- A logistics company must distribute cases of parts from 3 factories to 3 assembly plants. The monthly supplies and demands, along with the per-case transportation costs are:

Assembly Plant

1 2 3 Supply

__________________________________________________________________

A 10 14 18 300

Factory B 6 6 10 500

C 6 12 11 300

__________________________________________________________________

Demand 320 420 300

(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.

- The Charming City Vacuum Company wants to assign three salespersons to three sales regions. Given their experiences, the salespersons are able to cover the regions in different amounts of time. The amount of time (days) required by each salesperson to cover each region is shown in the following table:

** **Region (days)

Salesperson** **I II III

________________________________________

A 9 17 11

B 9 14 13

C 10 13 15

** **

However, because of his health reason, salesperson B does not want to be assigned to region III.

The Company wants to assign either salesperson B or salesperson C to region II. The objective is to minimize total time of covering the three sales regions.

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.

**Mathematical Operations Homework 4**

**Note: Do NOT solve the problem after formulating.**

- A congressman’s district has recently been allocated $46 million for projects. The congressman has decided to allocate the money to five 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 five projects and votes per dollar for each project are given below.

Project Votes/dollar

________________________

Parks 0.09

Education 0.07

Roads 0.10

Health Care 0.11

Family Welfare 0.08

In order to also 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 family welfare cannot be more than the amount spent on roads.

– All of the money must be allocated.

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.**

- A builder is developing a lakeside community. After considering possible advertising media and the market to be covered, the builder has decided to advertise in four media. He collected data on the number of potential customers reached, the cost per advertisement, the maximum number of advertisements available, and the exposure quality rating for each of the four media. These data are given in the following table.

Number of Maximum

Potential Number of Exposure

Customers Cost ($) per Advertisements Quality

Advertising Media Reached Advertisement Available Units

____________________________________________________________________________

Daytime TV (1 min ad) 3500 4000 8 80

Evening TV (30 sec ad) 5000 6000 6 85

Daily newspaper (full page ad) 2000 1300 10 55

Sunday newspaper magazine 3000 1900 10 66

(1/2 page color ad)

The builder has an advertising budget of $100,000 for the campaign. In addition, he wants the following restrictions: At least 10 television commercials must be used, at least 50,000 potential customers must be reached, and no more than $15,000 may be spent on Sunday newspaper magazine advertisements. What advertising media selection plan should be recommended to maximize the total exposure quality units?

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.**

**Notes:**

– Each student must solve Homework 4 problems individually by himself/herself **or **within the team he/she is a member of. Please do not ask for help from anyone else to answer any questions in this homework.

– Please include all details and steps performed to find your answers. Just writing the final answers will not get you full credit.

– Please post your answers in the Assignments section on the course website.

– A team can post their answers in the Assignments section on the course website by any one of the team members but the submission must contain names of all team members. A member whose name is missing from the submission will not receive any credit.

– A team submission will be graded based on the answers submitted by the team. If answer to one or more questions is missing because a member of the team did not do his/her part, then the team will not receive any credit for those questions.

– Please submit the homework answers in Microsoft Word or PDF format. Scans or photos of the answers are acceptable but they must be clear and legible. Other formats are not acceptable.

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