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MNS407 Management Science, Fall 2024

407 Module 1: Quiz#1 (Quiz) INSTRUCTOR

Vahid Keyhani National University, La Jolla CA

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This Quiz covers material in Chapters 1 to 3 as well as the basics of linear equations. You have one attempt to

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10/31/24, 6:38 PM 407 Module 1: Quiz#1 - MNS407 Management Science, Fall 2024 | WebAssign

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Consider the following linear program.

Identify the feasible region.

Min 4A + 5B s.t.

1A + 3B≥9 1A + 1B≥7 A, B ≥ 0

ℹ ℹ

ℹ ℹ

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

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at (A, B) =

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The O'Neill Shoe Manufacturing Company will produce a special-style shoe if the order size is large enough to provide a reasonable profit. For each special-style order, the company incurs a fixed cost of $3,300 for the production setup. The variable cost is $40 per pair, and each pair sells for $70.

(a) Let x indicate the number of pairs of shoes produced. Develop a mathematical model for the total cost (C) of producing x pairs of shoes.

C =

(b) Let P indicate the total profit. Develop a mathematical model for the total profit realized from an order for x pairs of shoes.

P =

(c) How large must the shoe order be before O'Neill will break even?

x =

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2. [–/1.25 Points] ASWMSCI15 1.E.012.DETAILS MY NOTES PRACTICE ANOTHER

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Suppose you are going on a weekend trip to a city that is d miles away.

(a) Develop a model that determines your round-trip gasoline costs.

Let

d = distance, m = miles per gallon, p = price per gallon, and C = total cost.

C =

(b) What assumptions or approximations are necessary to treat this model as a deterministic model? (Select all that apply.)

p does not vary by location.

m does not vary for the duration of the trip.

p varies by location and due to local taxes.

m varies depending on city and highway traffic conditions.

We use the same route on the outbound and the inbound leg of the trip.

(c) Are these assumptions or approximations acceptable?

A reasonable person would conclude that the real cost would deviate widely from the estimate and reject the approximation.

A reasonable person would hedge their bets and increase the budget by 20% over the estimated cost.

A reasonable person would assume average values for m and p and accept the approximation.

A reasonable person would note that neither m nor p can be known ahead of time and reject the approximation.

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What are the advantages of analyzing and experimenting with a model as opposed to a real object or situation?

Experimenting with a model is ---Select--- expensive than experimenting with a real situation.

Experimenting with a real situation takes ---Select--- time than experimenting with a model.

Experimenting with a real situation incurs ---Select--- risk than experimenting with a model.

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Kelson Sporting Equipment, Inc., makes two different types of baseball gloves: a regular model and a catcher's model. Letting

R = number of regular gloves C = number of catcher's mitts

leads to the following formulation:

The computer solution is shown below.

Optimal Objective Value = 3640.00000

Variable Value Reduced Cost

R 440.00000 0.00000

C 180.00000 0.00000

Constraint Slack/Surplus Dual Value

1 90.00000 0.00000

2 0.00000 3.00000

3 0.00000 28.00000

Variable Objective

Coefficient

Allowable

Increase

Allowable

Decrease

R 5.00000 7.00000 1.00000

C 8.00000 2.00000 4.66667

Constraint RHS

Value

Allowable

Increase

Allowable

Decrease

1 800.00000 Infinite 90.00000

2 280.00000 120.00000 146.66667

3 100.00000 18.00000 30.00000

(a) Determine the objective coefficient ranges. (Round your answers to two decimal places.)

regular glove to

catcher's mitt to

(b) Interpret the ranges in part (a). (Round your answers to two decimal places.)

Max 5R + 8C s.t.

R + C ≤ 800 Cutting and sewing

R + C ≤ 280 Finishing

R + C ≤ 100 Packaging and shipping

R, C ≥ 0

3 2

1 2

1 3

1 8

1 4

5. [–/1.25 Points] ASWMSCI15 3.E.006.DETAILS MY NOTES PRACTICE ANOTHER

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As long as the profit contribution for the regular glove is between $ and $ , the current solution ---Select--- optimal. As long as the profit contribution for the catcher's

mitt is between $ and $ , the current solution ---Select--- optimal.

(c) Interpret the right-hand-side ranges.

The dual values for the resources are applicable over the following ranges. (Round your answers to two decimal places. If there is no upper or lower limit, enter NO LIMIT.)

cutting and sewing

to

finishing

to

packaging and shipping

to

(d) How much will the value of the optimal solution improve (in $) if 10 extra hours of packaging and shipping time are made available?

$

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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:

Number of Fans

Number of Cooling Coils

Manufacturing Time (hours)

Economy 1 1 8

Standard 1 2 12

Deluxe 1 4 14

For the coming production period, the company has 240 fan motors, 340 cooling coils, and 2,600 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:

The computer solution is shown below.

Optimal Objective Value = 18320.00000

Variable Value Reduced Cost

E 140.00000 0.00000

S 100.00000 0.00000

D 0.00000 24.00000

Constraint Slack/Surplus Dual Value

1 0.00000 31.00000

2 0.00000 32.00000

3 280.00000 0.00000

Variable Objective

Coefficient

Allowable

Increase

Allowable

Decrease

E 63.00000 12.00000 15.50000

S 95.00000 31.00000 8.00000

D 135.00000 24.00000 Infinite

Constraint RHS

Value

Allowable

Increase

Allowable

Decrease

1 240.00000 70.00000 70.00000

2 340.00000 70.00000 100.00000

Max 63E + 95S + 135D s.t.

1E + 1S + 1D ≤ 240 Fan motors 1E + 2S + 4D ≤ 340 Cooling coils 8E + 12S + 14D ≤ 2,600 Manufacturing time

E, S, D ≥ 0

6. [–/1.25 Points] ASWMSCI15 3.E.012.DETAILS MY NOTES PRACTICE ANOTHER

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3 2600.00000 Infinite 280.00000

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

E

S

D

profit $

(b) Which constraints are binding? (Select all that apply.)

fan motors

cooling coils

manufacturing time

(c) Which constraint shows extra capacity?

fan motors

cooling coils

manufacturing time

How much extra capacity (in hr) is available?

hr

(d) If the profit for the deluxe model were increased to $145 per unit, would the optimal solution change? Use the information in the output to answer this question.

The optimal solution ---Select--- change because the new profit is ---Select--- the ---Select--- .

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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' 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.

Refer to the computer solution shown below.

Optimal Objective Value = 62000.00000

Variable Value Reduced Cost

S 4000.00000 0.00000

M 10000.00000 0.00000

Constraint Slack/Surplus Dual Value

1 0.00000 −0.05667

2 0.00000 2.16667

3 7000.00000 0.00000

Variable Objective

Coefficient

Allowable

Increase

Allowable

Decrease

S 8.00000 Infinite 4.25000

M 3.00000 3.40000 Infinite

Constraint RHS

Value

Allowable

Increase

Allowable

Decrease

1 1200000.00000 300000.00000 420000.00000

2 60000.00000 42000.00000 12000.00000

3 3000.00000 7000.00000 Infinite

(a) Suppose the risk index for the stock fund (the value of increases from its current value of 8 to 16. How does the optimal solution change, if at all?

The optimal solution becomes (9600, 3000).

The optimal solution does not change.

(b) Suppose the risk index for the money market fund (the value of increases from its current value of 3 to 4.5. How does the optimal solution change, if at all?

The optimal solution becomes (9600, 3000).

The optimal solution does not change.

C )S

C )M

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(c) Suppose increases to 16 and increases to 4.5. How does the optimal solution change, if at all?

The optimal solution becomes (9600, 3000).

The optimal solution does not change.

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CS CM

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The Kartick Company is trying to determine how much of each of two products to produce over the coming planning period. There are three departments, A, B, and C, with limited labor hours available in each department. Each product must be processed by each department and the per unit requirements for each product, labor hours available, and per-unit profit are as shown below.

Labor required in each department

Department Product (hours/unit)

Labor Hours Available Product 1 Product 2

A 1.00 0.30 100

B 0.30 0.12 36

C 0.15 0.56 50

Profit Contribution $33.00 $24.00

A linear program for this situation is as follows.

Let of product 1 to produce.

Let of product 2 to produce.

Mr. Kartick (the owner) used trial and error with a spreadsheet model to arrive at a solution. His proposed solution is as shown in the figure. He said he felt his proposed solution is optimal.

Consider the following tables a representation of an Excel spreadsheet.

Kartick

Data

Department Prod 1 Prod 2 Hours

Available

A 1.00 0.30 100

B 0.30 0.12 36

C 0.15 0.56 50

x = the amount1

x = the amount2

Maximize 33 x + 24 x s.t.

1.0 x + 0.30 x ≤ 100 Department A 0.30 x + 0.12 x ≤ 36 Department B 0.15 x + 0.56 x ≤ 50 Department C

x , x ≥ 0

1 2

1 2

1 2

1 2

1 2

x = 75 and x = 60,1 2

8. [–/1.25 Points] ASWMSCI15 2.E.055.DETAILS MY NOTES PRACTICE ANOTHER

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Per unit Contribution

$33.00 $24.00

Decisions

Prod 1 Prod 2

Quantity 75 60

Model

Department Hours Used

Unused Hours

A 93 7

B 29.7 6.3

C 44.85 5.15

Contribution $3,915.00

Is his solution optimal? Without solving the problem, explain why you believe this solution is optimal or not optimal.

His solution is optimal, it results in a maximum number of hours used.

His solution is optimal, it results in a maximum profit contribution.

His solution is optimal, it results in a minimum number of unused hours.

His solution is not optimal, it results in more hours used than what is available.

His solution is not optimal, it results in unused hours, thus more products can still be made.

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