Operation Research

CPSMA 3933 Operations Research

Pre-Final Exam (100 points)

Due date: Sunday, December 8th 11:59pm

Submit 1 Excel file (create a different sheet for each problem) via Blackboard by the deadline

(strict deadline).

NO COLLABORATION IS ALLOWED, THIS MUST BE YOUR OWN WORK.

1. A company produces two products, A and B. The sales volume for A is at least 80% of the total

sales of both A and B. However, the company cannot sell more than 100 units of A per day. Both

products use one raw material, of which the maximum daily availability is 240lb. The usage rates

of the raw material are 2lb per unit of A, and 4lb per unit of B. The profit units for A and B are

$20 and $50, respectively. Determine the optimal product mix for the company using Solver.

Save the table and solution in a separate Excel sheet within the same Excel file. – 15 points

2. An individual wishes to invest $5000 over the next year in two types of investment: Investment

A yields 5%, and investment B yields 8%. Market research recommends an allocation of at least

25% in A and at most 50% in B. Moreover, investment in A should be at least half the investment

in B. How should the fund be allocated to the two investments? Find the optimal solution using

Solver. Save the table and solution in a separate Excel sheet within the same Excel file. – 15

points

3. Four products are processed sequentially on three machines. The following table gives the

pertinent data of the problem.

Formulate the problem as an LP model and find the optimum solution using Solver. Save the

table and solution in a separate Excel sheet within the same Excel file. – 15 points

4. Consider the following set of constraints:

𝑥1 + 2𝑥2 + 2𝑥3 + 4𝑥4 ≤ 40

2𝑥1 − 𝑥2 + 𝑥3 + 2𝑥4 ≤ 8

4𝑥1 − 2𝑥2 + 𝑥3 − 𝑥4 ≤ 10

𝑥1,𝑥2,𝑥3,𝑥4 ≥ 0

Solve the problem for each of the following objective functions using Solver. Save the table and

solution in a separate Excel sheet within the same Excel file for each part (4 separate sheets

for this problem). – 32 points

Machine Cost per hr ($) Product 1 Product 2 Product 3 Product 4 Capacity (hr)

1 10 2 3 4 2 500

2 5 3 2 1 2 380

3 4 7 3 2 1 450

Unit selling price ($) 75 70 55 45

Manufacturing time (hr) per unit

(a) Maximize 𝑧 = 2𝑥1 + 𝑥2 − 3𝑥3 + 5𝑥4

(b) Maximize 𝑧 = 8𝑥1 + 6𝑥2 + 3𝑥3 − 2𝑥4

(c) Maximize 𝑧 = 3𝑥1 − 𝑥2 + 3𝑥3 + 4𝑥4

(d) Maximize 𝑧 = 5𝑥1 − 4𝑥2 + 6𝑥3 − 8𝑥4

5. Solve the dual of the following problem. Save the table and solution in a separate Excel sheet

within the same Excel file. – 23 points

Minimize 𝑧 = 5𝑥1 + 6𝑥2 + 3𝑥3

Subject to:

5𝑥1 + 5𝑥2 + 3𝑥3 ≥ 50

𝑥1 + 𝑥2 − 𝑥3 ≥ 20

7𝑥1 + 6𝑥2 − 9𝑥3 ≥ 30

5𝑥1 + 5𝑥2 + 5𝑥3 ≥ 35

2𝑥1 + 4𝑥2 − 15𝑥3 ≥ 10

12𝑥1 + 10𝑥2 ≥ 90

𝑥2 − 10𝑥3 ≥ 20

𝑥1,𝑥2,𝑥3 ≥ 0