Scheduling a multi-stage production system with startup delays

OME.GA. The Int. JI of Mgmt Sol.. Vol. 6. No. L pp. I83-187 C Pergamon Press Lid 1978. Printed in Great Britain

0305-0,183~78;0501-0183;$0!00/0

Scheduling a Multi-Stage Production System with Startup Delays JM M O G G

PE BIGHAM The University of Houston (Received January 1977: in re~'isedforra Norember 1977)

The problem treated is that of multi-stage scheduling as affected by the interdependence of successive stages. An aigoritlun is developed which incorporates startup delays as analytical functions of cycle times and production rates at adjacent stages. This is accomplished in part by imposing integer multiple restrictions on the relationship between the cycle times at adjacent stages. Total cost is then minimized fo~ a given set of integer multiples.

SCHEDULING A MULTI-STAGE SYSTEM WITH STARTUP DELAYS THE SYSTEM considered here is one in which N production stages are arranged in series, with the output of one stage serving as input to the next. Significant assumptions include deterministic parameters, an infinite time horizon, constant lot sizes, and finite production rates. The major obstacle to economic scheduling in this system is the interdependence of successive stages. This interdependence is accounted for by computing the delay in the initial startup at each stage, relative to that stages' predecessor, required to prevent interstage stockouts. Although several of the listed references discuss multi-stage scheduling under these assumptions, most add restrictions which make startup delays unnecessary [2, 6-8]. Jensen and Khan [5] do not assume away the stage interdependence problem; however, their algorithm involves only the selection of the best schedule from among a user supplied list of possible schedules. Startup delays are found in [5] as non-analytical functions of the cycle times and production rates at adjacent stages. In this paper, an integer multiple restriction is imposed on the relationship between the cycle times at adjacent stages. Analytical start-

up delay expressions are written, and total cost is minimized for a given set of integer multiples. A computerized algorithm for finding the optimal integer multiples values is then discussed. MODEL DESCRIPTION The multi-stage production model developed below has the following characteristics: (1) A single product is produced for which demand occurs constantly and continuously. (2) The production stages are arranged in series with raw materials at the input to the system and product units moving sequentially from one stage to the next until arriving at the output stage as the final product. (3) Each production stage has a fixed known finite rate of production which is greater than the demand rate for the product. Thus, • each stage must be periodically shut down and restarted so that the average production rate of the stage is equal to the demand rate. A setup cost is realized for each such operation. In addition, the uneven production of each stage causes in-process inventories to accumulate between the various stages with an associated inventory holding cost incurred as a result. 183

3,1o9g. Bigham--Multi-Stage Production System

184

The following notation is used: N = number of production stages (numbered in reverse order to the product flow). D = demand (units/day) for the finished product. Ri = production rate (units/day) of stage i while operating (all quantities are measured in terms of units of final product). hi = cost (S/unit-day) of holding one unit of inventory for 1 day at stage i (i = 2 . . . . . N).

Si = setup cost ($) for stage i. This is the sum of the costs of one startup and one shutdown operation at stage i. T/= cycle time (days) for stage i, (i.e. the length of time between two successive startups at stage i). t; = production on-time (days) per cycle at stage i. Each stage must produce in each cycle the quantity required to meet the product demanded during the cycle. Therefore, equating total production per cycle at the ith stage (t~R~) to demand per cycle (T/D), and solving for t~ yields ti

=

TiO/Ri.

where mi is a positive integer or the reciprocal of a positive integer. THE STARTUP-DELAY P R O B L E M Let the inventory held between stage i and its immediate predecessor, stage i + 1, be designated the stage i + I installation inventory [1]. The inventory profile associated with these units will cycle over a period of time equal to the least common multiple of the production cycle times at stages i and i + 1. This inventory profile will, in general, not have the familiar saw-tooth shape. To illustrate, consider the parameter values and cycle times listed in Table 1. The stage i + 1 installation inventory profile resulting from these values is plotted in Fig. 1, Part A. As the graph shows, this inventory is negative over part of each cycle. This is, of course, a physical impossibility since stage i can operate only on the semi-finished units passed to it by stage i + 1. The negative inventory positions illustrated in Fig. I may be avoided by delaying the initial production startup at stage i, relative to that

(1)

d; = the delay (days) in the initial production startup at stage i, relative to that at stage i + I, required to insure that no shortages occur in the in-process inventories held between the stages.

InventOry

levet

=Idle ~

2o IG

Time,

days iProducing

1\

(o3I

o

_,ok

l

/ \

1\

I '~ i

.. )-l. . . ,o ,, ,2 The scheduling methods presented here have ( N o delay) I ~Stage i as an objective the determination of the cyclic I i S t a q e i+t startup and shutdown schedules for each stage 60 (i.e. the schedule of production lot sizes) which i minimize total time-averaged cost. The only costs considered are the startup and shutdown costs incurred at each production stage, and '~l I II IX/ I\ the cost of holding the in-process and finished '°0 II2 3 4 ~I l6l 7 @ 9 I0 il] 12 (2t/2daydelay) goods inventories. No inventory shortages are L' Stage i Stage allowed, and an infinite time horizon and continuous review reorder policy are assumed. The setup and shutdown times required at each i stage are initially assumed to be zero. (It can be shown that. this assumption is not crucial.) In addition to the above, a modified version '~/I I ~ 6 rIMa 9 Io IIIk12 of the integer multiple restriction imposed by o t 2 3 4 (2 day delay) Stage i others [2, 6-8] is adopted. In particular, the Stage it l I only production schedules considered are those for which FIG. 1. Stage i + 1 inventory profile for startup delays of 0. 2½, 2 days (reference Table 1). Ti+, = mi Ti. (2) 4C

"'

'

i

60

j

I '/k"

k

i*l

Omega. I,bl. 6, No. 2 TABLE

1. PARAMETER VALUES OF HYPOTHETICAL

STAGES.

R, > R ~ - t : (TIMES IN DAYS. D = 15 PER day) C.~cle

Production

Production

Production

Stage

time

rate

on time

off time

j

Tj

Ri

ti

~ - ti

i i + 1

4.0 6.0

40.0 20.0

1.5 4.5

2.5 1.5

at stage i + 1. For example, suppose the initial start of production at stage i is delayed 2.5 days. The resulting inventory profile is shown in Fig. 1, Part B. Numerical integration techniques may now be used to calculate an average daily stage i + 1 inventory of 30 units. However, as Fig. 1, Part C shows, stage i + 1 stockouts could also have been avoided by a delay of only 2.0 days, with a resulting average daily inventory of 22.5 units. If stage i + 1 has the greater production rate, a startup delay at stage i may or may not be required, depending on the cycle times chosen. In general, the minimum required delay at any stage i is a function of the production rates and cycle times at stages i and i + 1 (the preceding stage); and, as illustrated above, the length of this delay directly affects the size of the time-averaged inventory held between stages i and i + 1. Systematic methods for finding minimum required delay times using what may be termed 'deterministic simulation' have been developed [-5]. However, no general closed-form (i.e. analytical) expressions have been derived for such delays. Clark and Scarf [-1] define the echelon of any stage i in a multi-stage system as the units in the system which have passed through that stage but which have not yet been sold. Under this concept, the inventory holding cost chargeable to stage i is the incremental holding cost incurred as a result of processing at that stage. The echelon inventory of any stage i will be depleted constantly and continuously by final demand, and replenished by the cyclic production runs at stage i. Thus, if the effects of startup delays are ignored, the average number of unit-days held per day in this inventory is given by the single stage result li = T I D ( R I - D)/2Ri.

(3)

The total average setup and holding cost incurred at all N stages may then be written as Yi"- 1~,(SJTi) + lihi} + D Y xi=- l ' )2~=i+ t h d i

(4)

185

where the last term of (4) accounts for the total additional holding cost resulting from startup delays at stages 1 through N - I. Minimization of (4) with respect to the T~ requires that the di be written as functions of these decision variables.

M I N I M U M REQUIRED STARTUP DELAY TIME Assume that for any two adjacent stages in the system described above there exists prime integers Ki and Ki+t such that Li.t = K i + I T / + I = KIT~,

(5)

where L~+I is the least common multiple of T~ and T~+~, and is the period of time over which the stage i + 1 installation inventory will cycle. Given the production rates at stages i and i + 1, general expressions for the minimum required startup delay at stage i are defined as below. If R~ > R~+I the points of relative minimum for the stage i + 1 installation inventory occur whenever stage i shuts down. The minimum required delay at stage i is found to be di = max~(k T~D/Ri÷ t) + [kT]/T~. t ] ( T ~ . t - t i ÷ t ) -((k -

I)Tj - t~)l

(6)

where the first two terms in (6) are the point in time at which cumulative production at stage i + 1 equals cumulative production at stage i and the last term is the point in time at which i shuts down for the kth time. Also l-a] denotes the greatest integer strictly less than a. If R~ < Ri+ 1, the points of relative minimum for the stage i + 1 installation inventory occur at those points in time at which stage i + 1 starts production. The minimum required delay at stage i is found to be dl = m a x { ( k - l ) T i . l - ((k - 1)Ti. tD/Ri) -[(k

-

I)T,..JT/](T/

--

tt) I (7)

where the first term in (7) is the point in time at which stage i + 1 starts production for the kth time and the last two terms the point in time at which cumulative production at stage i last equaled that at stage i + 1. Again, [a] denotes the greatest integer strictly less than

186

Moy9, Biyham--Mutti-Stazje Production System TABLE 2. ANALYSISOF HYPOTHETICAL 3 STAGE SYSTEm. D = 15 PER day

Stage

(i)

System description Si Ri Hi

1

$100

2 3

200 100

40/ Day 20 600

$2 1.50 I

Computed values T~ di

Average daily costs set-up hold

delay

Total cost

3.02

$33.03

$7.09

$25.54

$65.67

33.83 16.52 82.59

5.68 44.27 57.04

0 0 25.54

38.71 60.79 165.71

1.14

6.05 0 6.05 0 Totals

a. The further development of analytical expressions for d~ must consider the integer multiple assumption defined by (2) and the magnitude of T~ vis a vis T~+ t. Similar to (2) the cycle time at any stage i may be written as a function of the cycle time at stage 1, namely T~ = M~TI, to be substituted into (4), which then becomes a strictly convex function of "Ft. The optimum 7"1 to minimize cost may then be determined. AN A P P L I C A T I O N A computerized algorithm has been developed for finding the minimum cost set of integer multiples. Suppose that m'l is the smallest positive integer for which C(m'~)< C(rn't + 1.0). It may then be shown that m't is the optimal value for m~, given the current, values for rni, and the restriction that m~ be a positive integer. Similarly, if m'~ is the smallest positive integer for which C(I.O/m'~) < C(I.O/(m'~ + 1.0)),

the optimal value for rn I is then (1.O/m'i), given the current values for mi, and the restriction that m I be the reciprocal of a positive integer. The program written uses an iterative procedure to find C(m'O and C(1.O/m'~) for each possible set of integer multiples mi, i = 2 ..... N - 1. The least cost solution found to date is saved after each interation. To illustrate, consider the 3-stage system described by the parameter values listed in Table 2. Use of these values yields unrestricted cycle times of 3.05, 6.53, and 5.30 days for stages 1, 2, and 3 respectively. Given (2), these cycle times imply ml = 2.0 and m 2 = 1.0, which yield the integer multiple production schedule described in Table 2. Note that solution time for the above pro-

cedures expands exponentially in the number of stages considered. The authors have developed a heuristic search technique for finding optimal integer multiples. This method involves setting mi = 1.0 and then making the single incrementation or decrementation which yields the largest cost savings. This procedure is repeated as long as cost savings may be attained. Fifty randomly generated problems were solved by both techniques; enumeration yielded a total cost more than 2~ lower than that found by search in only three cases. SUMMARY AND C O N C L U D I N G REMARKS The multi-stage system considered herein is essentially identical to that described in references 1"2] and 1,5]. In 1,-23, integer multiple production schedules are found only for the case where Ri+l > Ri, thus avoiding the necessity of startup delays. In the present work, explicit expressions have been written for the d~ thereby allowing the removal of this restriction. Jensen and Khan 1,5-1 do not impose the integer multiple restriction, requiring instead that the user input finite lists of allowable cycle times for each stage. Their algorithm involves only the recursive selection of the least cost set of cycle times from among these user-defined possible sets. The performance of this algorithm is directly limited by the judiciousness with which possible cycle times are specified. No guidelines are developed in 1,5] for selection of allowable cycle times. REFERENCES 1. CLARK A & SCARF H (1960) Optimal policies for a multi-echelon inventory problem. Mgrnt Sci. 6, 475--490. 2. CROWSTO~ WB, WAGNER M & WILLIAms ) (1973) Economic lot size determination in multi-stage assembly systems. Mgmt Sci. 19(5), 517-527.

Omega, VoL 6, No. 2 3. CULLENDER M (1971) A scheduling algorithm for a multi-stage multi-product production system, Unpublished masters thesis. The University of Texas. 4. H^DLEVG & WHrrnN TM (1963) Analysis of Inventory Systems. Prentice-HalL Englewood Cliffs, NJ. 5. JENSENPA & KHAN HA (1972) Scheduling in a multistage production system with setup and inventory costs. AIIE Trans. 4(2). 6. SCHUSSELG (1968) Job shop lot release sizes. Mgmt Sci. 14(8), B449-B472.

187

7. TAH* HA & SKEITH RW (1970) The economic lot sizes in multi-stage production systems. AIIE Trans. 2(2). 8. THOMAS AB (1963) Optimizing a multi-stage production process. Op. Res. Q. 14(2), 201-213.

Dr JM Mogg. Associate Professor, University of Houston, Quantitative Management Science. Houston. Texas 77004. USA.

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