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IME 605, 2015-16

LP Formulation Problems

1.

Distributing Goods through a Distribution Network

The DISTRIBUTION LTD. CO. produces a product at two different factories, and then shipsthem to two warehouses from either factory. The distribution network available for shipping thisproduct is shown in Fig. 1, where F1 and F2 are the two factories, W1 and W2 are the twowarehouses, and DC is a distribution center. The amounts to be shipped from F1 and F2 areshown to their left, and the amounts to be received at W1 and W2 are shown to their right. Eacharrow represents a feasible shipping lane. Thus, F1 can ship directly to W1 and has three possibleroutes (F1 DC W2, F1 F2 DC W2, and F1 W1 W2) for shipping to W2.Factory F2 has just one route to W2 (F2 DC W2) and one to W1 (F2 DC W2 W1).The cost per unit shipped through each shipping lane is shown next to the arrow. Also shownnext to F1 F2 and DC W2 are the maximum amounts that can be shipped through these lanes.The other lanes have sufficient shipping capacity to handle everything these factories can send.Formulate the problem as a linear program to minimize the total shipping cost.

Figure 1. The distribution network for Distribution Ltd. Co.

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2. Job-Training Problem

A machine tool company conducts a job-training program for machinists. Trained machinists are

used as teachers in the program at a ratio of one for every ten trainees. The training program lasts

for one month. From past experience it has been found that out of ten trainees hired, only seven

complete the program successfully (the unsuccessful trainees are released). Trained machinistsare also needed for machining and the company's requirements for the next months are as

follows:

January 100

February 150

March 200

In addition, the company requires 250 trained machinists by April. There are 130 trained

machinists available at the beginning of the year. Payroll costs per month are:

Each trainee $400

Each trained machinist $700 (Machining or teaching)

Each trained machinist idle $500

Set up the linear programming problem that will produce the minimum cost hiring and training

schedule and meet the company's requirements.

3.

Roster Planning

The production manager of a chemical plant is attempting to devise a shift pattern for his

workforce. Each day of every working week is divided into three eight-hour shift periods (00:01-

08:00, 08:01-16:00, 16:01-24:00) denoted by night, day and late respectively. The plant must be

manned at all times and the minimum number of workers required for each of these shifts overany working week is as below:

Mon Tue Wed Thu Fri Sat Sun

Night 5 3 2 4 3 2 2

Day 7 8 9 5 7 2 5

Late 9 10 10 7 11 2 2

The union agreement governing acceptable shifts for workers is as follows:

Each worker is assigned to work either a night shift or a day shift or a late shift and once

a worker has been assigned to a shift they must remain on the same shift every day that

they work. Each worker works four consecutive days during any seven day period.

In total there are currently 60 workers. Formulate the production manager's problem as a linear

program.