Linear programming $25 | Mathematics homework help

 

You can use a calculator to do numerical calculations. No graphing calculator is allowed. Please DO NOT USE ANY COMPUTER SOFTWARE to solve the problems.

 

 

 

 

 

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 1NONBasic       0

 

slack 2Basic    160

 

slack 3Basic    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.

 

 

 

Note: Do NOT solve the problem after formulating.

 

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