decomposition for scheduling flexible manufacturing systems

Decomposition for Scheduling Flexible Manufacturing SystemsAuthor(s): Robert R. Inman and Philip C. JonesSource: Operations Research, Vol. 41, No. 3 (May - Jun., 1993), pp. 608-617Published by: INFORMSStable URL: http://www.jstor.org/stable/171859 .

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DECOMPOSITION FOR SCHEDULING FLEXIBLE MANUFACTURING SYSTEMS

ROBERT R. INMAN General Motors Research Laboratories, Warren, Michigan

PHILIP C. JONES Northwestern University, Evanston, Illinois

(Received January 1990; revisions received November 1990, October 199 1; accepted November 1991)

In some flexible manufacturing systems, tool magazine capacity naturally divides parts into families so that substantial setups occur only when switching production between families. For this situation (positive setup times and costs only between families), we show that the production planning problem decomposes into two simpler problems: an aggregate scheduling problem, and a disaggregation. First, we derive a disaggregation that determines the optimal intrafamily product schedules corresponding to a given family schedule in polynomial time. Second, we show how to aggregate so that the decomposition maintains optimality for family schedules satisfying the property that production does not begin until inventory is zero (the Zero-Switch rule).

In this paper, we address scheduling production when there are multiple part families. The model

has positive setup times (and nonnegative setup costs) between families and zero setup times and costs within families. The objective is to minimize the sum of average inventory carrying costs plus average setup costs while producing to meet known demand. We explicitly include setup time in addition to setup cost because, as Karmarkar (1987) noted, setup time is often important. This setup structure is also used in the literature on aggregate production planning. Bitran and Hax (1981), Bitran, Haas and Hax (1981, 1982), Bitran and Iirupati (1989), and Bitran, Haas and Matsuo (1986) use an hierarchical production planning (HPP) approach and consider setups between product families, but assume that changeover costs between parts of the same family are negligible. One situation in which a family setup structure occurs naturally is in flexible manufacturing systems.

Flexible manufacturing systems (FMS) can produce many different parts without stopping: If the tool magazine holds the tools required for the next part, that part does not require a setup; production can proceed with a lot size of one. Realistic manufacturing environments, however, may require more tools than the magazine can carry. In one Chicago area firm, for example, designs for existing parts produced on a laser punch-press specify over 900 tools, but the tool mag-

azine capacity is 200. Thus, producing all active parts requires tool changes that necessitate stopping the machine to manually replace tools. This flexible manufacturing system has families of parts defined by the tools in the magazine. Determining optimal tool and part families is an interesting research question beyond the scope of this paper; we will take part families as given.

The setup structure used in this paper, therefore, is both realistic and with precedent in the literature. The paper presents three main contributions:

1. an optimal disaggregation of family schedules into part schedules;

2. a demonstration that this disaggregation is an optimal, continuous-time disaggregation policy for HPP models; and

3. for the constant demand case, an aggregation heuristic that decomposes the N-product, G-family problem (N >> G) into a much smaller G-product scheduling problem. The aggregation is always feasible and is optimal over the class of family schedules satisfying the Zero-Switch rule (the Zero- Switch rule, see Maxwell (1964) is that production of any product does not begin until its inventory is zero).

We review the relevant literature in Section 1 and provide some definitions in Section 2. The problem

Subject classifications. Production/scheduling: decomposition for hierarchical production planning. Production/scheduling, flexible manufacturing: scheduling families of parts.

Area of review: MANUFACTURING, OPERATIONS AND SCHEDULING.

Operations Research 0030-364X/93/4103-0608 $01.25 Vol. 41, No. 3, May-June 1993 608 (? 1993 Operations Research Society of America



Decomposition for Scheduling / 609

naturally decomposes into two steps: aggregation and disaggregation. The disaggregation step of Section 3 assumes that we are given a family schedule and determines the best part schedules consistent with that family schedule. Since this optimal part disaggregation holds for dynamic demand, it can be used in HPP models, and we show that it is the optimal, continuous-time disaggregation for HPP models. In Section 4, we restrict our attention to constant demand and present the aggregation. Section 5 dem- onstrates the decomposition with an example that compares our results to an existing heuristic.

1. LITERATURE SURVEY

The model in this paper is closely related to the traditional economic lot scheduling problem (ELSP). The traditional ELSP is to optimally schedule production of many parts on one machine assuming: constant production and demand, no backlogging, sequence-independent setup times and costs, and an infinite time horizon. Elmaghraby (1978) gives a val- uable survey of the vast literature on the traditional ELSP. There is much less written that is applicable to economic lot scheduling in the presence of part families.

Research that is applicable to the ELSP with part families includes the work of Dobson (1992), who studies the problem with sequence-dependent setups, and the results of Ham, Hitomi and Yoshida (1985), who formulate a group technology ELSP in the presence of part families and provide a method for generating feasible schedules. Dobson's sequence- dependent problem is more general than that studied in this paper because it does not assume a family structure. Dobson obtains excellent results for the general problem, but his more general approach does not take advantage of the special structure (zero intrafamily setups) considered in this paper. The group technology ELSP model studied by Ham, Hitomi and Yoshida has two types of setups: intrafamily (between different parts within the same family) and interfamily (between different families of parts). It is slightly more general than the model discussed in this paper because they allow nonzero, intrafamily setup costs and times. They state, however, that zero intrafamily setup is a reasonable assumption and give equations explicitly for this case.

2. DEFINITIONS

A family setup occurs whenever production is switched to a part in a different family. Once the

family is set up, any part of that family can be produced without any additional setup. Switching to produce a part of a different family, however, requires a setup for that family. Intrafamily setup times and costs are zero, but interfamily setup times and costs are nonnegative. As in the traditional ELSP, we assume continuous demand and production (so infinitesi- mally small lots can be feasibly produced). This problem, the FMS-ELSP, has the following parameters:

Sg = the setup time for family g; Ag = the setup cost for family g; hg = the inventory holding cost of part j in

family g; djg(t) = the demand rate of part j in family g; pf(t) = the production rate of part j in family g; ng = the number of parts in family g; N = the total number of parts; G = the number of families; tm( g) = the start time of the mth lot of family g; and Tn(g) = the duration of the mth lot of family g.

The parts in family g will be labeled j = 1, 2, . . ., ng. In this paper, superscripts refer to families and sub- scripts refer to parts or lots. We assume that the FMS has sufficient capacity to meet demand.

We say the facility is producing family g whenever it is producing any of family g's parts. The family schedule is comprised of the production lots of each family. To maintain a correspondence between the family schedule and the part schedules in that family, we define the family inventory to be the total inventory measured in the common units of production time. That is,

ng

Ig(t) = E I9(t)/p9 for all t and g, j=1

where Ig(t) is the aggregate inventory of family g at time t and I(t) is the inventory of part j of family g. A part schedule is easily translated into a family schedule. In the next section we look at the converse, determining the best part schedule consistent with a given family schedule.

3. OPTIMAL DISAGGREGATION

3.1. Maintaining Feasibility

In this section, we show how to optimally disaggregate a family schedule buy determining the optimal part schedules one family at a time. To guarantee feasibility, we look ahead and calculate the part inventories moving backwards in time. As Erschler, Fontan and



6 1 0 / INMAN AND JONES

Merce ( 1986) showed, if we look ahead over the entire horizon, we can always find a feasible disaggregation. We adopt this strategy for the finite-horizon models discussed in subsection 3.4.

For infinite-horizon models, however, we cannot look ahead over the entire horizon. Instead, we assume that we are given a cyclic family schedule-a schedule that repeats itself every T time units. The cyclicity assumption is not necessary (see Inman and Jones 1987), but greatly reduces the complexity of the procedure and exposition. Note that cyclic schedules need not be rotation cycles. In a rotation cycle each product is produced only once per cycle; but generally, there may be more than one lot of a product in a cycle.

From the definition of family inventory, the inventories of all the parts in that family must be zero whenever the family inventory is zero. Each family, we assume, has zero inventory at least once every cycle. Continuous demand and production along with zero intrafamily setups imply that a feasible schedule exists even if the inventories of all parts in a family hit zero simultaneously. Note that total costs of any cyclic schedule not satisfying the zero family inventory assumption could be reduced by lowering inventory. We use the zero inventory points to compute optimal inventories of parts in the family by working backwards from the points of zero inventory.

We derive the optimal part schedules one part at a time and in a particular order. We label the parts in family g in decreasing order of the product hJpJ. Subsection 3.2 derives the optimal schedule for part 1; subsection 3.3 explains how the same arguments can be used to recursively derive optimal schedules for the remaining parts.

3.2. Scheduling Part 1

This subsection characterizes the optimal policy for part 1 (largest hfpf). In the optimal schedule, we show that part 1 is the last part to be produced in a lot of family g. We know the optimal part inventories at the end of this family lot because we calculate part schedules by moving backwards in time from a zero inventory point (when part inventories are known). Given a family lot and part inventories at the end of the lot, we determine the optimal corresponding part schedules. For the remainder of Section 3 we assume that, for each family g, the following condition holds:

Itm+ 1(g)

df(t) dt > O for all j E family g. (1) Thi g)+dt sg)

This condition says that each part must experience

positive demand between two successive lots of its family.

Define the cycle for family g as starting and ending when the family g inventory is zero. Number the family g lots in the cycle m = 1, 2, ... , M. The optimal part 1 inventory profile (displayed in Figure 1) is described in the following disaggregation theorem.

Disaggregation Theorem. There is an optimal schedule in which every family lot ends with a finite (i.e., not infinitesimal) lot of part 1 (during which time the facility is dedicated to producing part 1). Furthermore, the finite lot of part 1 that ends the family lot is the onlyfinite lot of part 1 produced during thefamily lot, and the finite lot either occupies the entire family lot (in which case the on-hand inventory of part 1 at the start of the lot may be positive) or is preceded by a period during which part 1 inventory is zero and part 1 is produced in infinitesimal lots to meet demand in JIT fashion (during this period, other parts are also produced).

Proof. We first establish the result for the last lot (lot M) of family g and then repeat the argument for all other lots in the cycle.

Family Lot M

We first show that family lot M ends with a finite lot of part 1. If not, there must be some other part j,

Either:

Part 1 Inventory

t m t m +1rm

Or:

Part 1 Inventory

tm tm + fm

Figure 1. Optimal part 1 inventory in a family lot.




j > 1 that is produced at the end of family lot M. Condition 1 implies that at least one finite lot of part 1 be produced prior to the end of family lot M (possibly in some prior family lot), for otherwise feasibility would be violated. We can then construct an alternative feasible schedule by switching an arbitrarily small time E of production ofj at the end of family lot M, with e of production of part 1's most recent previous finite production. This construction saves some of the inventory carrying cost of part 1, simultaneously increases the inventory carrying cost of part j, but has no other cost impact (provided that E is sufficiently small). The ratio of savings to cost is (glpf)/(hJpy), which, by the labeling of parts, is greater than or equal to 1, so the resulting schedule is no worse than the original. We have obtained a feasible schedule, no worse than the original, in which family lot M ends with a finite lot of part 1.

We next show that the each finite lot of part 1 must either start with zero on-hand inventory of part 1 or completely occupy the family lot. If not, let t1 denote the start time of the last finite lot of part 1, let t denote the start time of the most recent previous finite lot of part 1 (such a lot must exist because, by sup- position, the on-hand inventory of part 1 at time t1 is positive), and let t' denote its completion time. It is feasible to switch the production during the interval [t' - i, t'] with that of the interval [t1 - E, t1] for e

sufficiently small. After the switch, the last finite lot of part 1 starts earlier with less part 1 inventory on- hand and the most recent previous lot of part 1 has been made smaller. The parameter e can be increased while maintaining feasibility until either:

1. t, - e = tM;

2. on-hand part 1 inventory at the start of the last finite lot hits zero; or

3. tP - E = t.

If case 1 (the last finite lot of part 1 completely occupies family lot M) or case 2 occurs, we are done. If case 3 occurs, we have deleted the most recent previous lot of part 1 by moving its production to the last finite lot of part 1 in family lot M. In this event, the argument can be repeated until either case 1 or case 2 occurs. This shows that the last finite lot of part 1 either starts at time tM or starts with zero on-hand inventory. The argument can be repeated inductively to show that all finite lots of part 1 in family lot M begin with zero inventory, except possibly the first (if it begins at time tM).

Next, we argue that there is only one finite lot of part 1 in family lot M. If not, the schedule cannot be optimal. To see this, let t denote the start time of the

next-to-last finite lot, let T denote its duration, and let At denote the length of time between t and the start time of the last finite lot. We can construct a less expensive schedule by "dividing" the next-to-last lot into two parts as follows: switch the production schedules of the two intervals [t + r/2, t + -r] and [t + At/2, t + (At + T)/2]. The switch maintains feasibility, reduces inventory carrying costs of part 1 (by delaying production), while simultaneously in- creasing the inventory carrying costs of the parts whose production was shifted to an earlier time. The ratio of savings to cost is larger than (hfpY)/(hfpj) which, by the labeling of parts, is at least as large as 1.

It remains to show that, if the finite lot of part 1 is preceded by infinitesimal production (note that any infinitesimal lot must start at zero inventory, for otherwise it would be feasible to reduce cost by delaying the start time of the infinitesimal lot), then part 1 inventory at time tM is zero. If not, let the time at which the first finite lot starts be denoted by t1 > tM. Let t' denote the end time of the previous lot (in an earlier family lot) of part 1. We can then construct an alternate schedule that is no more expensive by switching the schedule of the interval [t, - E, t1] with that of the interval [t' - E, t']. So long as E is sufflciently small, the switch maintains feasibility. Inventory carrying costs for part 1 are decreased, while those of some part j > 1, where j is the part originally produced in the interval [t, - E, tj, are increased. The ratio of inventory savings to extra costs is greater than or equal to (hjpy)/(h,9p), which is greater than or equal to one.

The argument for family lot M used constructions that only affected family lot M and previous family lots. Hence, we can repeat the constructions in turn for family lots M - 1, M - 2, and so on until we reach the first family g lot in the cycle.

The practical implication of infinitesimal lots is to produce in lot sizes of one. Thus, infinitesimal lots translate to just-in-time (JIT) production in which parts are produced as needed. We refer to these infinitesimal lots as JIT lots. The disaggregation theorem tells us that, within family lot m, optimal production of part 1 is in lot sizes of one except for the last lot of part 1 in the family lot. The last lot must be a non- JIT lot to build-up inventory needed to satisfy demand between family lots.

3.3. Scheduling the Other Parts in the Family: Parts 2 Through ng

The strategy for determining the optimal schedule for parts 2 through ng is to recursively apply the disaggregation theorem to the next (as previously ordered)



612 / INMAN AND JONES

unscheduled part. To determine the optimal schedule for part 2 in a family lot, define a new problem by removing part 1 from consideration (since its schedule is completely known) and determine the schedule of part 2. That is, when part 1 is being produced in its non-JIT lot, the machine is dedicated solely to part 1 and consequently is not available for other parts. Thus, the time window available for part 2 production in a family lot ends when part l's non-JIT lot begins. To define the new problem, we therefore delete the time during which part 1 is produced in non-JIT mode. Furthermore, when part 1 is being produced in infinitesimal JIT lots to meet ongoing demand for part 1, the effect on production of other parts is to reduce their production rates. Since the facility is committed to producing part 1 for (dY/pY) 100% of the time during part l's JIT lots, the effective production rate for part 2, denoted by pig, becomes: p'g =

p2(1 - dY/pY). We then schedule production of part 2 by applying the disaggregation theorem to part 2 in the new problem.

Repeat this procedure in turn for all parts in family g. When parts 1 through j - 1 have been scheduled, the effective production rate for the most expensive unscheduled part (currently part j) is: p'g -

pf(1 - r4l), where

r= dl/pf for j > 0, and r6 = 0. (2) k=i

Figure 2 illustrates the procedure. Note that each part's production occurs in two different modes:

1. JIT production in which production of part j exactly meets ongoing demand for j without any increase in the inventory of j; and

2. non-JIT production in which production of part j exceeds the ongoing demand and inventory increases.

Recall that each part is produced in at most one non- JIT lot in any family lot. The following summarizes the disaggregation procedure.

Procedure Disaggregation Input: feasible family schedule For each family g:

1. sort the parts in each family in decreasing order of hJpJ.

2. schedule family g's parts using the disaggregation theorem.

Output: an optimal part schedule corresponding to the input family schedule.

a) first schedule part 1

1( t) X 1- 1 p-d

-i - : :

_s

b) then schedule part 2

12(t) 2d {

Sg

c) then schedule part 3

13(t) -d 3 3

Time

Figure 2. Recursively scheduling parts in order.

Observation: Given a cyclic (or finite horizon) family schedule, the optimal corresponding part schedule can be found in O[N log N] time.

This disaggregation provides the optimal part schedule corresponding to the family schedule. Note that the demand rates were not needed in the proof of the disaggregation theorem. Consequently, as explained in the next session, the disaggregation can be applied to HPP models having dynamic demand (assuming, of course, that the demand rate for any set of parts is always less than the maximum production rate so that JIT production is possible).

3.4. Optimal Continuous-Time Disaggregation for HPP

Since the disaggregation policy can be used for time- varying demand, it can be used in aggregate hierarchical production planning. For example, Hax and Meal (1975), Bitran and Hax (1981), Bitran and Tirupati (1989), Bitran, Haas and Hax (1981, 1982), and Hax and Candea (1984), aggregate items (or parts) into families, and families into types. Planning is performed in a three-level hierarchy: types are scheduled first, the type schedule is decomposed to




determine the schedule for the families of each type, and finally, the family schedule is decomposed to provide part schedules. These authors recognize that, in many practical manufacturing environments, there is a hierarchical setup structure, and consequently they consider setups only between different families-not between different parts of the same family. Similarly Bitran, Haas and Matsuo consider setup costs between families, assuming that changeover costs between parts in the same family are negligible. Hence, the setup structure in the FMS ELSP is the same as in the HPP literature. Unlike the FMS ELSP, the HPP assumes discrete time periods and instantaneous production, but allows dynamic demand in the form of step functions.

Hax and Candea use an Equal Run Out Time (EROT) policy to disaggregate the family schedule into part schedules, and show that the EROT is an optimal item disaggregation in their HPP. Since their model is in discrete time, their EROT can only specify how many of each part are to be produced in this period. In contrast, procedure Disaggregation is in continuous time and specifies exactly when within this period to produce the parts. The disaggregation is also an EROT policy, and will specify the same number of parts to be made in that period as well as when within the period they should be made. In continuous time, there are infinitely many policies that would specify the same part production as the EROT; but, as shown in subsections 3.2 and 3.3, procedure Disaggregation provides the best policy.

In addition, procedure Disaggregation always provides a feasible disaggregation. In HPP, family schedules are typically only disaggregated for the current period, and consequently, infeasibilities may occur later. Procedure Disaggregation guarantees feasibility by looking ahead over the whole horizon (see Erschler, Fontan and Merce). Thus, our disaggregation procedure provides an optimal, continuous-time EROT that guarantees feasibility in HPP models. Since the proof of the disaggregation theorem holds for variable demand rates in the form of step functions, the following observation holds.

Observation. Given a feasible HPP family schedule, procedure Disaggregation provides the optimal continuous time EROT item disaggregation.

The EROT policy in HPP will specify how many parts to produce each period; procedure Disaggregation will specify the same amounts of the same parts, but will also specify when within the period to produce which parts. If the time periods are small, then our continuous-time disaggregation has

little advantage over the discrete time EROT. If, however, the time periods are long, significant savings could be realized by using our procedure to disaggregate family schedules into item schedules in the con- text of hierarchical production planning.

4. AGGREGATE SCHEDULING FOR CONSTANT DEMAND

This section addresses how to find the family schedule. To do so, we now restrict ourselves to the constant demand FMS ELSP. For constant demand, we define an aggregate product for each family that aggregates all the parts within that family. Each aggregate product has five parameters: setup cost Ag, setup time Sg, demand rate Dg, production rate Pg, and unit holding cost Hg. The setup time and cost between families, Sg and Ag, are given in the problem data, and both are positive. Furthermore, we define

pg= 1, and ng

Dg= dg/pg for all families g; j=I

so that ng

D/pg = i dJg/4p for all families g. j=1

This definition ensures that scheduling aggregate product g will reserve exactly the amount of time needed to meet the demand of family g's parts.

It remains to determine the unit holding cost of the aggregate products. If we can define the unit cost of the aggregate products so that the cost of the aggregate schedule equals the cost of its optimal corresponding part schedule, then the problem reduces to optimally scheduling the aggregate products, because we know the optimal disaggregation. For the case of constant demand, (the assumption of the traditional ELSP), and family schedules that satisfy the Zero-Switch rule, we accomplish this complete decomposition.

For constant demand, as long as the aggregate schedule follows the Zero-Switch rule, we can define the unit cost of the aggregate products so that the cost of the aggregate schedule equals the cost of its optimal corresponding part schedule. ELSP heuristics almost universally enforce this intuitive assumption. Conse- quently, most attempts to solve the FMS ELSP in practice will provide a Zero-Switch (ZS) aggregate schedule.

Using the disaggregation results of the previous section, we know the optimal part inventories corresponding to any lot of a ZS family schedule, because




the part inventories at the start of each family lot are all zero. Consequently, we can define the holding cost of the aggregate family products to reflect the holding cost of the corresponding optimal part inventories.

We show that the proportion of the aggregate inventory attributable to a given part is constant for all family lots. That is, the part composition of any ZS lot of aggregate inventory is constant. This observation enables us to prove the decomposition theorem, which states that the average on-hand inventory of part j in the optimal part schedule is proportional to the aggregate schedule's average on-hand inventory. Since the family schedule has the ZS property and Pg = 1, the average family g inventory for lot m is:

I D9(1 - Dg)(t,+ l - tin).

2

Decomposition Theorem. In the constant demand case, given any lot of a ZS aggregate schedule for family g that starts at time ti, and whose next lot of g starts at time tm+l, the average on-hand inventory of any part j in family g, Ij, is:

-1I Ij - D9(1 - Dg)(tm+l -

2

where -y. is independent of the aggregate family schedule.

Proof. We derive yJ. In the remainder of the proof, all parameters refer to family g, so we drop the super- script for notational clarity. Since the aggregate schedule has, by assumption, the ZS property, every family lot starts with zero inventory. Hence, every part in the family has zero inventory at the start of production. The disaggregation theorem shows that optimal part inventories corresponding to a ZS aggregate schedule must be as illustrated in Figure 3. Let wrj denote the duration of the non-JIT lot of part j in family lot m. We calculate these wrj's so that the inventory of partj at time tm+l will again be zero (since the family schedule is ZS, we know that the inventory of all parts in family g will be zero at t,+1). The calculations must use the effective production rate which is reduced to account for concurrent JIT production of other parts. The non-JIT lot duration for part j is:

adjqj-1 = p(l - rj) -dj'

where

a = tm+ - tm - Tm,

j= [PI(1 - r1)- 1 1 j j ng, with i=1 L(l- ri1) - di_

ro= 0,

a)

Optimal Part

Inventory 3

m m+1

b)

X ~ ~~~~~ X + 7c + 7 j-1

t t m 7c

j+1 j+2m n 9 t +1

Figure 3. a) Optimal part inventories corresponding to lot m of a zero-switch family schedule; b) optimal inventory for part j.

and q0 = 1, so that qj can be calculated recursively:

qj= qj-[p( - N

l)'j (3) [pj(1 - rj-A) 1 j

The average inventory of partj in family lot m is then given by:

= 2(tm+i - tin) La + ? wrila + ridj.

By induction, it follows that

aqj = a + E 7ri, i=1

so we have:

a2djqj q_ =a d _1q~ 2( tM+ - tm)

Substituting back in for a and eliminating tn yields the relationship claimed in the statement of the theorem with:

(1 - D)(djqjqj1)

D (4)




Since yj is completely defined by given data, it is independent of the particular family schedule.

Note that y9 is independent of the schedule and can be calculated recursively in polynomial time with initial problem parameters. With this result, we can calculate the holding cost of the aggregate products so that the cost of the family schedule equals that of the best corresponding part schedule. Then we can fully define the aggregate product for each family and schedule these independently of the component parts. We schedule these aggregate products using any method that provides ZS schedules, and guarantee that the resulting family schedule costs the same as the best corresponding part schedule. Define the unit holding cost for family g, Hg, so that the cost of a ZS lot of the aggregate family equals that of the optimal corresponding part schedule.

Corollary 1. There exist constants, H', ... Hkg,

HG, such thatfor any ZSfamily schedule, the inventory cost of a family lot of aggregate product g using Hg as the unit holding cost, equals the optimal corresponding part inventory cost for family g.

Proof. We derive Hg and show that Hg is independent of the schedule specifics. From the decomposition theorem, the average inventory cost of the optimal part schedule corresponding to lot m of family g in a ZS family schedule is

ng I ng

Tjshg = 2)Dg(I - Dg)(tm+l-ti) _

tm j=l j=l

where 4yg is independent of the schedule. We want an aggregate holding cost coefficient independent of schedule and such that scheduling the aggregate family products costs the same as the optimal corresponding part schedule. Equivalently, we want to define Hg so that the cost of one lot of the family inventory is the same as the sum of the optimal component part inventories for the same period of time. That is:

( )Dg(l - Dg)(tm+ - tm)Hg

= 2Dg(I - Dg)(tm+l tm) E, hjg.y. 2 ~~~~~~~~~~j= 1

Clearly, ng

Hg= E hj>9, (5) j=1

which depends only on given problem parameters.

Since Hg is independent of the schedule, we can schedule the G aggregate products ignoring the com-

ponent parts. That is, the cost of the optimal multipart cycle, with JIT production as in Figure 3a, equals the cost of a single-product cycle when the parameters of the single family product are appropriately aggregated. This decomposes the FMS ELSP into a family scheduling problem and disaggregation problems. Sorting the parts in descending hjpj order takes O[N log N] time, and computing the Hg's takes O[N] time. The following summarizes this procedure.

Procedure Aggregation

Input: initial part data

1. For all families g, given Ag and Sg, and Pg = 1, define an aggregate product:

a. sort parts in each family in descending hjgpg order;

b. for all parts j = 1 to ng: calculate rj, qJ, and gg [eqs. 2-4];

c. calculate Hg [eq. 5]; and calculate Dg =

E djg/pj.

2. Schedule these aggregate products according to any ELSP heuristic that provides ZS schedules.

Output: feasible family schedule

The aggregation and disaggregation procedures com- bine to make up the decomposition. This decomposition is optimal in the following sense. Suppose we aggregate according to procedure Aggregation. If the optimal aggregate schedule is ZS (and if we are able to find it!), then the resulting part schedule from the disaggregation procedure is optimal. If, however, we obtain a suboptimal family schedule, or if the optimal family schedule is not ZS, then the resulting part schedules may not be optimal.

5. EXAMPLE OF DECOMPOSITION

To demonstrate, we apply the decomposition to an example problem given in Ham, Hitomi and Yoshida. They solved an example with nonzero, intrafamily setup costs and times. For the sake of comparison, however, we modify the example by imposing zero, intrafamily setup costs and times. Positive setup times, however, remain between families. In our notation (and after renumbering the parts so that hfpf 3 h5p5 3 hipi) the data are given in Table I. Re- solving the problem using their algorithm results in a feasible schedule with an average setup and inventory cost of 31.30. Using our procedure, we first aggregate by calculating the parameters of the aggregate family products. Recalling our definition of the aggregate




Table 1 Part and Family Data

Family g Ag Setup Time Part h p d

1 350 2 1 0.01 200 30 2 0.01 150 10

2 400 3 1 0.008 120 20 2 0.005 180 25 3 0.005 100 15

products, we immediately have:

D' = 0.2167 P1= A' = 350 S =2

D2=0.4556P2=I A2 =400 S2=3

and after computing ri and qJ for g = 1, 2 and j = 1, 2, 3, we get H1 = 1.8186 and H2 = 0.7247, so that the two aggregate products are now completely defined. It remains to schedule these aggregate products. Dobson (1987) provides one of the best methods for generating feasible and near-optimal schedules for the ELSP. Using his approach to schedule the two aggregate products results in an average setup and inventory cost of 27.07. This schedule (which in this simple problem coincided with the rotation cycle heuristic) is shown in Figure 4 with its corresponding

optimal part schedules computed by the disaggregation procedure. In the figure, the unshaded rectangles represent the positive setup times between families. The lightest shading indicates JIT production and the darkest shading indicates machine dedication to that part. The intermediate shades represent non-JIT production that is concurrent with another part(s)' JIT production. The decomposition in conjunction with Dobson's ELSP heuristic provides a feasible schedule 15.6% less expensive than that of Ham, Hitomi and Yoshida. Of course, Ham, Hitomi and Yoshida's heuristic can accommodate positive intrafamily setup times and costs, but for the FMS case of negligible intrafamily setups it is possible to do better.

6. SUMMARY

We have addressed the realistic problem of scheduling production on an FMS. The problem is to schedule N parts divided into G families with setups required only between families. Given an aggregate schedule, the optimal corresponding part schedule can be found in 0 [N log N] time. This disaggregation is optimal for both the constant-demand FMS ELSP,

s1 < 12.01 > Family 1 Schedule

2 < - 24.24 ------ Family 2 _ __ Schedule

<- 7.66 Family 1: Part 1

4.35 Part 2 -

6.03 Family 2: Part 1

7.24 Part 2

'- 11.97 -> Part 3

Family 1/ Part Inventories 2 /=1

Family 2

Part Inventories 3 2/

0 55.4

Time

Figure 4. Aggregate and optimal corresponding part schedules.




and dynamic-demand hierarchical production planning models. In the case of the FMS ELSP, if we restrict ourselves to aggregate schedules that have the ZS property, then the aggregate products can be defined so that the optimal aggregate schedule will provide the optimal part schedule as well. The N-part, G-family FMS ELSP then decomposes to one G aggregate product scheduling problem, and G intrafamily scheduling problems, thus realizing considerable computational savings.

REFERENCES

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BITRAN, G. R., AND G. TIRUPATI. 1989. Hierarchical Production Planning. Working Paper #89-08-03, Department of Management, The University of Texas at Austin.

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BITRAN, G. R., E. A. HAAS AND H. MATSUO. 1986. Production Planning of Style Goods With High Setup Costs and Forecast Revisions. Opns. Res. 34, 226.

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HAM, I., K. HITOMI AND T. YOSHIDA. 1985. Group Technology Applications to Production Manage- ment. Kluwer-Nijhoff Publishing, Hingham, Mass.

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INMAN, R. R., AND P. C. JONES. 1987. Decomposition of a Group Technology Economic Lot Scheduling Problem. Technical Report 87-04, Center for Manufacturing Engineering, Northwestern University, Evanston, Ill.

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MAXWELL, W. 1964. The Scheduling of Economic Lot Sizes. Naval Res. Logist. Quart. 11, 89.



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