The recreational center at UMass is considering increasing the staffing hours fo

Question

The recreational center at UMass is considering increasing the staffing hours for two different locations: the basketball court and the gym. Only one staff member is needed during staffing hours. Three staff members now have extra time. The hourly rates of the first staff member are $8 and $9.5 at the two locations, respectively. The hourly rates of the second staff member are $7.5 and $7 at the two locations, respectively. The hourly rates of the third staff member are $10 at the first location, whereas he cannot work at the second location. The increase of the staffing hours at the two locations should be at least 3 and 5 per day, respectively. The three staff members have available extra time of 2, 4, and 3 hours per day, respectively. Management wishes to know how to allocate the staff members to the locations to minimize total extra labor cost per day. Formulate (write down the decision variable definition, objective function, and constraints) but do not solve this problem as a transportation problem.

Explanation / Answer

There are three staff members available for extra times of 2, 4, and 3 hours respectively. There are two locations which require at least 3 and 5 extra hours. Allocation of staff to these locations costs are varying with the the staff and the location. The given data may be represented in the form of a transportation problem as follows:

Decision Variables are defined as the Xij representing the number of extra hours of ith staff allocated to the jth location.

Objective function is to minimize the total costs of allocating staff members to the locations.

Minimize Sigma (over i) Sigma(over j) Cij*Xij where Cij is hourly rate of allocating ith staff to jth location

Constraints are in terms of the availabilioty of hours with the staff members, and the requirements of hours at the locations.As the demand is given in terms of atleast values therefore it is recommended to have full utilization of available hours. Then there is non-negativity constraints as the number of hours can not be negative.

Hourly rates matrix Locations Staff Member Court Gym. Constraint Hrs. Available 1 8 9.5 " = " 2 2 7.5 7 " = " 4 3 10 ---- " = " 3 Constraint " >= " " >= " Minimum Hrs. 3 5

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