PDE4905 Engineering Simulation

Assignment 1

Option 3: Linear Programming

(5 Tasks)

1 Linear Programming: A Simple Problem

A small factory has two products (A and B). In order to keep the factory running,

at least 30 units of products must be produced each day. Product A costs 200 (units

of costs, for example, 200 dollars) to produce, while product B costs 100 (units) to

produce. Costs must be less 8000 units (e.g., 8000 dollars) as the budget each day. In

addition, the market regulations require that the number of Product B cannot exceed

twice the number of Product A.

If each of Product A gives a prot of 20 units (dollars) and each of Product B

gives a prot of 30 units (dollars), how many products of A and B should be produced

each day so as to maximize the overall prot?

Task 1

Write above the problem as a simple linear programming problem with the objective

and all appropriate constraints. Then, solve it using any method (e.g., graph method)

to nd the optimal solution. Represent the solution on a graph and sketch the feasible

region.

Task 2

Solve the above linear program using either Excel Solver or Matlab (linprog), and

then conrm if your solution obtained earlier is correct.

2 Linear Programming: Supermarket Scheduling

This part simulates the scheduling of sta for a busy supermarket.

A busy supermarket is open 24 hours daily. It has a set of 20 checkout counters,

though the actual number of checkouts needed ranges from 2 to 20 depending on the

time of day. The number of sta needed to provide a satisfactory service varies at

dierent hours as summarized in Table 1.

Table 1: Number of checkout counters needed.

Time period # Counters Needed

9:00 a.m. -10:00 a.m. 12

10:00 a.m. – 12:00 (noon) 8

12:00 – 2:00 p.m. 18

2:00 p.m. – 5:00 p.m. 8

5:00 p.m. – 9:00 p.m. 20

9:00 p.m. – 3:00 a.m. 4

3:00 a.m. – 9:00 a.m. 2

The supermarket employs 10 full-time sta and a large number of part-time sta.

Part-time sta typically can work a 6-hour shift every day and can start on the hour

between 9:00 a.m. to 2:00 p.m. or at 5:00 pm, 9:00 pm or 3:00 am. Full-time sta

will work at most 40 hours per week between 9:00 a.m. to 5:00 p.m. with a daily

wage of $90 for 8 hours per day, while part-time workers are paid at a standard rate

of $8 per hour (i.e., $48 per day for 6 hours). The company policy only permits that

up to 80% of the hours of any day to be part-time hours.

The main task is to schedule the sta so that the total sta costs are minimized

on a daily basis.

Task 3

Formulate the design problem in terms of a linear programming problem with a correct

objective and all appropriate constraints. Explain in detail why your mathematical

formulation is appropriate.

Task 4

Solve your formulated sta scheduling problem using either Excel Solver or Matlab

(intlinprog) or any other tools. Summarize the solution procedure, main results and

any ndings in a report.

3 A Practical Application

Find a real-world application such as a small shop, a bookstore and a small company

so that you can demonstrate how linear programming can be used.

Task 5

Discuss a real-world scenario and then convert the problem into a linear programming

problem. Use any simulated data and try to solve it using either Excel Solver (or

Matlab or any other tools) so as to demonstrate that your formulation make sense

and can yield a good solution.

Marking Criteria and Submission

Write a brief report (about 3000 to 5000 words) to summarize your formulations, the

main solution procedure and key ndings.

Marks/weights: 40% of the total marks will be given for Tasks 1 and 2, 50% for

Tasks 3 and 4, and 10% for Task 5.

For the nal submission, you need to submit both your report le and computer

source le(s) for solving the problems and generating the plots/graphs.

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