OPRE 315 Homework 4

1. 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                           900               1200                500                  25

   Factory            Seattle                           650              1050                700                  25

                         Baltimore           550               815                 472                  10

                           Detroit                 620                  905                   514                     25

             __________________________________________________________________

                          Demand             40                   20                 25      

(a) Formulate a linear programming problem to minimize total cost for this transportation problem.

(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 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 five jobs. The company needs to assign an installation crew to each of the five 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     Job 5

___________________________________________________

Red                  26        34        32        48         52

White               38        36        39        47         60

Crew    Blue                 25        40        30        41         55

Green               28        40        35        38         48

The company wants to assign Red Crew or White Crew to Job 3 because of customer preference. It does not want to assign Blue Crew to Job 1 because of quality considerations.

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

Formulate this assignment problem as a linear programming model by determining

(a) The decision variables

(b) The objective function

(c) All the constraints

Note: Do NOT solve the problem after formulating.

3. A congressman’s district has recently been allocated $52 million for projects. The congressman has decided to allocate the money to six ongoing projects. However, the congressman wants to allocate the money in a way that will gain him the most votes in the upcoming election. The details of the six projects and votes per dollar for each project are given below.

Project              Votes/dollar

________________________

Parks                      0.10

Education               0.09

Roads                     0.10

Health Care            0.12

Child Welfare        0. 08

County Library      0.08

In order to satisfy some local influential citizens, he must meet the following guidelines.

- None of the projects can receive more than 25% of the total allocation.

- The amount allocated to education cannot exceed the amount allocated to health care.

- The amount allocated to child welfare must be equal to or more than the amount spent on the

   county library.

- All of the money must be allocated.

Formulate a linear programming model for the above situation by determining

(a) The decision variables

(b) The objective function

(c) All the constraints

Note: Do NOT solve the problem after formulating.

4. An ad campaign for a trip to Europe 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.

The number of TV ads cannot be more than 2. 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 12 ads in all four media. The advertising budget is $42,000. 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) The objective function

(c) All the constraints

Note: Do NOT solve the problem after formulating.