# Chapter 1 An Introduction to Management Science

Chapter 1: An Introduction to Management Science

Question 1 (5 points): Micromedia offers computer training seminars on a variety of topics. In the seminars each student works at a personal computer, practicing the particular activity that the instructor is presenting. Micromedia is currently planning a two-day seminar on the use of Micro- soft Excel in statistical analysis. The projected fee for the seminar is $600 per student. The cost for the conference room, instructor compensation, lab assistants, and promotion is $9600. Micromedia rents computers for its seminars at a cost of $120 per computer per day.

Solution

Question 2 (5 points): Preliminary plans are under way for the construction of a new stadium for a major league baseball team. City officials have questioned the number and profitability of the luxury corporate boxes planned for the upper deck of the stadium. Corporations and selected individuals may buy the boxes for $300,000 each. The fixed construction cost for the upper- deck area is estimated to be $4,500,000, with a variable cost of $150,000 for each box constructed.

a)

the profit on each box is $(325,000 -150,000 )=$175000

so to cover the $4,500,000 fixed cost you need

so, there are 26 boxes

b)

The marginal revenue from a one-unit increases in sales is $325,000

The marginal cost (variable cost per box) is $175,000

which indicates a marginal profit of $175,000 for each additional box sold.

Since marginal profit is positive, and all 47 boxes can be sold

it is recommended to build all 47 boxes.

The total profit = 175000(47) –4,500,000 =

$3725000

Chapter 2: An Introduction to Linear Programming

Question 3 (6 points):: 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 ho- rizon. 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:

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

a) Feasible region is shaded in the graph

B) Coordinates of extreme points and the corresponding profit are listed

A C Z

Intersection of C1 and A-axis 1,700 0 8,500

Intersection of C2 and C-axis 0 1,680 6,720

Intersection of C2 and C3 800 1,200 8,800

Intersection of C1 and C3 1,400 600 9,400

Point (1400, 600) generates the maximum profit

c) profit line (objective function(z)) is drawn on the graph.

d) Which constraints are binding? Explain

Constraints c1 and c3 are binding, because optimal solution lies at their intersection.

Now the solution point changes to intersection of C2 and C3.

A = 800, C = 1200

Profit = 9200

Question 4

Solve the following linear programming problem using the graphical solution procedure:

Max 5A + 5B

s.t.

1A ≤ 100

1B ≤ 80

2A + 4B ≤ 400

A, B ≥0Let c = 5a + 5b

We are going to maximize c.

Now take the constraint equations…

a < 100

When graphed, this will be the region to the left of the straight line a = 100 and excluding the points lying on the line a = 100. We shade this region.

b < 80

When graphed, this will be the region below the straight line b = 80and excluding the points lying on the line b = 80. We shade this region.

2a + 4b < 400 => a + 2b < 200

First we graph the line a + 2b = 200

To do this, we take points (0, 100) and (200, 0), which lie on the line.

Now, putting x = 0, y = 0 (Test Point), we see that 0 + 0 < 200 is TRUE

Therefore, we shade the region which contains the origin.

The intersection of all the shaded regions is the feasible or solution region.

The maximum value of c lies at one of the corners of the feasible region.

The corners have the coordinates: (40, 80) and (100, 50)

Plug in all these values in c = 5a + 5b. We see that maximum value of c occurs when

a = 100, b = 50

The maximum value of c is 5(100) + 5(50) = 750.

Question 7. The Westchester Chamber of Commerce periodically sponsors public service seminars and programs. Currently, promotional plans are under way for this year’s program. Advertising alternatives include television, radio, and online. Audience estimates, costs, and maximum media usage limitations are as shown.

To ensure a balanced use of advertising media, radio advertisements must not exceed 50% of the total number of advertisements authorized. In addition, television should account for at least 10% of the total number of advertisements authorized.

Question 8

Question 8. The management of Hartman Company is trying to determine the amount of each of two products to produce over the coming planning period. The following information concerns labor availability, labor utilization, and product profitability:

Solutions

Question 9:Consider the following all-integer linear program:

Max 1×1 + 1×2

s.t.4×1 + 6×2 ≤22

1×1 +5×2 ≤15

2×1 + 1×2 ≤ 9

x1, x2 ≥ 0 and integer

a)

B&c)

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