A Spreadsheet Model for Simultaneous Management of ...



on establishing service

operatiNG hours

Dechter, Avi

Department of Systems and Operations Management

College of Business and Economics

California State University, Northridge

E-mail: adechter@csun.edu

Abstract

Establishing the days and hours it is open for service is a critical decision for every service organization. In making this decision the service organization should consider the trade-off between the cost of keeping its service facility open and the potential loss associated with leaving it closed. This paper proposes an integer linear programming model which explores this trade-off and may be used to obtain an optimal operating schedule.

Introduction

An important aspect of the planning of any service organization is determining its operating schedule, that is, the days and hours its customer service units are open for service. This decision is not a simple one as is evident from the fact that different organizations, even within the same industry -- banking, for example -- differ widely in their policies regarding hours of operation. While many services operate 24 hours a day, 7 days a week, many others limit the number of hours they operate per day as well as the numbers of days they operate per week. The operating schedule decision is part of a group of service policies commonly referred to as the management of capacity and demand. Capacity management and demand management techniques are particularly important in service operations and have received much attention in the service research literature (e.g., Heskett, Sasser & Hart, 1990, Showalter & White, 1991, Crandel & Markland, 1996). As pointed out by Klassen (1997), the operating schedule decision affects both the capacity and the demand of the service organization. It is therefore surprising that there is practically no discussion on this subject in the service literature.

Ideally, at least from the customers’ point of view, a service operation should be available for service whenever demand for its services might exist. For most service organizations, however, such a policy would not be practical because the cost of keeping the operation open may not be justifiable during periods where demand is relatively low. When deciding not to offer service in periods when there might be demand, service providers must believe that the resulting savings in operating costs are greater than the potential cost of not meeting this demand (for example, lost sales). Unfortunately, this kind of analysis cannot be done separately for each hour of the day (or each day of the week) primarily for two reasons. First, the “natural” demand for a particular period may not be completely lost if service is not offered during that period but rather shifted to other periods. Second, service employees’ work-assignments are not normally done to individual periods, but rather to work-shifts, or tours, involving a minimum number of periods (e.g., five days a week, eight-hour shift per day, etc.). For this reason, deciding to close the service during a particular period may not result in proportional savings in labor costs. Calculation of the effect of such a decision on labor costs requires the adjustment of the entire work schedule. Therefore, the decision regarding which hours to keep the service open cannot be done independently of the decision of how employees are scheduled to work.

Employee work-shift scheduling has been one of the most effective tools for capacity management in service operations. With this method, employees are scheduled to work separate shifts, which may vary from one employee to another in their starting times and, within limits, in their lengths (or even the timing of relief breaks), so that the aggregate capacity provided by all of the shift assignments corresponds more closely to the demand. Numerous models and solution techniques have been proposed for obtaining optimal work-shift schedules (e.g., Bechtold & Jacobs, 1990, Thompson, 1995, Aykin, 2000). The input data for these models consists of a measure of the demand in each of a set of consecutive planning periods (typically 15 minutes to one hour long) in a given planning horizon, and the specification of the permissible shifts. The objective is to determine the number of employees that should be assigned to each shift so that the demand in each period can be met while the total labor cost is minimized. The implication of the constraint that all demand must be met is that the service facility must be open regardless of how large or how small the demand is. In this paper we show how such a scheduling model might be modified to include the question of operating hours in addition to the question of the employee shift scheduling, thus creating a tool for a more rational determination of an operating schedule.

A crucial aspect of an operating hours model is that of modeling how varying the operating hours affects the demand. The simplest case is that the demand in periods selected for closure is lost. This would be a reasonable assumption when the customers of the service organization are occasional, for example, a fast-food restaurant in an airport or a shopping mall. However, when the customers are recurring, for example at the neighborhood grocery store or post office, it would probably more reasonable to assume that at least some of the demand of the periods selected for closure will shift to other periods. The modeling of such a situation is also discussed.

The remainder of this paper is organized as follows. In the next section we discuss a variation of the work-shift scheduling model we use as the starting point of our hours of operations model. In the third section we modify the model to include the selection of the operating hours under the assumption of lost demand. In the following section we discuss the modeling of the demand shifting which may occur as a result of changing operating hours. We conclude summarizing the paper and pointing to possible extensions of this research.

The Work-shift scheduling model

The first integer programming formulation of the work-shift scheduling problem was suggested by Dantzig (1954), who suggested that the problem could be modeled as a set-covering problem. A variation of this model is given below:

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where J represents the set of permissible shifts, T is the set of planning periods in the planning horizon, dt is a measure of the demand (e.g., number of calls) during period t, bt is the contribution to the service capacity of each employee scheduled to work during period t, ajt is equal 1 if period t is a work period for shift j and 0 otherwise, ct is the cost of scheduling an employee to work during period t, and xj is an integer variable indicating the number of employees assigned to shift j.

Many authors have pointed out that in this formulation the number of variables could be very large and proposed other formulations. However, we use this formulation because it is the simplest and most succinct. To illustrate the use of this model, consider the following highly simplified example. The expected number of calls to a call center during each hour in day (24 hours) is given in figure 1. An employee can handle up to 8 calls per hour. An employee may be scheduled to any 8-hour long shift that starts and ends within the same day. Each hour an employee is scheduled to work costs $20. There is also a fixed cost of $100 for every when the call center is open (i.e., at least one employee is scheduled to work). An optimal solution for this instance was obtained using ILOG OPL Studio (ILOG, 1999). An optimal assignment of employees to shifts is shown graphically in figure 2. The minimum number of employees needed is 15. A graph contrasting the demand and the capacity provided by this schedule is given in figure 3.

[pic]

Figure 1

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Figure 2

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Figure 3

The total cost of this schedule is $4,800 (fixed cost: $2,400, variable cost: 20(15(8 ( $2,400). Since there is some demand during each of the 24 hour-long periods, the facility must remain open throughout the day and the fixed cost is incurred in each of these periods. In the following section relax this requirement and let the model determine in which periods, if any, the service should remain closed.

the OPERATING HOURS model

In order to add the selection of hours of operation to the work-shift scheduling model, it has to be augmented in several ways. First, we add a set of binary variables, yt, t(T, such that yt = 1 if the service facility is open during planning period t, and 0 otherwise. Second, the cost function should include not just the variable cost incurred when employees are assigned to work, but also any fixed costs required just to keep the facility open and the cost of not meeting the demand during periods when the facility is closed. Finally, constraints must be added to ensure that employees are not assigned to work when the facility is closed. The modified model is:

[pic]

where, in addition to the notation defined above, ft is the fixed cost of keeping the service facility open during period t, rt is the cost of losing one unit of demand during period t, and Ut is an upper bound on the number of employees that could be assigned to work during period t. The purpose of the second set of constraints is to guarantee that no employee is assigned to work during periods when the service facility is closed.

Consider now the example from the preceding section with the additional data that the cost of losing one unit of demand (one call) in any period is $10. An optimal solution for this instance suggests that the facility should be closed during the first 6 periods. The number of employees needed is now only 14 and an optimal schedule is shown in figure 4. A graph contrasting the demand and the capacity provided by this schedule is given in figure 5. The total cost of this schedule is $4,350 (fixed cost: $1,800, variable cost: $2,240, and lost demand cost: $310), that is $450 less than the solution obtained by the original model.

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Figure 4

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Figure 5

In this example, it is apparent that the demand in the first six periods is too low to justify the sum of the fixed cost associated with keeping the service open and the cost of the additional employee that is needed. However, this is true for the given demand pattern and cost structure. Changing the inputs may change the solution. For example, if the fixed cost is doubled (from $100 to $200), leaving everything else unchanged, it is optimal to keep the service open only for 9 hours each day. Doubling the lost sales cost (from $10 to $20), on the other hand, would make it optimal to restore hour 6 to the operating schedule. Figure 6 shows the optimal number of hours in the operating schedule for different combinations of fixed costs and lost sales costs.

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Figure 6

Modeling demand shifting

As pointed out in the introduction, in many cases the assumption that all the demand in the periods selected for closing is lost would not be correct. Rather, it is often the case that at least part of this demand will be shifted to other periods. In such cases, the model of the last section would not be appropriate and must be modified to reflect the expected redistribution of the demand. For this purpose, what is needed, is some characterization of the effect of closing the service in a given time period on the “natural” demand for this time period. Several examples of such a characterization follow:

1. The demand in a period selected for closing is shifted to the first open period following the closed period.

2. The demand in a period selected for closing is shifted to closest open period (either before or after the closed period).

3. The demand in all the periods selected for closing is distributed equally among all the open periods in the planning horizon.

4. The demand in all the periods selected for closing is distributed among all the open periods in the planning horizon in the same proportion as the “natural” demand for these periods.

We now demonstrate how the first characterization (demand moves to next open period) may be represented in our model. The key is to assume that demand may “flow” from each period to the immediately succeeding period. (The demand of the last period flows to the first period representing shifting to the next cycle, e.g., the next day.) To guarantee that the demand of closed periods ends up in the first open periods, two constraints must be satisfied. First, The net outflow of each closed period must be equal to the original demand of that period. This will cause the remaining demand in this period to be zero. Second, no flow out of an open period is allowed. Let zt be the flow out of period t, t = 1,2,…P, and, as before, let dt be the demand in period t, and yt, t = 1,2,…P, be a set of binary variables such that yt = 1 if the service facility is open during planning period t, and 0 otherwise. Then the following set of constraints will guarantee that the demand in every period selected for closing is shifted to the next open period.

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For example, if the service is closed for periods 1, 2, 3, 13, 23 and 24, the constraints above will cause the demand of figure 1 to be modified as shown in figure 7.

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Figure 7

Adding the flow variables and the new constraints to the operating hours model and solving (for the original cost structure) we obtain a new solution depicted in figures 8 and 9.

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Figure 8

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Figure 9

As can be seen in figures 8 and 9, under the new assumption of shifting demand it is again optimal to open the service for just 18 out of the 24 hours. However, these are different hours than in the case of lost demand. Instead of closing in hours 1 trough 6, it is recommended to close in hours 1 through 5 and in hour 24. The demand from these hours is shifted to hour 6. As figure 6 shows, 14 employees are needed in this case. The total cost of this schedule is $4,040 (fixed cost: $1,800, variable cost: $2,240). There is no lost sales cost because demand is not lost here. It would be a simple matter to modify the model to reflect an assumption that a certain percentage of the demand is lost while the rest shifts to other periods.

Similar methods may be used to represent other characterizations of demand shifting. These will be explored in a future study.

conclusion

Many issues enter the selection of an operating schedule for a service organization, including competitive considerations, governmental restrictions, and employee union agreements. However, the overriding consideration is an economic one: finding the best balance between the costs required to keep the service open and the cost of not meeting the demand as it occurs. This balance depends on the various costs involved, on the way capacity is modified to meet varying demand (e.g., shift scheduling), and on the way customers react to changes in the operating schedule.

This paper shows how an integer linear programming model may be used to combine all of these factors in order to find an optimal operating schedule. We then use the model to explore the effects of changing some of the inputs on the operating schedule.

Our model is highly simplified and is limited in many ways, particularly in that it is restricted to considering the operating schedule in a single day. The same ideas, however, would certainly apply to longer time frames and to the question of which days of the week should be included in the operating schedule in addition to the question of the operating hours each day.

REFERENCES

Aykin, T. (2000) A comparative Evaluation of Modeling Approachs to the Labor Shift Scheduling Problem”, European Journal of Operational research, No. 125, pp. 381-397.

Bechtold, S.E. & Jacobs, L.W., (1990) “Implicit Modeling of Flexible Break Assignments in Optimal Shift Scheduling”, Management Science, Vol. 36, No. 11, pp. 1339- 1351.

Crandall, R.E. & Markland, R.E., (1996) “Demand Management - Today's Challenge for Service Industries”, Production and Operations Management, Vol. 5, No. 2, 1996, pp.106-120.

Dantzig, G.B., (1954) “A Comment on Edie’s ‘Traffic Delay at Toll Booths’”, Operations Research, Vol. 2, No. 3, pp. 339-341.

Heskett, J.A., Sasser, W.E. & Hart, C.W., (1990) Service Breakthroughs. New York: The Free Press, 1990.

Klassen, K.J., (1997) “Simultaneous Management of Demand and Supply in Services”, Ph.D. Dissertation, The University of Calgary, Alberta, Canada, 1997.

Showalter, M.J. & White, J.D., (1991) “An Integrated Model for Demand-Output Management in Service Organizations: Implications for Future Research”, International Journal of Operations and Production Management, Vol. 11, No. 1, 1991, 51-67.

Thompson, G.M., (1995) “Improved Implicit Optimal Modeling of the Labor Shift Scheduling Problem”, Management Science, Vol. 41, No. 4, pp. 595-607.

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