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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

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    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

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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



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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. Hospital operating room scheduling, which involves staff scheduling
(capacity management) as well as scheduling patients for elective surgical procedures
(demand management), is an example.




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Bowman, E.H., "Production Scheduling by the Transportation Method of Linear
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Crandall, R.E. & Markland, R.E., "Demand Management - Today's Challenge for Service
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Heskett, J.A., Sasser, W.E. & Hart, C.W., Service Breakthroughs. New York: The Free
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Klassen, K.J., “Simultaneous Management of Demand and Supply in Services.” Ph.D.
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Mabert, V.A. & Showalter, M.J., "Measuring the Impact of Part-Time Workers in Service
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Rising, J.R., Baron, R. & Averill, B., "A System Analysis of a University Health-Service
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Showalter, M.J. & White, J.D., "An Integrated Model for Demand-Output Management
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