SPECIAL SECTION - OPERATIONS MANAGEMENT

Elwood S. Buffa, Editor University of Carifornia - Los Angeles

DISAGGREGATION IN MANUFACTURING AND SERVICE ORGANIZATIONS:

SURVEY OF PROBLEMS AND RESEARCH LeRoy J. Krajewski, The Ohio State University

Larry P. Ritzman, The Ohio State University

ABSTRACT This paper addresses a class of problems, referred to as “disaggregation prob-

lems,” which lie between planning at the top level and the more detailed decisions of inventory control and scheduling at the bottom level. Most real-world problems are sufficiently complex to warrant a sequential or topdown approach to problem solv- ing. However, researchers have paid scant attention to disaggregation until very recently. The resulting lack of an interfacing mechanism diminishes the utility of solution procedures for aggregate planning, inventory control, and scheduling. In order to draw attention to this gap, a taxonomy of disaggregation problems is developed for both manufacturing and service organizations. The purpose is to iden- tify and classify problems, describe representative research, and identify unresolved issues.

INTRODUCTION

Considerable research during the last two decades has concentrated on aggregate production planning, inventory control, and scheduling. Aggregate planning proce- dures, which have been proposed for both manufacturing [3] and service organizations [ 5 5 ] [69], help determine monthly (or quarterly) output, inventory and manpower aggregates. Inventory and scheduling procedures, using the aggregate decisions as input, tend to focus on such short-term decisions as (1) the sizing and timing of production (purchase) orders for specific items, (2) sequencing of individual jobs (orders), and (3) short-term allocations of resources to individual activities and opera- tions [5] [ 161 [21] , The total process of going from aggregate plans to more detailed plans is referred to as disaggregation. This paper at tempts to categorize representative research of specific disaggregation problems to demonstrate that, although much work has been done in the past, additional research in disaggregation is needed.

Disaggregation is an important issue in manufacturing and service organizations. Depending upon the nature of the production system, disaggregation decisions in a manufacturing organization may exist on one or more of the following three levels:

1 ) Given aggregate decisions on output and capacity, determine the timing and sizing o f specific final product production quantities over the time horizon (sometimes referred to as a master schedule).

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2) Given the timing and sizing of final product production quantities, determine the timing and sizing of manufactured (or purchased) component quantities.

3) Given the timing and sizing of component quantities, determine the short- term sequences and priorities of the jobs (orders) and the resource allocations to individual operations.

The taxonomy shown on the left side of Figure 1 provides the framework within which the three levels of disaggregation in manufacturing organizations are discussed in the section on manufacturing organizations.

Disaggregation problems in service organizations possess the complicating charac- teristics of stringent response time requirements, time dependent demand rates, and the inability to use finished goods inventories to smooth production rates. Disaggrega- tion decisions in service organizations also exist at three levels:

1 ) Given aggregate deicisions on output and capacity, allocate manpower and other resources to specific operations over the time horizon (sometimes referred to as the staff sizing problem).

2) Given the allocation of resources to specific operations, determine the shift schedules and crew assignments of employees.

3) Given the shift schedule assignments, determine short-term adaptations, reallocations between operations, and priorities of the service requirements.

The right half of Figure 1 provides a basis for describing disaggregation problems in the section on service organizations. Although there is a certain degree of uniqueness in the structure of service sector disaggregation problems, that section surveys research in a wide variety of organizational settings.

A GENERAL MODEL

The definition of disaggregation in the introduction for manufacturing and ser- vice organizations does not consider the uniqueness of the two problem structures. Although all levels of disaggregation decisions in a manufacturing organization cannot be made to look exactly like those in a service organization, disaggregation levels 1 and 2 in the manufacturing sector and disaggregation level 1 in the service sector bear remarkable similarities. In this section the authors present a general mathematical programming model which serves to demonstrate this similarity as well as provide a basic model with which to compare research reported in the sections on manufacturing and service organizations. To facilitate later reference, the model presented below will be called GDM (General Disaggregate Model).

Optimize: Z = 9 (Xit, Sit, lit, Bit, Hjt, Fjt, Ojt, Wjt)

Subject to:

i#k

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Qi

z ieL

Pjjt - Wjt - Ojt < 0

1 i fXi t>O

0 if Xit = 0

Qit =

i e N, (3) j E J t = 1 , 2 , . . . , T

j e J (4) t = l , 2 , . . . , T

j e J ( 5 ) t = 1 ,2 , . . . , T

j e J (6) t = 1 ,2 , . . . , T

i e L (7) t = 1,2, ..., T

where:

Xit = output of product (service) i in period t.

Iit = on hand inventory level of product i in period t (manufacturing setting only).

Sit = subcontracted output of product (service) i in period t.

Dit = market requirements for product (service) i in period t .

Pi = production (or procurement) lead time for product (service) i. In service set- tings Qi would normally equal I time period.

aik = the number of units of product i which are required for one unit of immediate parent product k. For final products the value of q k is zero for all i,k. (In service settings, the ctik would be zero for all i,k).

Bit = amount of product (service) put on backorder in period t.

rim, = number of manhours required per unit of product (service) i at operation j in the mth period since production started on i. (In service settings, rim, is usually 0 or 1 for all i ,mj since output is normally measured in manhours.)

rim, = total setup time required by product i at operation j in the mth period since production started on i. (In service settings, rimj is zero for all i,m,j.)

$it = binary variable wluch assigns a setup time for product i whenever Xit > 0.

Pijt = production output of product (service i at operation j in period t, expressed in manhours.

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W. = regular time manhours assigned to operation j in period t.

0. = overtime manhours assigned to operation j in period t.

Hjt = manhours of labor hired for operation j at the start of period t.

Fjt = manhours of labor released from operation j at the start of period t.

Jt

Jt

0 = proportion of the regular time workforce which can be used on overtime.

L = set of all products (services) to be controlled. In manufacturing settings, each raw material, component, subassembly, and final product has its own identity.

T = length of the planning horizon.

J = set of all operations where we assume there is only one type of skill at each operation.

N. = set of all products (services) which require resources at Operation j. J

Although most of the constraints are self-explanatory, several need elaboration. Constraint ( 2 ) is the basic inventory identity relationship with the added feature of the recognition of demands placed on the inventory of product i by higher order com- ponents and subassemblies. In the service sector, ( 2 ) merely identifies the amount of service i backordered in period t. Constraint (3) defines the production output of product (service) i at operation j in period t. For any given i and m, there is at most one j for which rimj (or rim,) is greater than zero. Finally, (4) insures that the work planned for operation j in time period t does not exceed the manpower capacity planned for department j in period t .

Although service sector variables such as Xit, Bit and Sit are expressed in terms of manhours rather than units of product, GDM demonstrates the similarities in the two sectors. The Xit values determine the entire planned production schedule, from master schedule through component production, in a manufacturing setting. In a service setting these same values represent the planned output of each service. The We values are the staff sizes at each operation. However, there are differences in emphasis between the two sectors. Level 1 disaggregation decisions in manufacturing organiza- tions emphasize final product production schedules with the manpower capacities to support these schedules. Level 1 disaggregation decisions in service organizations make the selection of manpower capacities paramount. In the manufacturing setting it is the Xit values for final products which are further disaggregated in level 2 , whereas in the service sector the Wjt values become the inputs to level 2 disaggregation.' The sections on manufacturing and service organizations review representative research done to date in the manufacturing and service sectors respectively, following GDM and the tax- onomy in Figure 1.

1'

' Level 2 decisions in the service sector, shift scheduling, is not included in GDM because (i) i t is not a major problem in most manufacturing settings, and (ii) the complexities imposed by a generalized shift scheduling structure in GDM would detract from its usefulness in relating to both the manufacturing and service oriented literature.

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MANUFACTURING ORGANIZATIONS

Figure 1 shows that disaggregation problems in manufacturing organizations can be found in single stage as well as multistage production systems. Since single stage systems are really a special case of multi-stage systems, the authors begin with them.'

Single Stage Systems

A single stage system consists of a single productive phase and stocking point. The complexity of disaggregation in this type of system depends upon the inde- pendence of the inventoried products.

Single Product

Disaggregation in a single product, single stage environment is limited to level 1 and 2 decisions since level 3 decisions are trivial. Essentially, the problem is to deter- mine production (or order) quantities, such that the annual costs of setup (ordering), holding inventory, and backorders (or stockouts) are minimized. With reference to GDM, L contains only one product, pi is usually deterministic, aik, = 0 for all i and k, and only constraint (2) is recognized.

Procedures for this problem mainly fall into the category of mathematical inven- tory t h e ~ r y . ~ Recently much attention has been focused on discrete lot sizing methods such as Wagner-Whitin's [34] dynamic version of the Economic Order Quan- tity [EOQ] model. A dynamic program is developed for determining discrete

values where demand, inventory holding charges, and setup costs can vary over the planning horizon of T periods.

Presumed coniputational difficulties with Wagner-Whiten's model have led to a number of heuristic approaches. Gorham [14] discusses a method called Least Total Cost which goes through the product requirements step by step, accumulating a lot size until that future period t where the total cost of carrying the inventory through t equals the setup cost. Orlicky [28] presents the Periodic Order Quantity [POQ] and the Lot-for-Lot approaches. The POQ model is a variant of the EOQ model whereby the economic time interval between replenishment orders, 7 , is determined by dividing

Xi, t-Qi,

1

T the EOQ by Z Dit/T. The order quantity, Xi,t-Qi, is the sum of Dit over the interval

t= I [t , t t ~ ] . The Lot-for-Lot method simply makes the lot size equal to the demand each period, or Xi,t-pi = Dit.

' A vast amount of literature has been written on the scheduling aspect of level 3 disaggrega- tion decisions in nianufacturing organizations. Because of space limitations we cannot begin to d o justice to the topic. As such, we refer the reader to any one of a number of excellent references on this topic, such ns I S ] .

I t is not the purpose of this paper to provide yet another review of a topic which is already documented s o well. In addition to the references cited below as well as many others, the reader can refer to [ 161 and 1321 for reorder point systems and dynamic inventory models.

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The importance of these single product, single stage methods is not only that they solve a relatively simplified production planning problem: but they can also be applied to the much more complex multiproduct, multi-stage level 2 decisions where total optimization is difficult, if not impossible, to achieve in practice.

Multiproduct

Multiple products introduce the complexities of resource constraints and non- trivial level 3 disaggregation decisions. Most research on this problem is focused on level 1 decisions. Manne [24] computes economic lot sizes using linear programming. He assumes Pi = 1, “ik = 0 for all i, Bit =Sit =O for all i and t , J contains only one department and Wjt is predetermined for all 1. A set Si of production sequences s:{Xil ,Xi2, . . . , X ~ T \ is specified a priori for each product i. The objective is to deter- mine the fraction of the total requirements for the i-th product to be supplied by the s-th alternative sequence of inputs, fis, such that overtime expenditures are minimized. This results in a large scale programming model. Dzielinski and Gomory [9] followed by Lasdon and Terjung [23] apply decomposition methods to reduce the coniputa- tional burden.

Whereas the above methods are optimizing, Winters [35] develops a heuristic procedure for determining specific values for Xit, given an aggregate production con- straint. The lot size for each product is computed and reorder points are determined. A priority function, consisting of the ratio of the reorder point to the expected end-of-period inventory level, is used to select the products to produce. The lot sizes of the high priority items are accumulated until they reach the aggregate production constraint.

A distinguishing feature of the work of Manne and Winters [24] is that aggregate capacities are predetermined. Others, however, include aggregate capacity determina- tion in the level 1 decision. Newson’s [25] model is similar to GDM except that aik = 0 for all i and k, Bit =Sit = 0 for all i and t, and Ili = 1 . Wagner-Wllitin’s [34] model is used to find Xit values that minimize setup, production and holding costs for each product i given (2) and (7). The values of Pijt can now be computed using (3). These values are used in (4), and along with (5) and (6) the aggregate planning problem is solved. The dual prices on (4) are used to modify the Wagner-Whitin solution to further reduce total costs. The procedure iterates in this way until no further improve- ment is possible. O’Malley, et d, [27] outline a procedure similar to Newson’s except that the iterative aspect of modifying the Xiis is not pursued. Zoller [37] first solves the aggregate planning problem and then disaggregates it to determine optimal individ- ual sales quantities where the objective function is nonlinear.

Several studies attempt to link level 1 and level 3 decisions. Gelders and Klein- dorfer [ 121 use a branch and bound technique. Given a set of jobs to perform in a one machine system, along with their due dates, and fixed regular time resources, the

‘The lot sizing methods discussed in this section are not all inclusive. Other approaches h v e been proposed by [ 131 [29] for example. Berry 1 1 1 compares the performance of several of the commonly used lot size methods.

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objective is to find the production schedule which minimizes total tardy and inventory holding costs as well as aggregate overtime costs. Green [IS] and Shwimer [30] extend the system under study to include multiple machines and use heuristics to link aggregate production plans to detailed production schedules. Hax and Meal [19] devel- op a hierarchial planning system for a process manufacturing firm. Composed of a number of planning subsystems, their procedure disaggregates the aggregate plan to arrive at detailed schedules for each product. For other disaggregation studies in pro- cess manufacturing, refer to [ 101 [ I 11 [20] [26].

Multistage Systems

Perhaps the most difficult disaggregation decisions in manufacturing occur in the multistage system where there are a number of production phases and stocking points. Production decisions for a given product (any inventoried item), can affect the produc- tion decisions of other inventoried products. The nature of this interaction depends upon how the final products are assembled.

Linear Assembly Trees

A linear assembly tree is one in which for all i E L, aik > 0 for one and only one k # i and aik = 0 for every other i' E L, i' # i. Thus the objective given to the final product demand pattern is to determine the lot sizes-xit (level 2 disaggregation) at each of the production stages arranged serially. Zangwill [36] represents this structure as a single source network and applies the theory of concave cost networks.' Taha and Skeith 1311 allow for delay between production stages and backorders for the final product. Their procedure assumes that lot sizes at any stage are an integer multiple of the lot sizes at the succeeding stage, an assumption later proved correct by Crowston, et al, [8] under certain conditions.

Nonlinear Assembly Trees

A nonliiiem assembly tree is more general than the linear assembly tree because aik > 0 for each i and at feasr one k f i , and there may exist i' E L, i' f i such that ai'k > 0. Each inventoried product i can have more than one successor (parent) and each successor can have more than one predecessor. The literature can be catorgized accord- ing to the number of final products considered.

Single Final Product

An implicit assumption in this section is that the individual final products in a manufacturing organization can be analyzed independently of each other. Another assumption is that each product i E L can have only one parent k. Even with these

' A very complctc rcfercnce t o other studies along these lines can be found in Clark (41. Even though rcfercnce is made to niultiechelon systems, some o f the references relate to multi- stage systems ;is defined here.

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simplifying assumptions, the analysis is complex and often heuristic methods are used. The work focuses on level 2 decisions.

Crowston, et al, [ 6 ] discuss a case where (1) the final product is subject to stochastic seasonal demands and ( 2 ) demand forecasts can be revised during the plan- ning horizon. Dynamic programming formulations become unwieldy and heuristics must be used to find the Xit values if delivery of the final product must be made during the selling season.

In [8] , Crowston, et al, propose a dynamic programming model to determine the optimal lot sizes at each stage when demand is constant over an infinite horizon. A sequel to this study by Crowston [7] demonstrates that heuristic models can do just about as well as the dynamic programming models with about one-fourth the compu- tational effort. As evidenced by the introduction of heuristics, the problem can get complex; however, considerations for capacity limitations and multiple products are still absent.

Multiple Final Products

Multiple final products and nonlinear assembly trees probably represent the largest set of practical disaggregation problems in existence. In [ 171 Haehling von Lanzenauer proposes a model similar to GDM to address the link between level 1 and level 2 decisions. In his model !+ = 1 and setup costs are ignored. The objective is to determine integer values for Xi*, Wjt and O,!, such that contribution to profit is maximized. In [18] a bivalent linear programming model is presented for level 2 and 3 decisions.

One of the major shortcomings in normative models such as those proposed above is that problems of practical size are not yet computationally feasible. However, a descriptive methodology which has proven to be quite useful in practice for level 2 decisions is that of Material Requirements Planning (MRP).6 In essence, MRP is a sophisticated information system and deterministic simulation model which takes the master schedule for final product production quantities from level 1 decisions, and translates that schedule into the time-phased net requirements, as well as the planned timing and sizing of production lot releases, of each component required by the schedule over the planning horizon. Many of the single product, single stage lot sizing approaches discussed in the section on single product can be used to determine the

values for the components, once net requirements are known. M K P has massive data processing capabilities and provides the information to plan material coverage and adjust open order priorities; this is a useful input level 3 disaggregation decisions.'

New Research Directions

In terms of the taxonomy in Figure 1 , all manufacturing problems are special cases of the multiproduct, nonlinear assembly tree case. Many of the procedures and

See Orlicky [ 281 for an excellent presentation of MRP. 'Biggs 121 analyzes the interaction of lot sizing decisions and detailed schcdulcs via simula-

tion using MRP to determine the order releases.

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models discussed in the preceding sections can be applied t o a more general case. However there are some areas that need additional research. Foremost is the need for a practical procedure for level 1 decisions, ie., generating master schedules. Man-ma- chine interaction models may be useful here. Another area is the specific timing, sizing and allocation of resources to operations discussed as part of level 3 decisions. In this respect a topic worthy of attention is the determination of the number of shifts to have in a particular manufacturing setting. Other research questions are discussed by Wagner [33] .' As we shall see in the next section, similar gaps in the research exist for service sector organizations.

SERVICE ORGANIZATIONS

As mentioned in the introduction, disaggregation decisions in the service sector appear o n three levels.' Level 1 decisions must recognize that manpower planning cannot rely on inventories to buffer operations from demand variability. If backorders are not acceptable, the only economical buffering mechanism is t o systematically vary manpower levels over the horizon t o meet demand. Although an aggregate plan may stipulate overall levels of manpower hires, overtime, and backorders, it is not suffi- ciently detailed as to short time periods and individual operations. Therefore level 1 disaggregation decisions must determine staff sizes for smaller time periods at each operation, within the overall constraints imposed by the aggregate plan.

The level 2 decisions assign employees t o 2 or more alternate shift schedules, subject to the contraints of desired staff sizes determined in the first level. A shiff is the set of time periods during which an employee can be o n duty during the day. A shift schedule, on the other hand, is the set of on-duty periods assigned to an employ- ee over sonie total time horizon. A shift schedule always covers a time horizon longer than a shift; it can cover a full day, week, or many weeks, depending on the situation.

By their very nature, service sector disaggregation decisions must be judged in terms of many criteria and constraints. These include: (1 ) bounds imposed by the aggregate plan, ( 2 ) service standards, (3) wage costs, (4) legal constraints (such as the Fair Labor Standards Act and safety regulations), (5) labor contracts, (6) company policies, (7) amount of departure from previous plans, (8) administrative convenience, (9) attitudinal differentials caused by transfers between operations, and ( I 0) workload imbalances.

Staff Sizing

Since level 1 decisions on staff sizing are a necessary prelude to lower level disaggregation in any service setting, the authors treat them first before proceeding with the rest of the taxonomy in Figure 1 . With respect t o GDM, the best values of W- Jt must be found where "ik = 0 for all i, Pi = 1, and #it = lit = 0 for all i,t.

*See Krltjewski [22] for research needs o n the interface between level 2 and level 3 disaggre-

' Due to space limitations. level 3 decisions are not considered further in the paper. gation decisions.

I9771 DlSAGGREGATION SUR VEY I I

Staff sizing procedures can be divided into two segments, depending on whether employees must be permanently assigned to a particular operation or whether there is flexibility to assign them to more than one operation over the time horizon.

Permanent Assignments

Given Dit estimates, a variety of mathematical programming queuing and simula- tion approaches have been devised t o determine staff levels Wjt by shift and day of the week. This section is restricted t o cases where each person is permanently assigned to an operation.

Mathematical programming methods for determining Wjt include linear program- ming [53] [ 6 8 ] , integer quadratic programming to minimize squared deviations be- tween manpower availability and workload requirements [50] , nonlinear programming to minimize the weighted sum of dollar and social costs [39] , dynamic programming [54] , and mixed integer quadratic programming using the projection technique of decomposition [70] . The time horizon of these models varies. Larson’s model for patrol allocation is a I-period model, whereas Vitt’s model has an 84-period horizon so as t o plan vacation schedules. Many o f the models ignore probabilistic variations in demand,’ with Dit presumably being increased judgmentally beyond its expected value t o handle randomness.

Since most disaggregation problems involve probabilistic variations’ ’ in Dit, queuing and simulation models can also be used t o assess cost-service tradeoffs. Such models help identify Wjt values which are sufficiently large so that the probability of a customer being delayed by more that a units of time does not exceed a threshold probability 0. Although some single-server queuing models have been proposed, multi- server queuing models are normally more appropriate (441 [49] [54] 1631. When demands for service are spatially disbursed, and the server must travel t o the customer, supplementary models have been developed t o relate service time t o staff levers, Dit values, and geographical characteristics [56] . Simulation models have also been ap- plied, particularly when mean arrival rates are time dependent, queue disciplines are not first-come-first-served, equipment capacities must be recognized, and multistage systems are encountered [57] (61 1 .

Variable Assignments

Employee assignments can be systematically varied from one operation t o an- other in some problems. In addition t o a basic nursing staff assigned t o a hospital floor (operation), Wolfe and Young [7 I ] propose allocating nurses t o a “float pool” opera- tion, which in turn is assigned t o other operations t o handle short term fluctuations in workloads. This concept has been successfully implemented [59] , although the impact

I Larson combines a multiserver queuing model with his dynamic programming algorithm

I ‘Exceptions are found in the transportation industry where the demand for operator crews to more adequately cope with randomness.

is determined by the timetable of scheduled trips which is known under conditions of certainty.

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on employee attitudes has not been fully ascertained [ 6 5 ] . Ignoring such behavioral considerations, Abernathy, er al, [38] apply a simulation model to demonstrate that a sufficiently large float pool can yield significant economies. Other examples of variable assignment staffing, using a variety of solution procedures, can be found in police departments [44], fire departments [ 5 6 ] , and banking [58].

Single Stage Shift Schedules

Whenever service demands go beyond a 5-day week and 8-hour day, mere specifi- cation of staff sizes is not adequate. To develop an operational plan, level 2 disaggrega- tion decisions must specify on-duty times for each person over the relevant time horizon. As shown in Figure 1, shift scheduling' can be classified as either a single or a multistage problem. Single stage problems involve one operation o r several indepen- dent operations; the customer must pass through only one phase of sewice. In a multistage problem, customers are routed through more than one operation, so that the demand pattern imposed on any ,one operation is a complex function of (i) customer arrivals to the system, (ii) customer routing patterns, and (iii) the staffing of other operations.

Shift scheduling problems can be further subdivided into futed and rotating schedules. With fuced (as opposed to rotating) schedules, each employee works the same days and shifts week after week, over the whole time horizon. The following integer programming model serves to present the basic structure of single stage, fixed schedule problems.

Minimize: cn Yn (8)

Subject to: Z!=l an Yn > W (9)

Yn>O, integer, n = 1,2, . . ., N (10)

where:

Yn = number of employees assigned to shift schedule n, cn = cost per person assigned to shift schedule n, possible adjusted judgmentally

to recognize intangible considerations,

an = column vector with element i equal to 1 if shift schedule n calls for an employee to be on duty at operation j E J during time period t, 0 otherwise,

W = column vector with element i equal to Wjt, and N = number of different shift schedules being considered.

Each column vector an represents a candidate shift schedule n, with a row element for each possible combination of j and t. The essence of the problem lies in

'Bodin [43] has recently proposed standardized terminology for shift scheduling, al- though a variety of approximate synonyms (tour, watch trick and rotation) are found in the literature.

19771 DISAGGREGATION SUR VEY 13

determining how many employees to assign to each shift schedule, so as to minimize costs and meet desired staff sizes W. This model takes on the dimensionality of a large scale programming model.

Fixed Schedules

Fixed schedules have the advantage of simplicity and ease in implementation. Solution procedures to such problems can be divided into two categories, depending on the number of possible shifts used to construct shift schedules.

Few Shift Options

In some cases time periods as long as a shift or even a day are used, thereby reducing the number of an vectors, This arises when (i) the start times of acceptable shifts are predetermined, (ii) the demand for service is fairly constant throughout the time period, or (iii) service delays as long as a shift or a day can be tolerated.

Rothstein considers a 1 -shift, 1 -operation problem of minimizing the number of employees not receiving two consecutive days off per week. Due to the problem’s unique structure, integral solutions are guaranteed. Tibrewala, e t al, [66 ] and Baker [41] consider a variant of this problem whereby 5 consecutive w o k d a y constraints must be met, with the objective of minimizing the number of employees required. Their manual procedure can optimally solve this integer programming problem. Final- ly, Mabert and Raidels [58] consider a problem where shift schedule n is enlarged to handle both time periods and multiple operations. Due to the size of the problem, which allows transfers between operations (variable assignments), two heuristic algo- rithms are proposed. One of them has been successfully implemented in a bank.

Many Shift Options

Another class of fixed schedule procedures is designed to cope with hour-by- hour demand variations, overlapping shifts, and relief periods. The number and size of the an vectors is expanded. Time periods can be as short as 15 minutes. Dantzig originally modelled this problem [47] as a linear program. He suggests that most of the optimal Yn’s will be zero and the others can be integerized heuristically with little effect on the objective function. Segal [63] offers an approximate solution with a network-flow model, which is solved with a 2-phase iterative procedure using the out-of-kilter method. In addition to heuristic procedures, Henderson and Berry [5 I ] [52] provide a branch and bound algorithm to find optimal solutions for problems of practical size. They show that low cost solutions are generated even when the number of an vectors is trimmed to a relatively small number (say 50).

Arabyre, er al, 1401 survey different approaches attempting to optimize the allo- cation of airline crews to fhghts.13 Column vector an usually implies a round trip

’ 3The higher level plans being disaggregated are the flight timetables, rather than an aggre- gate plan as conventionally defined. However, the problem of allocating airline crews is mathemati- cally identical to shift scheduling.

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consisting of several legs of flight segments. Since each leg is to be covered just once, the problem is reduced to a Boolean programming problem where the Yn values must be 0 or 1 . A variety of methods can be used to reduce the size and number of an vectors. Methods for solving the resulting model include: heuristic algorithms, branch and bound, implicit enumeration, various integer programming codes, and combinato- rial enumeration.

Rotating Schedules

Rotating schedules call for each bracket of employees (a set of employees with identical work periods over the time horizon) to rotate each week to the work periods assigned to another bracket during the previous week. By the end of the time horizon, all brackets will work through the same pattern of work and recreation (days off) clusters, thereby providing non-preferential treatment to all employees. Designing ro- tating schedules is more difficult than fixed schedules, owing to the additional con- straints forcing rotation. The solution, procedures proposed have multiple phases. In the first phase, the day-off pattern is constructed in a fashion similar to the procedures of the permanent assignment and fxed schedules sections. Subsequent phases then assign shifts to the day-off pattern.

Bennett and Potts [42] attempt to maximize the number of recreation clusters (consecutive days off) consisting of 2 or more days as well as spread the clusters evenly over the time horizon. Maire-Rothe and Wolfe [59] also recognize multiple criteria in their heuristic approach to staffing each of 7W days in a W week cyclic graph. Heller, et al, [50] provide a computerized package of sequential procedures. Given the Wit values for each shift and day of the week, cyclic graph analysis is applied to one shift at a time to determine recreation clusters. Implicit enumeration is coupled with the notion of a separation matrix to construct feasible schedules, which are then ranked with a lexicographic scheme to recognize the multiplicity of criteria. Given the highest ranking schedule found, a branching procedure enumerates all feasible multi-shift schedules. The most attractive rotating schedules, again measure with a lexicographic scale, are printed out for management's consideration.

Another impressive set of procedures and perspectives is provided by Bodin [43], who structures the problem for sequentially solving 3 submodels. Given appro- priate Wjt values, a grouping model is used to find the best collection of recreation clusters. The' pattern model is next applied to intersperse recreation clusters, with a shift model finally applied to assign shifts to clusters so as to satisfy Wjt requirements without resorting to broken clusters.

Multistage Shift Schedules

One of the most difficult, albeit less frequently encountered, disaggregation problems in the service sector is multistage shift scheduling. Such a problem is encoun- tered by a post office, where several categories of mail must be sorted sequentially at a series of operations. One major source of complexity is that staff assignments at one stage affect the Dit values for other stages. Additional complications are the multipli- city of ( 1 ) criteria and constraints, (2) mail categories (types of services), and (3)

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dispatch schedules. A dispatch schedule is the set of times (unique to each niail category) at which sorted mail can be loaded onto trucks and transported to its next destination. With the existence of dispatch schedules, staffing levels providing good service at low cost are not necessarily those which closely match the pattern of mail arriving to the whole system.

Level 1 and 2 disaggregation procedures for solving this problem have been designed for the fixed schedule case, relying on heuristic algorithms coupled with simulation [61] [64]. The objective is t o minimize mail flow time while contending with a variety o f other criteria and constraints.

New Research Directions

Several new directions appear t o be promising. The first is t o develop procedures with greater generality, particularly if they are transparent and understandable to management. Current procedures tend to reduce problem dimensionality in one way or another. For example, procedures capable of handling many shift options are lim- ited t o single stage problems with permanent assignments required. Procedures t o design rotating schedules in a multistage system have yet t o be developed. Multiple criteria decision methods [45] have received scant attention in disaggregation.

Other directions are (i) research on how best t o revise shift schedules over time (so as to accomodate new hires, temporary employees, or vacations), (ii) comparative work on the relative merit of techniques proposed for similar problems, and (iii) research assessing the impact of alternate staff scheduling policies o n employee atti- tudes and behavior.

CONCLUSION

An assumption providing the motivation for this paper is that decisions made at any level in an organization must be compatible with higher level decisions. The process of decomposing high level aggregate plans into more detailed plans is called disaggregation. In trying t o provide a useful framework for defining and identifying disaggregation problems, this paper presents a taxonomy of disaggregation problems in the manufacturing and service sectors. It describes a small, but representative, segment of literature to demonstrate today’s state of the art. It shows that although much work has been done, a great deal still remains. Specific recommendations for future research are made at the ends of the sections on manufacturing and service organizations. In general, more research is needed which (i) recognizes the interface between the various disaggregation decisions and (ii) provides procedures useful for practicing managers.

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