce the cleaning ﬂuid at its Cincinnati plant for $1.20 per gallon. Even though the Cincinnati location is its only plant, Solutions Plus has negotiated with an industrial chemicals company located in Oakland, California, to produce and ship

up to 50,000 gallons of the locomotive cleaning ﬂuid to selected Solutions Plus customer

locations. The Oakland company will charge Solutions Plus $1.65 per gallon to produce the cleaning ﬂuid, but Solutions Plus thinks that the lower shipping costs from Oakland to some customer locations may offset the added cost to produce the product.

TABLE 6.8

Location Gallons Required Location Gallons Required

Santa Ana 22,418 Glendale 33,689

El Paso 6,800 Jacksonville 68,486

Pendleton 80,290 Little Rock 148,586

Houston 100,447 Bridgeport 111,475

Kansas City 241,570 Sacramento 112,000

Los Angeles 64,761

TABLE 6.9 FREIGHT COST ($ PER GALLON)

Cincinnati Oakland

Santa Ana — 0.22

El Paso 0.84 0.74

Pendleton 0.83 0.49

Houston 0.45 —

Kansas City 0.36 —

Los Angeles — 0.22

Glendale — 0.22

Jacksonville 0.34 —

Little Rock 0.34 —

Bridgeport 0.34 —

Sacramento — 0.15

The president of Solutions Plus, Charlie Weaver, contacted several trucking companies

to negotiate shipping rates between the two production facilities (Cincinnati and Oakland) and the locations where the railroad locomotives are cleaned. Table 6.9 shows the quotes received in terms of dollars per gallon. The — entries in Table 6.9 identify shipping routes that will not be considered because of the large distances involved. These quotes for shipping rates are guaranteed for one year.

To submit a bid to the railroad company, Solutions Plus must determine the price per

gallon they will charge. Solutions Plus usually sells its cleaning ﬂuids for 15% more than

its cost to produce and deliver the product. For this big contract, however, Fred Roedel, the director of marketing, suggested that maybe the company should consider a smaller proﬁt margin. In addition, to ensure that if Solutions Plus wins the bid, they will have adequate capacity to satisfy existing orders as well as accept orders for other new business, the management team decided to limit the number of gallons of the locomotive cleaning ﬂuid produced in the Cincinnati plant to 500,000 gallons at most.

Managerial Report

You are asked to make recommendations that will help Solutions Plus prepare a bid. Your report should address, but not be limited to, the following issues:

1. If Solutions Plus wins the bid, which production facility (Cincinnati or Oakland)

should supply the cleaning ﬂuid to the locations where the railroad locomotives are

cleaned? How much should be shipped from each facility to each location?

2. What is the breakeven point for Solutions Plus? That is, how low can the company

go on its bid without losing money?

3. If Solutions Plus wants to use its standard 15% markup, how much should it bid?

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