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A dynamic programming approach to work force scheduling
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A Dynamic Programming Approach to Work Force Scheduling FLOWliRS and Preston [~1] reported the results of a simulation stud? of the application of the search decision rule tSDRI to a work force scheduling problem for a manufacturing firm. The results indicated that the SDR methodology was easily implemented, did not require restrictive mathematical assumptions, was eomputationally efficient and produced superior results over the simulated time period 165 weeks) when compared to the actual compan? decisions. The purpose of this note is to demonstrate that the same work force scheduling problem may be formulated and solved optimally as a dynamic programming problem, and to compare its solution by dynamic programming to the SDR methodology. In this case, as is often true in management science applications, there are alternate ways to formulate and solve the same problem. The special nature of this scheduling problem resulted in restricting the decision variable to include only the work force size in each period. These decision variables become the state variables for the dynamic programming formulation. To determine the possible values of the state variables for the specified thirteen week planning horizon, the maximum and minimum number of hours scheduled dr, ring the planning horizon are first determined. Then the integer number of employees necessary to allow the maximum scheduled hours to be worked completely on regular time is compvted by dividing the maximum scheduled hours by the 37.75 net productive hours per week per employee and the result rounded up to the next integer: value. The minimum work force size is determined analogously using the minimum scheduled hours and rounding the rest.It to the next lower integer. The minimum and maximun work force size so determined are then compared to the beginning work force size for the planning period. If the beginning value is larger than the computed maximum, the maximum is reset to the beginning value. If the minimum computed value is greater than the beginning work force size, the minimum is reset to equal the beginning work force size. The resulting limits on the decision variable are used for each period in the planning horizon to determine the number of states considered, The same objective function as was used in the SDR analysis is also used for the dynamic programming solution. Standard backward induction procedures for dynamic programming problems were programmed for the same computer (IBM 370/145) that was used to solve the original problems with the SDR methodology, and the analogous
fourteen runs for the thirteen week planning horizon were made. The results achieved by the dynamic programming algorithm included a total cost for the 65 week period of 5194.004.71 as compared to the SDR value of 5193.799.23 and an average execution time of 1.5 seconds per run for dynamic programming as compared to It seconds per run of t h e ' S D R procedure. The explanation of these results is relatively straightforward. The SDR methodology, while not optimizing each 13 week period, nonetheless finds "frozen" solutions for four periods at a time Isee Flo~.ers and Preston [~1]) which result in a total cost 0.[ per cent lower than the 13 week dynamic programming algorithm. The true optimum solution for the problem when solved as a 65 week dynamic programming problem results in a total cost of 5193.210.81. The computational results for the dynamic programming result in a clear advantage as compared to the SDR method. The SDR tries a number of values in its searches that are not promising (i.e. values above the maximum and below the minimum work force size) and the SDR has a step size used to change the search variable values which must be initially set to a fairly large value to explore a substantial area of the response surface and then later reduced when no further improvement with that step size is possible. This results in a larger number of function evaluations than are necessary for the implicit enumeration approach of dynamic programming.
REFERENCES I. FLOWERS AD and PRES'rO,~ SE (1977) Work force scheduling with the search decision rule. Omega 5(4). 473--479.
A Dale Flowers David L Dukes Walter D Curtiss (April 1978) Department of Operations Research Case Western Reserre Unit'ersity Cleveland Ohio 44106 USA
.$lemoranda
A Dynamic Programming Approach to Work Force Scheduling FLOWliRS and Preston [~1] reported the results of a simulation stud? of the application of the search decision rule tSDRI to a work force scheduling problem for a manufacturing firm. The results indicated that the SDR methodology was easily implemented, did not require restrictive mathematical assumptions, was eomputationally efficient and produced superior results over the simulated time period 165 weeks) when compared to the actual compan? decisions. The purpose of this note is to demonstrate that the same work force scheduling problem may be formulated and solved optimally as a dynamic programming problem, and to compare its solution by dynamic programming to the SDR methodology. In this case, as is often true in management science applications, there are alternate ways to formulate and solve the same problem. The special nature of this scheduling problem resulted in restricting the decision variable to include only the work force size in each period. These decision variables become the state variables for the dynamic programming formulation. To determine the possible values of the state variables for the specified thirteen week planning horizon, the maximum and minimum number of hours scheduled dr, ring the planning horizon are first determined. Then the integer number of employees necessary to allow the maximum scheduled hours to be worked completely on regular time is compvted by dividing the maximum scheduled hours by the 37.75 net productive hours per week per employee and the result rounded up to the next integer: value. The minimum work force size is determined analogously using the minimum scheduled hours and rounding the rest.It to the next lower integer. The minimum and maximun work force size so determined are then compared to the beginning work force size for the planning period. If the beginning value is larger than the computed maximum, the maximum is reset to the beginning value. If the minimum computed value is greater than the beginning work force size, the minimum is reset to equal the beginning work force size. The resulting limits on the decision variable are used for each period in the planning horizon to determine the number of states considered, The same objective function as was used in the SDR analysis is also used for the dynamic programming solution. Standard backward induction procedures for dynamic programming problems were programmed for the same computer (IBM 370/145) that was used to solve the original problems with the SDR methodology, and the analogous
fourteen runs for the thirteen week planning horizon were made. The results achieved by the dynamic programming algorithm included a total cost for the 65 week period of 5194.004.71 as compared to the SDR value of 5193.799.23 and an average execution time of 1.5 seconds per run for dynamic programming as compared to It seconds per run of t h e ' S D R procedure. The explanation of these results is relatively straightforward. The SDR methodology, while not optimizing each 13 week period, nonetheless finds "frozen" solutions for four periods at a time Isee Flo~.ers and Preston [~1]) which result in a total cost 0.[ per cent lower than the 13 week dynamic programming algorithm. The true optimum solution for the problem when solved as a 65 week dynamic programming problem results in a total cost of 5193.210.81. The computational results for the dynamic programming result in a clear advantage as compared to the SDR method. The SDR tries a number of values in its searches that are not promising (i.e. values above the maximum and below the minimum work force size) and the SDR has a step size used to change the search variable values which must be initially set to a fairly large value to explore a substantial area of the response surface and then later reduced when no further improvement with that step size is possible. This results in a larger number of function evaluations than are necessary for the implicit enumeration approach of dynamic programming.
REFERENCES I. FLOWERS AD and PRES'rO,~ SE (1977) Work force scheduling with the search decision rule. Omega 5(4). 473--479.
A Dale Flowers David L Dukes Walter D Curtiss (April 1978) Department of Operations Research Case Western Reserre Unit'ersity Cleveland Ohio 44106 USA