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Mathematics III ExaminationHomework
Answer all questions (take-home examination)
“I have neither given nor received help on this exam.” _______ (student’s initials)
Do all the following problems.
I. Choose the best answer for each multiple choice. Please use CAPITAL letters to indicate your answer and write neatly. (20 points)
1. ____ 3.____ 5. ____ 7.____ 9. ____
2. ____ 4.____ 6.____ 8.____ 10.____
1. Which of the following is a valid objective function for a linear programming problem?
A. Max 5xy
B. Min 4x + 3y + (2/3)z
C. Max 5x2 + 6y2
D. Min (x1 + x2)/x3
2. Which of the following statements is NOT true?
A. A feasible solution satisfies all constraints.
B. An optimal solution satisfies all constraints.
C. An infeasible solution violates all constraints.
D. A feasible solution point does not have to lie on the boundary of the feasible region.
3. Innis Investments manages funds for a number of companies and wealthy clients. 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. According to Innis’s 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. Which is the objective function for this problem? Mathematics III Examination Homework Help
A. Max 8S + 3M
B. Max 8S + 3M – 1.2X
C. Min 8S + 3M
D. Min 8S + 3M – 1.2X
4. A solution that satisfies all the constraints of a linear programming problem except the nonnegativity constraints is called
A. optimal.
B. feasible.
C. infeasible.
D. semi-feasible.
5. Slack
A. is the difference between the left and right sides of a constraint.
B. is the amount by which the left side of a ≤ constraint is smaller than the right side.
C. is the amount by which the left side of a ≥ constraint is larger than the right side.
D. exists for each variable in a linear programming problem.
6. Given a linear programming model:
Max 4x1 + 6x2
s.t. x1 + 2x2 < 6
2x1 + 4x2 < 18
x1, x2 > 0
Which of the following statement is true?
A. The model has only one optimal solution.
B. The model has no feasible solutions.
C. The model has multiple optimal solutions.
D. The model has unbounded solution. Mathematics III Examination Homework Help
7. A constraint that does not affect the feasible region is a
A. non-negativity constraint.
B. redundant constraint.
C. standard constraint.
D. slack constraint.
8. To find the optimal solution to a linear programming problem using the graphical method
A. find the feasible point that is the farthest away from the origin.
B. find the feasible point that is at the highest location.
C. find the feasible point that is closest to the origin.
D. None of the alternatives is correct. Mathematics III Examination Homework Help
9. Let xij = the production of product i in period j. To specify that production of product 1 in period 3 and in period 4 differs by no more than 100 units, which of the following constraints are correct?
A. P13 – P14 < 100; P14 – P13 < 100
B. P13 – P14 < 100; P13 – P14 > 100
C. P13 – P14 < 100; P14 – P13 > 100
D. P13 – P14 > 100; P14 – P13 > 100
10. Whenever all the constraints in a linear program are expressed as equalities, the linear program is said to be written in A. standard form.
B. bounded form.
C. feasible form.
D. alternative form.
II. Problem Solving
1. Consider the following linear programming problem:
a. Write the problem in standard form. Identify slack/surplus variables. (10 points)
b. The optimal solution of the above LP model is (180/7, 150/7). What are the values of the slack and surplus variables at the optimal solution? (12 points)
2. Consider the following linear programming problem:
a. Identify the feasible region. (12 points)
b. Are any of the constraints redundant? If yes, then identify the constraint that is redundant. (10 points)
c. Find all the extreme points – list the value of x1 and x2 at each extreme point. (6 points)
d. What is the optimal solution? (5 points) Mathematics III Examination Homework Help
3. RVW (Restored Volkswagens) buys 15 used VW’s at each of two car auctions each week held at different locations. It then transports the cars to repair shops it contracts with. When they are restored to RVW’s specifications, RVW sells 10 each to three different used car lots. There are various costs associated with the average purchase and transportation prices from each auction to each repair shop. Also there are transportation costs from the repair shops to the used car lots. RVW is concerned with minimizing its total cost given the costs in the table below.
a. Given the costs below, draw a network representation for this problem. (10 points)
| Repair Shops | Used Car Lots | ||||||
| S1 | S2 | L1 | L2 | L3 | |||
| Auction 1 | 550 | 500 | S1 | 250 | 300 | 500 | |
| Auction 2 | 600 | 450 | S2 | 350 | 650 | 450 |
b. Formulate this problem as a linear programming model. (15 points)
(Note: You do not need to solve the model.)
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