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Simultaneous Demand Smoothing and Work-Shift Scheduling in a Service Operation Avi Dechter Department of Management Science College of Business Administration and Economics California State University, Northridge Email: avi.dechter@csun.edu Abstract In trying to balance short-term capacity with uneven demand, service operations often engage in both demand management, i.e., an effort to smooth the demand, and capacity management, i.e., an effort to adjust capacity to meet the fluctuations in the demand. A common practice is to smooth the demand to some extent and then schedule service personnel to meet the smoothed demand. This paper demonstrates that such approach is not optimal, as it may result in over-smoothing the demand. By consulting a model where demand smoothing and personnel scheduling are determined simultaneously, the quality of these decisions can be improved. Introduction Because of the characteristic fluctuations of the demand for services, and due to the fact that services are produced and consumed simultaneously, service organizations struggle constantly with the problem of matching their short-term capacity and demand. The strategies used to respond to this situation are classified as either capacity management - adjusting capacity to better respond to the variations in the demand - or demand management - changing the demand pattern to better match the available capacity. Examples of capacity management are scheduling of employees to varying work shifts and the use of part-time employees. Examples of demand management are the use of an appointment system and the use of differential pricing schemes for demand smoothing. For a thorough discussion of demand and supply management see, for example, Heskett et al. (1990). When the demand is hard to control (e.g., a call center) the focus is on capacity management (e.g., employee scheduling). When the capacity is largely fixed (e.g., a hotel) the focus is on demand management (e.g., differential pricing and reservations). In many cases, however, services have some control over both the demand and the capacity. Typically, in these cases, the decisions concerning the two strategy types are done sequentially. For example, a medical clinic typically determines the employees work schedule and then use an appointment system to control the arrivals of patients in accordance with the availability of the employees. Conversely, a tax preparation service might use incentives to mitigate the fluctuations in demand during the tax season and then schedule it’s tax preparers to meet the expected modified demand. Showalter and White (1991) and Klassen (1997) argue that capacity and demand management decisions should be made simultaneously. In this paper we examine this idea by adding a demand smoothing component to a well-known capacity management model to create a model where decisions about capacity and demand management are considered simultaneously. The paper is organized as follows. We start by introducing a simple work-shift scheduling model which serves as our example of a capacity management model. Next, we explore the notion of demand smoothing and introduce a transportation-type model of 1 demand smoothing which serves as an example of demand management. The two models are then combined to create a single model where both decisions are made simultaneously. We then use the joint model to develop some insights regarding the optimal combinations of capacity and demand decisions. We conclude the paper by suggesting some possibilities for further study. Note: all the models and examples in the paper were created and solved using Microsoft Excel and the Excel Solver. For the sake of clarity, we chose to present them in a sprearsheet format rather than a standard mathematical notation. We use a dotted background for cells represented “decision variables” and a slanted lines background for cells representing the “objective function”. The Work-Shift Scheduling Model One of the most common ways of meeting uneven demand is the use of varying work- shifts. Under this strategy employees are assigned to different shift types, each involving a different pattern of “on” and “off” work periods, in such a way that the number of employees who are “on” in each period is sufficient to meet the demand in this period. The objective is to minimize the total cost of the shift assignment. When the cost of assigning a worker to a shift is the same for all shift types, the objective is to minimize the amount by which the capacity provided by the schedule exceeds the demand (i.e., minimize idle time). The problem has a straightforward integer linear programming formulation. An Excel model representing an instance of this problem is shown in figure 1. This is an instance of the well known two consecutive days-off scheduling problem (e.g., Baker and magazine 1977) where, for a repetitive seven-day demand cycle, there Period 1 2 3 4 5 6 7 Demand -> 50 30 25 12 34 22 22 Assigned 1 1 1 1 1 1 9 S 2 1 1 1 1 1 0 H 3 1 1 1 1 1 0 I 4 1 1 1 1 1 0 F 5 1 1 1 1 1 25 T 6 1 1 1 1 1 0 7 1 1 1 1 1 16 Scheduled -> 50 50 25 25 34 25 41 Total Idle Idle -> 0 20 0 13 0 3 19 55 Figure 1 – The Work-shift Scheduling Model are seven shift-types, each involving five consecutive workdays followed by two days off. The model exhibits an optimal assignment of employees to the seven shift types to meet the given demand pattern. A total of 55 idle capacity units are need. There are two principal ways of reducing the idle time (in general, the cost of scheduling). The first is to allow additional types of work shifts (e.g., consisting of fewer than 5 work periods per cycle), thereby increasing the schedule flexibility (see Mabert and Showalter, 1990, for detailed analysis of this strategy). The second approach is to smooth the demand. For example, figure 2 shows the result of optimal scheduling of the employees when the demand pattern of figure 1 is smoothed out. The smoothed demand 2 Period 1 2 3 4 5 6 7 Demand -> 28 28 28 28 28 28 27 Assigned 1 1 1 1 1 1 6 S 2 1 1 1 1 1 6 H 3 1 1 1 1 1 5 I 4 1 1 1 1 1 6 F 5 1 1 1 1 1 5 T 6 1 1 1 1 1 6 7 1 1 1 1 1 5 Scheduled -> 28 28 28 28 28 28 27 Total Idle Idle -> 0 0 0 0 0 0 0 0 Figure 2 - Work-shift Scheduling with Maximally Smoothed Demand is simply the average daily demand rounded to integers. Not surprisingly, the total idle time went down from 55 to 0. The cost of scheduling (idle time), as this example demonstrates, may be reduced or even eliminated by smoothing the demand. Cost reduction, however, does not necessarily require maximum smoothing and may be realized by smoothing the demand to a lesser degree. In order to explore this relationship between the smoothness level of the demand pattern and the cost, we must address the question of how demand smoothing (to any degree) is gained and what is its cost. These issues are addressed in the following section. Measuring the Cost of Demand Smoothing Demand smoothing is aimed at reducing the fluctuations in the level of demand. There are three principal ways that service organizations use to smooth their demands: influencing, scheduling, and delaying/denying. Influencing is primarily done by differential pricing and by promoting off-peak demand; scheduling is done by requiring advance reservations; delaying/denying is the practice of forcing customers to wait (or causing them to leave) until capacity becomes available. While being very different from one another, all three strategies result in shifting some demand from one period to another. For that reason we use the paradigm of demand shifting as a generic representation of demand smoothing. The cost of smoothing may be measured by the minimum amount of demand which needs to be shifted in order to transform a given demand pattern into a smoother demand pattern. We use a transportation problem formulation for calculating the cost of smoothing the demand. This approach is similar to the well-known transportation-type formulation of production scheduling problems (e.g., Bowman 1956). Figure 3 shows a transportation problem representation of the model as used for calculating the cost of transforming the original demand pattern of the previous section into the perfectly smooth pattern discussed in that section. In this model, the “sources” represent the given demand is each period and the “destinations” represent the target demands in the same period. Each cell in the transportation table represents the amount of demand to be shifted from the “source” period to the “destination” period. The “unit transportation cost” in each cell is simply number of periods separating the “source” period from the “destination” period. For example the cost of shifting demand from period 3 to period 5 is 2. (The cost of shifting demand period 5 to period 3, however, is 5, since we assume, without loss of 3 generality, that only forward shifting is allowed.) The objective is to minimize the sum of the products of the quantities shifted and the corresponding “costs”. We refer to this To-> P E R I O D Original From 1 2 3 4 5 6 7 Demand 0 1 2 3 4 5 6 1 28 1 16 5 50 6 0 1 2 3 4 5 2 28 2 30 P 5 6 0 1 2 3 4 E 3 25 25 R 4 5 6 0 1 2 3 I 4 12 12 O 3 4 5 6 0 1 2 D 5 23 6 5 34 2 3 4 5 6 0 1 6 22 22 1 2 3 4 5 6 0 7 22 22 Smoothed Total Demand-> 28 28 28 28 28 28 27 Shifting-> 88 Figure 3 – The Demand-Shifting Model quantity as the total shifting. The total shifting in our example is 88. The analysis, so far, suggests two alternatives: schedule employees to meet the original unmodified demand or smooth the demand and then schedule employees to meet the smoothed demand. If we choose the first alternative we incur 55 idle days but no smoothing cost. If we choose the second alternative, we incur the cost of shifting 88 demand units but no idle time. The choice between the two alternatives depends on the cost of idle time vs. the cost of demand shifting. Let Ci be cost of one idle employee-day and Cs be the cost of shifting one unit of demand forward one day. If Ci/Cs < 8/5, then the first alternative costs less than the second; if Ci/Cs = 8/5, then the two alternatives are equally costly; and if Ci/Cs > 8/5, then the second alternative costs less than the first. Clearly, these two are not all the strategies possible. In their classification of demand management strategies, Crandel and Markland (1995) refer to the first strategy, where there is no attempt to change the demand as the “match” strategy, and to the second, where the capacity is kept constant to meet average demand, as the “control” strategy. A third strategy, included in the classification is the “influence” strategy, where the variability of the demand is reduced but not necessarily eliminated. Rising et al. (1973) provide an example of the use of “partial” smoothing. In order to consider the options represented by this strategy, we need to modify our demand-shifting model to accommodate partial smoothing. This is done in the following section. The Cost of Partial Smoothing A model that includes the notion of partial smoothing requires that we use a quantitative measure of the degree of smoothness of a given demand pattern. While many such measures are possible, we elect to measure the degree of smoothness by half the sum of the absolute changes in demand from one period to next over one complete demand 4 cycle1. We refer to this measure as the smoothing index. A perfectly smooth demand would have a smoothing index of 0. As figure 4 shows, the original demand in our example has a smoothing index of 50 and the maximally smoothed demand has a smoothing index of 1. The difference between these two numbers, 49, is the maximum possible reduction in the value of the smoothing index. For patterns that are less than maximally smoothed, we define the Percent Smoothing as the ratio of the reduction in the smoothing index to the maximum possible reduction. For example, the third pattern in figure 4 has a smoothing index of 20 and, therefore, a smoothing percentage of 61.22% ((50-20)/(50-1)). A graphical comparison of all three patterns of demand is shown in figure 5. Period 1 2 3 4 5 6 7 Total Original Demand 50 30 25 12 34 22 22 195 Increase 28 0 0 0 22 0 0 50 Decrease 0 20 5 13 0 12 0 50 Smoothing Index: 50 Percent Smoothing: 0.00% Maximally Smoothed 28 28 28 28 28 28 27 195 Increase 1 0 0 0 0 0 0 1 Decrease 0 0 0 0 0 0 1 1 Smoothing Index: 1 Percent Smoothing: 100.00% Partially Smoothed 30 35 30 35 25 20 20 195 Increase 10 5 0 5 0 0 0 20 Decrease 0 0 5 0 10 5 0 20 Smoothing Index: 20 Percent Smoothing: 61.22% Figure 4 – The Smoothing Index and the Percent Smoothing 50 45 40 Original Demand Denand 35 30 Maximally Smoothed 25 Partially Smoothed 20 15 10 1 2 3 4 5 6 7 Period Figure 5 – Demand Patterns of Varying Degrees of Smoothness 1 We divide the total change in two because, by definition, the sum of the increase in demand is equal to the sum of the decreases in demand. 5 In order to explore the potential benefits of partial smoothing, we need to have a model for evaluating the cost of smoothing the demand to a given degree. We accomplish this by modifying the demand-shifting model of the previous section. The modified model combines the demand-shifting model of figure 3 with the smoothing index model of figure 4, which calculates the smoothing index of the smoothed demand. In the combined model, in addition to the demand shifting quantities, the amounts of increase or decrease from one period to the next in the smoothed demand are also adjustable. This model can be used in one of two ways: to find a least cost (as measured by the total shifting) demand shifting plan that would result in desired degree of smoothing, or to find the smoothest demand pattern possible for a given amount of total shifting. Figure 6 shows a maximum smoothness (i.e., minimum smoothing index) demand- shifting plan when the total shifting is restricted to 40 units. The resulting pattern has a smoothing index of 16 and a smoothing percentage of 69.39%. To-> P E R I O D Original From 1 2 3 4 5 6 7 Demand 0 1 2 3 4 5 6 1 37 8 5 50 6 0 1 2 3 4 5 2 29 1 30 P 5 6 0 1 2 3 4 E 3 17 8 25 R 4 5 6 0 1 2 3 I 4 12 12 O 3 4 5 6 0 1 2 D 5 26 4 4 34 2 3 4 5 6 0 1 6 22 22 1 2 3 4 5 6 0 7 22 22 Smoothed Total Demand-> 37 37 22 21 26 26 26 Shifting-> 40 Increase-> 11 0 0 0 5 0 0 Smooting Decrease-> 0 0 15 1 0 0 0 Index-> 16 Smooting Percentage-> 69.39% Figure 6 – The Demand Smoothing Model By varying the limit on the total shifting (alternatively, varying the minimum smoothing requirement), the demand smoothing model of figure 6 can be used to generate a demand pattern of a desired degree of smoothness. The resulting demand pattern is then used in the work-shift scheduling model to generate a minimum cost schedule. The expectation is that the reduction in the cost of idle time due to the smoother demand will outweigh the increase in the cost of demand shifting needed to smooth the demand. The amount of demand shifting, as well as the resulting amount of idle time, for varying degrees of demand smoothing are shown in figure 7. As expected, the amount of demand shifting needed increases with the degree of smoothness. However, smoother demand does not 6 60 90 80 50 70 Total Shifting 40 60 Idle Time 50 30 40 20 30 20 10 10 0 0 0 20 40 60 80 100 0 20 40 60 80 100 Sm oothing Percentage Sm oothing Percentage Figure 7 – Total Shifting and Idle Time as Functions of the Degree of Smoothness result necessarily in a less costly schedule, as can be shown in the graph on the left. Moreover, the sequential process, where the smoothing of the demand is done independently of the scheduling process, may result in scheduling decisions which are sub-optimal in that calls for more smoothing that is really needed for achieving a desired reduction in idle time. These deficiencies may be eliminated, by using a decision model that would directly connect the demand shifting decisions with the work-shift scheduling decisions. Such a model is discussed in the next section. The Joint Demand/Capacity Management Model As demonstrated in the last section, using demand smoothing as a device to connect the demand shifting model with the work-shift scheduling model is not optimal. In this section the two models are instead combined into one model where demand shifting and work-shift scheduling are done simultaneously. In the joint model the re-distributed demand, which is the output of the demand shifting model, is also the input of the weekly work-shift scheduling model. An Excel version of this model is shown in figure 8. In the joint model the demand shifting variables and the work-shift assignment variables are all decision variables. This model can be used in several ways. For example, the solution shown in figure 8 represents a minimal demand-shifting solution required to eliminate idle time altogether. The amount of shifting needed is 18 units. The resulting demand pattern has a smoothing index of 34 (smoothing percentage 32.65%). This is in sharp contrast with the sequential approach of the preceding section where complete elimination of idle time required the shifting of 60 units of demand (smoothing percentage 89.8%). By relaxing the restriction on the amount of idle time to various levels and calculating the corresponding amounts of demand-shifting required one can uncover the “efficient frontier” of this model. The resulting graph for our example is shown in figure 9 together with a graph depicting the “inefficient” relationship between the amount of smoothing and the amount of idle time in the sequential procedure discussed above. The selection of one of the points on the efficient frontier depends, as before, on the relative costs of demand shifting and of idle time. Adding a cost function to this model is very simple and then the objective becomes minimizing the total cost. 7 To-> P E R I O D Original From 1 2 3 4 5 6 7 Demand 0 1 2 3 4 5 6 1 39 9 2 50 6 0 1 2 3 4 5 2 30 30 P 5 6 0 1 2 3 4 E 3 22 3 25 R 4 5 6 0 1 2 3 I 4 12 12 O 3 4 5 6 0 1 2 D 5 34 34 2 3 4 5 6 0 1 6 22 22 1 2 3 4 5 6 0 7 22 22 Modified Demand-> 39 39 22 17 34 22 22 Total Shifting: 18 1 1 1 1 1 1 17 A S 2 1 1 1 1 1 0 S H 3 1 1 1 1 1 0 S I 4 1 1 1 1 1 0 I F 5 1 1 1 1 1 17 G T 6 1 1 1 1 1 5 N 7 1 1 1 1 1 0 Available -> 39 39 22 17 34 22 22 Idle -> 0 0 0 0 0 0 0 Total Idle: 0 Figure 8 – The Joint Demand-Shifting/Shift-Scheduling Model 60 50 Total Dema nd Shifting 40 30 Sequential Procedure 20 10 Efficient Frontier 0 0 10 20 30 40 50 60 Idle Time Figure 9 – The Relationship between Total Demand Shifting and Idle Time 8 Conclusion In trying to balance their capacities and demands, many service organizations engage in both capacity and demand management. A common practice is to smooth the demand as much as possible and then schedule employees and facilities to meet the smoothed demand. As we demonstrate in this paper, this strategy is not optimal, as these organizations may over-smooth their demands. By consulting a model where demand shifting and employee/facility scheduling are determined simultaneously, the quality of these decisions can be improved. We demonstrated these ideas by using two generic models for capacity management and demand management. However, linking the capacity management decision model with the demand management decision model in actual applications requires further study. Situations where this approach appears more likely to be successful are those where the service organization has the power to schedule both its service personnel and its customers as well. 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