1. (a) Briefly describe the important parts of each step needed to make a decision using decision sciences models.
(b) What are the different methods to solve a system of linear equations? Briefly describe the steps needed in each method.
(c) What is an unbounded linear programming problem? How do we find if a given linear programming problem has unbounded optimal solution? Give a real world example of a linear programming problem where unbounded optimal solution may occur.
(d) Describe the important aspects of, and give a real world example for, the deterministic models. Describe the important aspects of, and give a real world example for, the stochastic models. What are the differences between the deterministic models and the stochastic models?
2. Given the following linear programming problem
Maximize 40x + 30y
Subject to
x + y < 80
3x + 2y < 180
2x + y > 20
x, y > 0
(a) Graph the constraints.
(b) Find the coordinates of each corner point of the feasible region
(c) Determine the optimal solution.
3. Given that the optimal solution of the following linear programming problem is x = 15 and y = 10, state the problem in standard form and do a constraint analysis for the optimal solution.
Maximize 10x + 8y
Subject to
2x + 3y ≤ 60
6x + 5y ≥ 100
x ≤ 15
y ≤ 12
x, y > 0
4. A company produces two products, A and B, which have profits of $80 and $70, respectively. Each unit of product must be processed on three assembly lines, where the required production times are as follows:
|
|
|
Hours/Unit
|
|
|
|
Line 1
|
Line 2
|
Line 3
|
|
Product A
|
11
|
4
|
6
|
|
Product B
|
5
|
8.5
|
4.8
|
|
Total Hours Available
|
600
|
400
|
380
|
The company wants to produce at least 30 units of product A and at least 30 units of Product B.
Formulate a linear programming model for the above situation by determining
(a) The decision variables.
(b) Determine the objective function. What does it represent?
(c) Determine all the constraints. Briefly describe what each constraint represents.
Note: Do NOT solve the problem after formulating.
5. The Charm City Aluminum Company produces three grades (high, medium, and low) of aluminum at two mills. Each mill has a different production capacity (in tons per day) for each grade, as follows:
|
|
Mill 1
|
Mill 2
|
|
High Grade
|
5.8
|
2.2
|
|
Medium Grade
|
2.4
|
4.1
|
|
Low Grade
|
3.8
|
8.6
|
The company has contracted with a manufacturing firm to supply at least 10 tons of high-grade aluminum, 8 tons of medium-grade aluminum, and 6 tons of low-grade aluminum. It costs the company $6,500 per day to operate mill 1 and $7,400 per day to operate mill 2. The company wants to know the number of days to operate each mill in order to meet the contract at the minimum cost.
Formulate a linear programming model for the above situation by determining
(a) The decision variables.
(b) Determine the objective function. What does it represent?
(c) Determine all the constraints. Briefly describe what each constraint represents.
Note: Do NOT solve the problem after formulating.
6. Determine whether the following linear programming problem is infeasible, unbounded, or has multiple optimal solutions. Draw a graph to find the feasible region (if it exists) and explain your conclusion.
Maximize 100x + 200y
Subject to:
x + 2y > 40
x < 16
y > 15
x, y > 0