The sensitivity of jobshop due date lead time to changes in the processing time

Computers ind. Engng Vol. 35, Nos 3-4, pp. 411-414, 1998 © 1998 Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain PIh S0360-8352(98)00121-1 0360-8352/98 $19.00 + 0.00

Pergamon

The sensitivity ofjobshop due date lead time to changes in the processing time M. Hamdy Elwany

Amr El Baddan

Professor o f Industrial Engineering Faculty o f Engineering - Alexandria University

Research Assistant Faculty o f Engineering - Alexandria University

Abstract. This paper provides a procedure for calculating the sensitivity of the production lead time to the average job processing time for a single machine problem under a general priority rule using simulation. The aim of this work is to develop sensitivity routines, for queues with any nonpreemptive priority rule, to calculate the sensitivity of the due date lead time to the average jobs processing time. Though GI/G/1 queuing model is given, the same methodology can be extended to GI/G/m queuing model. The new algorithm is validfor the SPT, LPT and FCFS priority rules. © 1998 Published by Elsevier Science Ltd. All rights reserved.

Keywords: Jobshop scheduling, simulation, queuing models. 1 NTRODUCTION AND LITERATURE REVIEW. The majority of the scheduling literature pertains to jobshop scheduling (see Panwalkar and Iskander 1977 and Graves 1981 for a survey) focuses on the priority sequencing decisions of the jobs waiting at a particular machine in the shop, which one should be worked on next to optimize one of the performance measures. Exceptions to this rule include (Baker 1984 a, and Jones 1973), who designed simulation studies considering various decisions and performance measures. More recent researches have analyzed sequencing decisions jointly with other dynamic decisions such as, due-date setting (Baker 1984b, Wein 1991b), due date setting andjobrelease (Weinand Chevalier 1992), pricing (Klienrock 1967, Balachandran 1972, Marcahnd 1974) and lot sizing (Karmarkar 1987). There have also been studies focusing on multi-criterion scheduling (Nelson et al. 1986) but these are restricted to the static, deterministic, single machine case. Other scheduling literature studies consider the scheduling problem solution techniques such as: mixed integer programming (Lasserre 1992), combinatorial optimization using problem specific heuristic and local search methods that are frequently associated with A.I. (Storer et al. 1992). Few papers, such as the work of (Wein and Ou 1991), intended to study the effect of various system parameters on the scheduling problem. 1-1 Gradient Estimating Techniques (GETs) for Discrete Event Simulation Models. GETs, evaluate the sensitivity of one or more performance measures during simulation. The GETs include: Perturbation analysis PA, finite perturbation analysis FPA, infinitesimal perturbation analysis IPA, smoothed perturbation analysis SPA, finite difference FD, f'mite difference with common random numbers FDC, and, score function SF, or likely hood ratio LR. 1 -2 Finite difference. The simplest derivative estimation technique is the finite difference one. It requires the performance of two separate simulation runs for each sensitivity measure. The centered FD estima.tor is. APLT

_ PLTI

- PLT2

1

1t0

40 1 -3 Score function. The score function, SF, or sometime called the likely hood ratio, LR, is based on the transformation of random variables by scaling them. 411

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1 -4 Perturbation analysis. Perturbation analysis aims to calculate the sensitivity measure in one single run. "The basic idea of perturbation analysis is the reconstruction of an arbitrary perturbed sample path from the nominal path" (He and Li 1988). Thus the perturbed path differs from the nominal one in two aspects, the event times are shifted and the sequence of the events occurrence can be changed. The perturbation analysis PA algorithm has its origin in a paper by (He et al. 1979). 2 SENSITIVITY ANALYSIS FOR A SINGLE MACHINE CASE USING SIMULATION 2 -1 Model Description. The jobshop is modeled as a single server queue. Orders arrive dynamically to the shop in a stochastic manner. The interarrival time distribution of jobs is, FA with a mean arrival rate ~.. The jobs effective processing time distribution is, Fs and the mean job effective processing time is, 0. 2 -2 Analysis of the Problem. Perturbation analysis is performed by constructing a perturbed path while performing the simulation run for the nominal path, with a mean service time of 0. The perturbed path is defined as the realization of the system but for a different mean service time of 0-A 0. That means that the same number and classes of customers requiring service at the nominal path will be used in the construction of the perturbed path with average processing time 0-A 0 2 -2 -1 Factors affecting the average due date lead time (Busy period analysis). The due date lead time of any arbitrary customer Ti (equation (2)), where, 1, is indexed on the departure sequence of the customers, served during this busy period is the difference between his arrival time and his departure time. t

2

T/= ~ S j(O)-(tl-tl) j=l

Knowing that, (n), customers are served during this busy period the average due date lead time is given by equation(3). _

1rn

I

]

I Y= L

E sj(O)-

n

I j=l

A, I-2

i=l

T = I-(n-jCB)+1)SjCB)-~..(n-i)A, n LJ-I ,.t

Equation (4) is a reformulation of equation (3) in terms of interarrival and service times, which are the two random variables driving the simulation model. "]", is the departure sequence index and, i, is the index for the arrivals. Where, n

!

E,S,(O)= n

6

n

I.I

E,(,-

n*l

A' =

~l~efore; ~

(n - 0 A'

j(o)+

Selected papersfrom the 22nd ICC&IE Conference

n

n-1

+l)xs/o)-

n

]

,-,x:"°A"x,J:°)s.

413

8

]

To calculate the partial derivative of the due date lead time with respecttothe average jobs processing time (0), we differentiate equation (8) with respect to (0). The differentiation of (j(0) Sj(0)) with respect to, 0, as seen from equation (10) is composed of d_f_T_ 1 (n+ 1) 8Sj(O) 80 n ~.~ 80

-~j.~A(j( Ü) s j( O))]

two terms. The first term, ~,j(0) 6/60 (Sj(0)) depends only on the change in the service time of the customers and thus is not affected if a change in the event order occurs. The second term ~Sj(0) * 8/60 0(0)) includes the change in the position of the customer in the departure stream and thus contains the effect of event order change. For the original IPA the event order change is neglected and thus this term equals zero. Since the evaluation of the second term of equation (10) is very hard analytically we will develop a computer algorithm to evaluate it. This term can only be assessed if one can predict the new position for each customer in the departure stream at the perturbed path and thus calculate for each customer

a(o)sj(o)) = j=l

Aj(O)=j

j(o) j=l

+ ¢~0

sj(O) j=l

Nominal -j Perturbed

2 -3

10

I1

Construction of the Perturbed Path.

To perform the perturbation analysis for the queuing system one must first calculate the perturbed k

12

Tk(8) = ~ Sj(O)-(tk'tl) j=!

value of the service time for each customer. Since perturbation analysis aims to construct a perturbed path for the system using the same realization of the random variables used for the nominal path, under a different parameter value, one can predict the customers perturbed service time as follows. Adapting the inverse transform method, which is one of the most widely applied methods in generating random variables. This method uses a uniformly distributed random variable U(0,1) to generate random variables of an arbitrary distribution. The perturbed service time will be calculated based on this method in random number generation. Let (ub u2, u3, u4, . .., uj) be realizations of the uniformly distributed random variable U(0,1). The service time of an arbitrary customer Cj by the inverse transform method is generated as follows. Where, Fs, is the cumulative probability distribution of the service time. Thus one way to calculate the perturbed service time is to save uj and then use it to generate the perturbed service

s J(O ) =

-I

13

(uJ)

time as follows:

S j(O

+ AO)=

F ( u S ( O + ,~0 )

/)

14

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The disadvantage of this method is that it requires the previous assessment and saving of, uj, which requires additional programming effort. Their are special cases where we can predict the perturbed value o f the service time directly from the nominal one. 3

NUMERICAL ALGORITHM.

VERIFICATION

OF

THE

DERIVATIVE

ESTIMATING

The numerical verification experiments are divided into two sets. The first of which are used to compare the new algorithm with queuing models for which exact analytical solutions are available. These are the M/G/1 queuing models with, FCFS, queue discipline. The second set of experiments is performed by comparing the performance of the new algorithm with the finite difference with common random numbers, FDC, one. This set is used for queuing systems for which exact analytical solutions are unavailable. We test the GI/G/1 under FCFS, SPT and LPT rules. An antithetic sampling variance reduction technique was utilized to reduce the simulation run length and to reduce the variance. Considering M/G/1 queuing system with : arrival rate 8 per unit time, the service time S=2X, where X, is a random variable that has any arbitrary distribution G(X) with a mean value of 1. The utilization factor is varied with a typical values of 0.9, 0.8, 0.7, 0.6, 0.5 and 0.3. The error in the SPT case is less than + / - 2% for both M/M/1 and M/G/1 models. The error in the LPT case is from +4% to-11% for both M/M/1 and M/G/1 models. Notations: Ai ti

= =

Si Fs0 TI n E(T)

= = = -=

j

=

i

=

0

=

WR

--

Interarrival time between customer C, and C~+t The arrival time of customer C~. The service time of customer C~. The probability distribution of, S,, under a parameter value of, 6. Due date lead time for the 1~hcustomer served in the busy period. The number of customers served in a busy period. The expected value of the due date lead time. Index for the departure sequence of customers. lndex far the arrival sequence of customers. The average job processing time. The waiting time of the I~h customer served in the busy period.

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