Instructions

This is a group term project. No individual projects will be marked.

Goal of the Term Project

The overarching goal of this term project is to give students an opportunity to (i) discover a shift scheduling problem, which is a widespread class of problems, and (ii) learn to solve

Management Science Problems with LINGO, a user-friendly algebraic modelling software.

Statement

Maliki manages a service center in the City of Sudluvy. The center opens at 7:00 am everyday  and closes at 7:00 pm. Each day, Maliki has a number of employees on call; the exact number  depends on each day’s needs. To better manage the operations of the center, Maliki divides the daily 12-hour work period into 24-half-hour periods.

You have been hired by Maliki to build an optimal schedule for next week in order to minimize   the total number of employees on call each day. Since Maliki has forecasted the same number of required employees for each day of the week, you can build the schedule for only one day. Table 1 below shows the forecasted number of required employees for each half-hour period.

Maliki’s center has 10 different shifts per day.  The start time and duration of each shift is shown in Table 2.

Part I

You first task is to formulate a mathematical model for this problem and to solve it with LINGO. For more efficiency, you are asked to use the concept of sets in LINGO.1 Define the following

two sets:

J = The set of indices of the allowed shifts; and

I = The set of periods

Remarks:  Provide a generic compact mathematical model by appropriately defining your decision variables and any parameters. Call this problem P1. Use the data only in LINGO.

Summary the optimal solution supplied by LINGO in a table and provide appropriate comments.

Now, solve a linear relaxation of P1, what do you observe? Provide detailed comments. Among others, compare the solution times and the number of solver iterations.

Part II

After several complaints from its staff, Maliki decided to allow the employees to take one break  per day; however, the required number of employees per time period still needs to be met. The    start time and duration of the breaks by shift is shown in Table 3. What change should be made   in P1 to reflect this new situation? Call this new problem P2. Solve P2 with LINGO as well. How does the optimal solution change? In a table, illustrate the differences between the optimal solutionsto P1 and P2, and provide appropriate comments.

Part III (A challenge)

Something happened to Maliki, he suddenly became a good manager. He decided to make the   break period flexible, however the duration didn’t change. This is good news to Katherine; she  didn’t want to take her break at the sametime as a few colleagues she didn’t stand, especially a guy by the name Peter XLow.  The break time window for each shift is shown in Table 4.

Provide a new formulation to account for this change. Call this problem P3. Solve P3 with

LINGO. Illustrate the differences between the three solutions in a table and provide appropriate analysis.

Remark: This part is difficult, you need a few more sets! You’ll need my help for sure!

Part IV

Solve a model P4, in which the break time is fixed at the end of the break window for each shift. Again, provide detailed analysis and compare the optimal solutions with the previous ones.

Part V

A few employees complained that the break policy was not fair. For instance, employees assigned to Shift 7 only worked 5 hours and had ahalf-hour break. Maliki agreed with them and decided that the break duration for Shift 7 would be reduced to 0.25 hours; similarly, the break   duration for Shift 10 to 0.5 hours. What changes should be made in the problem formulation or   structure to account for this? Call this problem P5. Solve P5, and provide a detailed analysis of    the optimal solution as well as comparisons with the previous solutions.

To submit:

•   Your paper

•   Your LINGO files