# (Solved Homework): Operation Research Many applications of linear programming involve determining the minimum cost…

Operation Research

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Many applications of linear programming involve determining the minimum cost for satisfying work force requirements. Formulate an LP for the following problem of this type. A post office requires different numbers of full-time employees on different days of the week. The number of ful1-time employees required on each day is given in the table below. Union rules state that each ful1- time employee must work five consecutive days and then receive two days off. For example, an employee who works Monday through Friday must be off on Saturday and Sunday and an employee who works Thursday through Monday must have Tuesday and Wednesday off. The post office wants to meet its daily requirements with only full-time employees. Formulate an LP that the post office can use to minimize the number of full-time employees that must be hired. (Let xi be the number of employees hired to begin their work week on day i.)

The following summary table has been derived from the information in the problem statement:

This is a good clarification.

 Table 1 – Summary Table Day Full-Time Employee Starts Shift Minimum Number of Workers For Day Limitations 1 = Monday 17 Work: Mon-Fri Off-Work: Sat-Sun 2 = Tuesday 13 Work: Tue-Sat Off-Work: Sun-Mon 3 = Wednesday 15 Work: Wed-Sun Off-Work: Mon-Tue 4 = Thursday 19 Work: Thu-Mon Off-Work: Tue-Wed 5 = Friday 14 Work: Fri-Tue Off-Work: Wed-Thu 6 = Saturday 16 Work: Sat-Wed Off-Work: Thu-Fri 7 = Sunday 11 Work: Sun-Thu Off-Work: Fri-Sat

Decision Variables

These are the decision variables for this problem:

 Table 2 – Decision Variables Decision Variable Description X1 Number of employees starting on Monday X2 Number of employees starting on Tuesday X3 Number of employees starting on Wednesday X4 Number of employees starting on Thursday X5 Number of employees starting on Friday X6 Number of employees starting on Saturday X7 Number of employees starting on Sunday

The results from each of these decision variables specifically show the number of workers beginning their 5-day shift on that particular day. For instance, the result for X1 represents the number of employees starting their 5-day long shift on Monday; the result for X2 represents the number of employees starting their 5-day long shift on Tuesday.

Objective Function

The goal of this problem is to minimize the number of employees to fulfill the Post Office’s daily workforce size demand. Since the decision variables quantify the number of employees starting on each day, there is no error of duplicity. Therefore, the following objective function has been made:

Min z =X1+X2+X3+X4+X5+X6+X7

Constraints

The main constraint in this problem is the specific number of employees needed per day, as shown:

One can interpret the chart as this: “For Monday, X1, X4, X5, X6, and X7 employees will be working. And, the required number of employees working on Monday is 17 people. Therefore, the sum of X1, X4, X5, X6, and X7 must be at least 17.” It should be noted that X1 ≠ X1, because X1 means the number of employees beginning their 5-day long shift on Monday, and X1 refers to the group of employees beginning their 5-day long shift on Monday. In other words, X1 is the number of people in X1.

By using the information in Figure 1, the S.T. Equations have been created, as shown below:

Another constraint that should be taken into consideration is sign restriction, as shown below:

X1, X2, X3, X4, X5, X6, X7 ≥0

This constraint is necessary because the essence of the problem can never allow a negative number of people. With this restriction, it prohibits the optimum solution from having any negative values for the decision variables.

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