Question

. A manufacturing company produces diesel engines in four factories located in Tucson, Seattle, Baltimore, and Detroit. Three trucking firms purchase these engines for their plants located in Nashville, Miami, and Charleston. The supplies and demands, along with the per engine transportation costs in dollars are given below:

                                                                          Plant                 

                                             Nashville             Miami         Charleston         Supply

             __________________________________________________________________

                        Tucson                800               1100                400                  35

   Factory            Seattle                 550               950                 600                  35

                        Baltimore           450               715                 372                  20

                           Detroit                 520                  805                   414                     35

             __________________________________________________________________

                          Demand              50                   30                 35      

                                                                                                                

(a) Formulate a linear programming problem to minimize total cost for this transportation problem by defining the decision variables, objective function, and all the constraints.

(b) Solve the linear programming formulation from part (a) by using either Excel or QM for Windows. Find and interpret the optimal solution and optimal value. Please also include the computer output with your submission.

The following questions are mathematical modeling questions. Please answer by defining the decision variables, objective function, and all the constraints. Write all details of the formulation. Please do NOT solve the problems after formulating.

2. A carpet company sells and installs floor covering for commercial buildings. The company was awarded a contract for four jobs. The company needs to assign an installation crew to each of the four jobs. Currently, four installation crews are available for assignment. Each crew is identified by a color code. The following table shows the time required (in hours) for each crew to complete each of the five jobs:

                                          Time Required in Hours

                                   Job 1    Job 2     Job 3     Job 4    

_____________________________________________

Red                  34        32        30        48      

White               38        34        36        47      

Crew    Blue                 28        42        35        37      

Green               28        35        35        40       

The company wants to assign Red Crew or Blue Crew to Job 2 because of customer preference. It does not want to assign Green Crew to Job 4 because of quality considerations.

The objective is to minimize the total cost of all assignments.

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.

3. Rachel wants to set up a trust fund for her two children using $1,600,000. The trust fund has three investment options: a bond fund, a stock fund, and treasury bills fund. The projected returns over the life of the investments are 4.2% for the bond fund, 6.2% for the stock fund, and 4.5% for the treasury bills fund. She wants to invest at least 25% of the total amount in the bond fund, at least 30% in the stock fund, and at least 20% in the treasury bills fund. She also wants the amount invested in the stock fund to be more than or equal to the amount invested in the bond fund. She wants to know how much money should be invested in each of the three alternatives to maximize the total projected returns.

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.

4. An ad campaign for a trip to Norway will be conducted in a limited geographical area and can use TV ads, radio ads, magazines ads, and newspaper ads. Information about each medium is shown below.

                                                         

Medium	Cost Per Ad	Number of People Reached
TV	12,500	42000
Radio	2000	8300
Magazine	950	3400
Newspaper	2000	8500

The number of TV ads cannot be more than 3. The total number of Radio and Newspaper ads must be more than the total number of TV ads. There must be at least a total of 10 ads in all four media. The advertising budget is $50,000. The objective is to maximize the total number of people reached.

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.

Answer 1

Solution:

Decision variables: Xij = Quantity to transport from each of the factories indicated by i to each of the plants indicated by j. Where i=1,2,3,4 for each of the factories located in Tucson, Seattle, Baltimore, Detroit and j=1,2,3 for each of the plants located in Nashville, Miami and Charleston

Objective: Minimize Z = 900X11 + 1200X12 + 500X13 + 650X21 + 1050X22 + 700X23 + 550X31 + 815X32 + 472X33 + 620X41 + 905X42 + 514X43

s.t.

X11 + X12 + X13 <= 80

X21 + X22 + X23 <= 100

X31 + X32 + X33 <= 40

X41 + X42 + X43 <= 100

X11 + X21 + X31 + X41 = 160

X12 + X22 + X32 + X42 = 60

X13 + X23 + X33 + X43 = 100

Xij >= 0