Real-Time Lot-Sizing Subject to Random Setup Times

Copyright © IFAC 12th Triennial World Congress. Sydney. Australia. 1993

REAL-TIME LOT-SIZING SUBJECT TO RANDOM SETUP TIMES M.H. Burman and S.B. Gershwin Laboratory [or Manufacturing and Productivity. Massachuse tts Institute o[Technology. Cambridge . MA 02139. USA

Key Words Dynamic Programming. Manufacturing Processes, Management Systems.

co mplexity of this problem (Oitran and Hax . 1977) . Traditionally, the preceden t has been to let lot sizin g dominate lot sequencing. For exa mple. cas ual obse rvati ons of traditional planning operat io ns indicate that co rporate manage m ent establis h lot sizes usin g classic m odels such as the E LSP (E lmaghraby, 1978) and due dates at a hi gh level. leav in g the task of lot seq uen cing (Melny k and Cart er . 1986) as th e res ponsibility of the fa cto ry. Burman ( 1992) separates the two functions of lot sequencing (Level 1) and lot sizing (Level 2) into two different leve ls with lot sequencing dominating. This paper hi ghli ghts the lot-sizing policy developed in that paper. That paper also discusses ho w to extend the methods discussed here for d ealing with one rand o m phenomenon (set up times) to ot her sources of variability.

Abstract A real-time , closed-loop set up ("o ntr ol algorithm for a manufacturing facility with due dates and significant setup times is pro posed. The problem is formulated as a dynamic program and num erically analyzed for a simple example. Extensions fo r prototyping are briefly discussed.

1

Introduction

A real-time method of dynamic lot sizing at a system bottleneck subject t o significant, sequencedependent, random setup times is developed based o n the scheduling requirements of a medical device manufacturing facility . Demand fo r a number of part types , all with the same fix ed due date, which must be met with a high level of co nfidence, are provided to facility every week along with limited future demand estimates. Once received , demand is assumed to be frozen . Inventory may be produced for selected part types. However , it is undesirable to produce excessively large inventories associated with large lots. Dynamic lot sizing is oft en treated as a variation of the eco nomic lot siZing problem (ELSP) (Wagner and Whitin , 1958). Zoller and Rob rad e (1988) summarize he uristic methods for dynamic lot sizing. Most of these methods adjust lot sizes based on long-term demand and capacity fluctuations . This approach overloo ks the details required for responding to shorterterm, random events , as d o ne in this paper. Algorithms must account fo r uncontrollable disruptive events in real time (Gershwin, 1993). However , efforts to optimize stochastic systems frequently lead to the well known curse of dimensionality. Hierarchical production planning can be used to the redu ce the

2

Problem Description

An C List of part t ypes (pi, i = 1, ... . n) which approximately minimizes expected total set up time is provided (Burman , 1992) . Figure 1 shows a Gannt chart summary of a typical week of production , where the horizontal distan ce represents the time spent on a particular activity (i.e. , black bars indicate setup time and white bars . production time). As eac h set up is co mpleted , t he time required fo r it. whic h was a n unkn own rand o m quantit\'. beco mes il. known qna ntity. Usin g t his new info rm ati o n . adj ustme nts a re

Ord. r . Arr1. ..... •

Se tup

o

Delivery Date Proctuat,10D

Figure 1: Two-Level Method

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made to the schedule in real time by altering the lengths of the future white bars in Figure 1 while maintaining the part type sequence of the L List. The purpose of this model IS to determine when to switch the part type being produced (I.e., adjust the current white bar). Meeting current requirements with a high level of confidence is very important. If a lot is too large , there will not be enough time to complete the rest of the current reqlllrements . However. if larger lots are produced. fewer setups are required. allowing more production. Lot sizes are required which establish an acceptable balance among the system capacity , the probability of completing current requirements, and inventory costs . in light of system uncertainty (e.g., random setup times). The success of the controller is measured according to the percentage of weeks during which current requirements are completed . We assume sufficient capaCIty and lo w inventory costs. A logical question that could he asked is , H\Vhen possible , why not just overbuild and keep very large inventories? " First . orders are only provided a limited number of wet'ks In advance and may vary substantially. Second. there are o ver 700 part types produced by the facility. Third. while the explicit cost of even large amounts of in ven tory is low. the logistics of tracking such inventory are conSId ered undesirable by management. Therefore . fo r most part types, producing inventory is an acceptable policy as long as it is limited strictly to what is included in the current demand projections. A number of additional assumptions are made to simplify the formulation of the model: the setup sequence is fixed; all setup times have the same discrete probability distribution: productio n times are deterministic: demand projections are pr ovided only one week into the future: and failures are nonexistent.

3

Decision Point i+1 i

D.ci.ion Point iN.xt W•• k.'. Production Requirement for Part Typ. i

Figure 2: Decision Points .

activity. Upon the completion of Setup i, the system can produce only the current requirements for Part Type i and immediately set up for Part Type i+ 1; or it can produce the next week's requirements for Part Type i and then change setups (Figure 2) . We model this as a dynamic program .

3.1

System Parameters and Variables

To work with the dynamic program, a number of system parameters must be defined. System Parameters

T N

The amount of time available each week for production and setups. The maximum number of setups required within in a week. Penalty coefficients for not meeting current production . Production time for current demand (Dt) or for projected demand of the following week (Di) for Part Type i (m 1,2) . Setup time i . This is a random variable which can take on two values which we call Long Setup and Short Setup. Probability mass function for S.

=

The Dynamic Program PiS)

The lot-sizing algorithm developed in this section determines the size of each lot in real time . while maintaining the part type sequence dictated by the L List. Determining a closed-loop policy for the factory lends itself to a dynamic programming formulation (Bertsekas, 1987). Since setups are the only random activities in the system and there is a finite number of them , a discrete dynamic programming model indexed to setup completions is formulated. Let i represent the index of the position in the L List of the part currently being produced. or the part for which the bottleneck is currently being setup to produce. Let i a i.so signify the ith point in time when a setup has been completed . Suppose a setup has just been completed (Figure 2, Decision Point i). Since processing times are assumed to be deterministic, the controller has as much information at the completion of Setup i as it does after the production of the current requirements of Part Type i. Setup i + 1 is the next random

System State Variables The system state contains all the relevant information for deciding whether to switch machine configurations or continue production of the same part type. The system state variables upon the completion of setup i are:

Ti Ri ;ri

3.2

The time of the ith setup completion. The number of future lots completed at Ti. (Ti ,Ri)

Control Function

The control function is a mapping from the system state to a control action. The control actions are to

512

produce only the present requirement (Dt) and the length of the random setup time (Si) . Consequently:

build inventory or to switch setups: Build inventory with the current machine configuration . Switch setups to the next part type in the £ List.

r + C(tJi(;ri)) =

1"+1

Ti

+ D~ + tJi(zi)D; + Si

with probability P( Si) where Si can be a

~hort ~ e tup

or a long

(4)

~etup.

and we always have :

(1 )

The d ynamics of accumulated future setup reductions is governed by :

because the control at state cannot be to build more if there is not enough time to complete both the current and future requirement for Part Type i.

(5)

:r'

3.3

Then :

Objective Function

(Ti+! , Ri+!)

The goal of the dynamic program is to develop a control policy which results in good system performance. System performance is measured as a function of the number offuture and current requirements completed . We are only conct'rned with completed future jobs because incomplete jobs do not eliminate any future setup requirements . The objective func tion is designed to reward a good system performance by increasing linearly with the number of future lots completed (Ri), and decreasing quadratically with the number of current jobs not completed by the end of the week (I). We would like! to be zero . The objective function is : R' - P 1 !- P 2( 1)2

{

gi(zi)

if Ti+! > T & Ti < T

zi

3.5

(6)

General Program Formulation

Since the objective function has a non-zero value only at the end of the week (2), the cost-to-go function (J) at Ti is written as a co nditional expectation (E) of the objective function value at the last decision point i· of the week . That is : Optimal Cost-to-Go Functio n J.,(Ti , Ri) =

,max,. E [gi·(Ti· , Ri")li , 1",R i ]

(2)

I' , ... ,1'

0

otherwise

s.t . T i " + 1 -> T and T i • < T i and (Ti,R ) satisfy (6) .

{

N -i N + 1- i

if T_Ti if T_Ti

2 <

Dt D~

(3)

(7)

We generate the solution to (7) by recursively solving the following equatio ns :

A discussion on the choice of PI and P2 coefficients is presented in Section 5.1 (Optimal Threshold Approximation) . The condition in (2) indicate that at some point between the completion of setup i and setup i + 1, production time runs out . Therefore , pi is the last part type produced in the current week . Condition (3) determines whether there is actually enough time (Dt) to produce Part i. In other words, once setup i has been completed, Dt time units are required to complete job i. The two equations establish the terminal system state and allow the policy cost to be determined .

3.4

+ ti, Ri + tJi) = Zi + (ti,tJ i ) + (D~ + S~ , 0) + tJi(zi)(D; , 1)

(Ti

where

I

= fi(Zi,tJi , t i )

J.,(1" , Ri)

=

maxE [i.'+I(fi(Ti , Ri,tJ i, ti))j , i

= 1 to n

1"

maxE [J., +I (Ti

+ D; + tJi D; + Si, Ri + tJi)j

1"

(8)

4

Numerical Analysis

An example demonstrates how an o ptimal co ntroller behaves (Table 1). T en part t y pes require processing in the current week and there is a projection for the same amount in th e followin g week . Each part type requires exactly 5 units of processing time each week. There is 100 time units available on the bottleneck machine in one week of production . Setup times are significant and require 2.5 time units with probability 0.5 or 5 time units with equal probability. Different values for the penalty coefficients PI and P 2 were tested.

System Dynamics

The random variable t i represents the amount of time between decision points i and i + 1 (Figure 2). If the control action is to build an extra job, t i is the time to produce the present requirements (Dt) plus the time required to produce a second job (D~) plus the random length of the setup time (Si). If the control action is not to build an extra job, t i is the time to

513

than setup time. Rather than formulating a new dynamic program , another he uristic based on an interpretation of the threshold solution was used. The heuristic allo ca tes an amount of time to all required future activities (setup time , failure time , primary production time, scheduled downtime and time for variability in all these quantities) based on a desired level of confidence for completing current requirements . More time must be allocated to each activity for a higher level of desired confidence. By subtracting the time allocated to future activities from the time left remaining in the week, an amount of available time for discretionary activities (e.g. , employee training, process validation , future production requirements , etc.) is obtained. As events occur (e.g., setups are completed , jobs are completed , failures are repaired, etc.), the difference between the actual time required to complete an activity and the time previously allocated to the activity becomes known. Therefore, as time passes. unplanned production time will usually increase. A list of all possible discretionary activities are provided to the system in what we call the P Set. The resource may be used for any P set activity as long as it requires less than the amount of accumulated unplanned production time (Figure 4). The amount of discretionary time establishes an equivalent threshold as established earlier in the paper

lOO

T cV D'",

10 5 fo r m

J

P(S)

l

OS 0.5 ()

1.2 i ::: 1 to 10 fo r S = 2.5 fo r S = 5 u therwise.

:::

Table I: Example Parameters Results A threshold policv nists if when i setups are completed , there is an associated 0' where if T Ti (the amoun t of time left in a week) exceeds oi one should build inventory. OtherWise. one should switch setups. That is.

J.L'(x')

=:

Build in Vellt o ry if (T

T') :C:

Switch setups If (I' _ T') fo r i .:::. 1 to 1U.

0' .

< o'

By assuming that the su',i ch sdups action in the anomal o us regio n where 'J" -= 12 is an unimportant end effect. a thr es hold policy represented by the thres hold approXImatIOn ( FiJl;ure :l) is suggested as a controller for the example.

9

...

1

XI

5

... J

Previously Allocated Time

f-'" .."

.... t:

7

~

~ o

'0 100

Figure 4: Allocation of Discretionary Activities From fO

80

70

60

so

40

JO

10

10

P Set

0

Time Left Within The We<>k (T,)

c:::::IBulld More&::mJlndlfTerence II1II

-

c ..... r ~tup­

Threshold Approximation

6

Figure 3: Control Policy

Currently, discussions are und erwav with a number of Fortune .500 companies for furth er research and implementation projects. These extensions include: testing different cos t functions to account for different business situations; more rigor o usly including failures and other random activities in the analysis; extending the model to multiple serial and parallel machine cases; accounting explicitly for up- and downstream control; and providing more general rules for optimizing the selection of P set activities. The results of our work demonstrate that closed-loop feedback is a reasonable methodology for factory scheduling subject to uncertainty.

Numerical simulations demonstrated both the superiority of a close loo p linear approximation of the optimal threshold as wpll as th e more generally known superiority of closed - vs. ope n-loop control policies (Burman, 1992).

5

Conclusion

Prototype

A prototype software program for an eventual implementation was developed based on the results of this research. The prototype had to account for many additional sources of randomness in the system other

514

Acknowledgements The authors would like to thank the Leaders for Manufacturing Program at MIT, and John Matson, Geo rge Brandenburg a nd Don Nociolo at Johnson & J o hnson for their support, advice and assistance in conducting this research .

References [1] D. Bertsekas. Dynamic Programmmg: DetermmIstic and Stochasil.c Models. Prentice Hall, New Jersey, USA, 1987 . [2] G. Bitran and A. Hax . O n the design of hierarchical production planning systems. DeciSion Sciences, volume 8:pages 28 - 55, 1977. [3] M. Burman . A real-time dispatch policy for a system subject to sequence-dependent, random setup times . Master 's thesis , MIT, 1992 . The eco no mic-lo t scheduling [4] S. Elmaghraby. problem (ELSP): Review and extensions. Management SCience , volume 24(No. 6): pages 587597. 1978. [5] S. Ge rshwin . Manufactunng Sys tems Engineering. Prentice Hall , Inc., 1993 . In Preparation. [6] S. Melnyk and P . C arter . Scheduling, sequencing and dispatching: Alternative perspectives . Production and Inventory Management, pages 58-67, Second Quarter 1986. [7] H. Wagner and T . Whitin. Dynamic version of the economic lot size model. Management Science, volume 5(No. 1 ):pages 89 - 96, October 1958 . [8] K. Zoller and A. Robrade . Efficient heuristics for dxnamic lot sizing. Int ernatIOnal Journal of ProductIOn Research, volume 26(No. 2):pages 249265, 1988.

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