Posted: November 9th, 2015

Linear Programming

Linear Programming

1. (a) What is an assignment problem? Briefly discuss the decision variables, the

objective function and constraint requirements in an assignment problem. Give a

real world example of the assignment problem.

(b) What is a diet problem? Briefly discuss the objective function and constraint

requirements in a diet problem. Give a real world example of a diet problem.

(c) What are the differences between QM for Windows and Excel when solving a linear

programming problem? Which one you like better? Why?

(d) What are the dual prices? In what range are they valid? Why are they useful in

making recommendations to the decision maker? Give a real world example.

Answer Questions 2 and 3 based on the following LP problem.

Let     P1 = number of Product 1 to be produced
P2 = number of Product 2 to be produced
P3 = number of Product 3 to be produced
P4 = number of Product 4 to be produced

Maximize 80P1 + 100P2 + 120P3 + 70P4        Total profit
Subject to
10P1 + 12P2 + 10P3 + 8P4 = 3200       Production budget constraint
4P1 + 3P2 + 2P3 + 3P4 = 1000       Labor hours constraint
5P1 + 4P2 + 3P3 + 3P4 = 1200       Material constraint
P1 > 100         Minimum quantity

needed for Product 1 constraint
And P1, P2, P3, P4 = 0             Non-negativity constraints

The QM for Windows output for this problem is given below.

Linear Programming Results:
Variable    Status    Value
P1    Basic    100
P2    NONBasic    0
P3    Basic    220
P4    NONBasic    0
slack 1    NONBasic    0
slack 2    Basic    160
slack 3    Basic    40
surplus 4    NONBasic    0
Optimal Value (Z)    34400

Original problem w/answers:
P1        P2         P3         P4          RHS        Dual
Maximize    80       100       120        70
Constraint 1    10         12         10         8   <=    3200         12
Constraint 2           4           3          2          3   <=    1000          0
Constraint 3           5           4          3          3   <=    1200          0
Constraint 4          1            0          0          0   >=      100       -40
Solution->         100            0      220          0   Optimal Z->    34400

Ranging Results:
Variable    Value    Reduced Cost    Original Val    Lower Bound    Upper Bound
P1    100    0    80    -Infinity    120
P2    0    44    100    -Infinity    144
P3    220    0    120    87.5    Infinity
P4    0    26    70    -Infinity    96

Constraint    Dual Value    Slack/Surplus    Original Val    Lower Bound

Upper Bound
Constraint 1    12    0    3200    1000    3333.333
Constraint 2    0    160    1000    840     Infinity
Constraint 3    0    40    1200    1160     Infinity
Constraint 4    -40    0    100    0    120

2. (a) Determine the optimal solution  and optimal value and interpret their

meanings.
(b) Determine the slack (or surplus) value for each constraint and interpret its

meaning.

3. (a) What are the ranges of optimality for the profit of Product 1, Product 2,

Product 3, and Product 4?
(b) Find the dual prices of the four constraints and interpret their meanings. What

are the ranges in which each of these dual prices is valid?
(c) If the profit contribution of Product 2 changes from $100 per unit to $130 per

unit, what will be the optimal solution? What will be the new total profit? (Note:

Answer this question by using the ranging results given above).
(d) Which resource should be obtained in larger quantity to increase the profit

most? (Note: Answer this question using the ranging results given above.).

4. The Portfolio Manager of Charm City Pension Planners, Inc., has been asked to

invest $1,000,000 of a large pension fund. The management of the company has

identified five mutual funds as possible investment options. The details of these

five mutual funds are given below:

Mutual Fund
1           2           3

4             5
Annual return (in dollars)     12%       10%      8.5%       10%      11%
Risk amount (in dollars)       9.8%       8%      7.2%       7.1%     7.3%

To control the risk, the management of the company has specified that the total

risk amount cannot exceed $200,000. In addition, the management wants to invest at

least $150,000 in mutual fund 2 and at least $125,000 in mutual fund 3.

With these restrictions, how much money should the portfolio manager of the company

invest in each mutual fund so as to maximize the total annual return?

(a) Define 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. A charity wants to contact people to collect donations. A person can be

contacted morning or evening, by phone, or door-to-door. The average donation

resulting from each type of contact is given below:

Phone          Door-to-Door
____________________________________
Morning           $35                    $60
Evening            $40                    $70

The Charity has 150 volunteer hours in the morning and 120 volunteer hours in the

evening. Each phone contact takes 6 minutes and each door-to-door contact takes 15

minutes to conduct. The Charity wants to have at least 550 phone and at least 400

door-to-door contacts.

Formulate a linear programming model that meets these restrictions and maximizes

the total average donations 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. The Charm City Truck Rental Inc. has accumulated extra trucks at three of its

truck leasing outlets, as shown in the following table:

Leasing Outlet    Extra Trucks
1. Atlanta          70
2. St. Louis        115
3. Greensboro          60

The firm also has three outlets with shortages of rental trucks, as follows:

Leasing Outlet    Truck Shortage
A. New Orleans          80
B. Cincinnati             50
C. Baltimore             45

The firm wants to transfer trucks from those outlets with extras to those with

shortages at the minimum total cost. The following costs of transporting these

trucks from city to city have been determined:

To (cost in dollars)
From       A     B     C
1      75    80    45
2    115    50    55
3    100    60    40

For this transportation problem:

(a) Define the decision variables.
(b) Determine the objective function. What does it represent?
(c) Determine all the constraints. Briefly describe what each constraint

represents.

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