Scheduling Jobs with Precedence Constraints on Jobshops and Shops' Coordination

Copyright © IFAC Large Scale Systems, Beijing, PRC, 1992

SCHEDULING JOBS WITH PRECEDENCE CONSTRAINTS ON JOBSHOPS AND SHOPS' COORDINATION! Sun Xiaoqing and Zheng Yingping lnsliluJe of AuJomalion, Chinese Academy of Sciences , Beijing 100080, PRC

ABSTRACT. Scheduling is one of the most important issues in the planing and operation of manufacturing systems. In practice, schedul ing is often c~lemented by some sirrple heuristics or simulation methods. They are either time consuming or lack of validity. The interactions of jobs as they c~te for limited resources(machines, pallet,etc.) is not visible. The dynamic changes in the systems are not easily accommodated. In the job-shops, operations are not only interacted by limited resources, but also processing pieces them self. This paper presents a method for scheduling jobs with different operations on jobshops and coordinations of jobshops with weak correlation. Each job consists of a number of operations with partial precedence constraints with application of Lagrangian relaxation technique. The methods also considers "timeout" for other usage. The objective function is the total weighted quadratic "tardiness" of the schedule combining weighted job's makespan. A two layer hierarchical structure for shops' coordination and jobshop schedul ing is proposed. As the problem discussed is NP-hard, we will present an efficient, near-optimal algorithm rather than the optimal solution. The major contribution of the paper has been three folds: (1)P.B. Luh's work is exterded to the job-shop's scheduling and sirrpler algorithm has been devised; (2) The problem is considered in continuous time domain; (3) Coordination of jobshops is also considered.

Keywords: Manufacturing systems; Discrete Event; Scheduling. Decomposition of such a jobshop into weakly coupled, small jobshops is necessary. This is often performed by applying group technology(GT). The coordi nat i on of these j obshops is a very important and challenging task. In this paper, non-preerrptive scheduling of jobs with due dates on job-shops are considered. Each job is made up a number of operations with partial precedence constraints. Some jobs may be combined into one job, which resembles assembly operations; and a job may be divided into more jobs, which accommodates di sassembl y operat ions. There may be timeout between successive operations for inspect ions or other operat ions wh i ch does not require the use of machines. The partial precedence here means some operations are to be processed in a particular order, some are not. Throughout the paper, two assumptions are made: 1). Processing times and times for 'timeout' are known; 2). The precedence constraints are known. The objective is to minimize an objective function combining weighted quadratic tardiness and weighted quadratic makespan for some particulars jobs. The reason for adding the makespan in the objective function is to minimize the in-system inventory and shops' coordination.

IIITRmUCTI(JI

The Sc:hecl.Jl i~ Problell

Many efficiency and productivity problems are closely related to scheduling operations. Manufacturing, transportation, cOflllU'lication systems are typical examples. The direct result of a consistently good schedul ing is to reduce excess inventory and resource's waiting time. The importance of finding a good schedul ing method lies in its wide appl ications and irrprovements resulted in the applicable systems. The difficulty is that most scheduling problems belong to a class of NP-hard combinatorial optimization problems, the optimal solutions of which are beyond the tractabil ity of existing mathematics. In Practice, sirrple heuristics which can not accommodate changing environment and its performance is not guaranteed, and expensive c~ter simulations are often appl ied. The dynamics of the system is non-visible such as system break down with application of the two methods. Algorithms which are commonly appl ied in practice can not accommodate a jobshop with a large nuttier of machines and products.

1The research is jointly sponsored by the National CIMS Project and the Laboratory of C~lex Systems, Institute of Automation, Chinese Academy of Sciences.

169

For medium and small sized job-shops, the algorithm proposed is efficient. For large sized shops, it is time consuming. Based on group techniques, a large shop is often divided into some small shops. By shortening the makespan of the jobs which are processed in roore than one jobshop, it is easier to coordinate job-shops. A two layer hierarchical framework is proposed to schedule jobshops and to coordinate them. One of the drawbacks of LR method in schedul ing manufacturing systems would be lack of consideration of buffer size. By introducing jobs' makespan to the objective function, some improvement can be made with this regard.

Organi zat i on of the Paper

Applying integer programming technique, the problem is formulated in Section 2. In Section 3, the Lagrangian relaxation technique is appl ied conbined with list scheduling concept to solve the problem. The frame work for coordinating shops is given in section 5. Conclusion remarks are given in Section 6. PROBLEM FORtIJLATIOII

To express the scheduling problem, integer programming formulation is a comroon method. Influenced by the work of Debra, et al,(1989), the following continuous-time integer programming formulation is presented_ Some variable definitions are given first. 3 i j {t)a 0,1 integer, 1 if operation j of job i is active at time t, 0 otherwise. 1 if tdbij,bij+tij+Si}' 0 otherwise. beginning time of job i operation j. completion time of job i. completion time of job i, operation j. due date of job i. objective function to be minimized. time horizon under consideration. a 0, 1 integer, 1 i f mach i ne m i s available at time t, otherwise. number of operations of job i. number of jobs. length of timeout of job after operation j, 0 otherwise_ processing time of job i, operation j. tardiness of job i. value of importance of job i. if (j, l) EPi, then operation j must be processed before operation l. beginning time of operation (i,j)_ a step funct ion, 1 if x ~O, 0 otherwi se. the weight on job i which couples some jobshops in the objective function. I[Bi , Ei J The feasible interval in which job i can be scheduled.

The Relevant Research A parallel machine scheduling problem with precedence constraints is studied in [Lenstra, et al., 1978] and [Lawler, et al., 1982]. the objective function used is the minimum of the makespan. The problem is shown to be NP-complete which is simpler than scheduling on jobshops. Heuristics have been proven to have advantages over analytical approach in terms of efficiency. Most heuristics are based on list scheduling concept. Jobs and . operations are listed according some criteria, and then scheduled based on the list. If makespan is used as the objective function of an representative schedul ing problem, non-preemptive, nonprecedence constrained parallel identical machine scheduling, the LPT ordering which schedules job with the largest processing time to the earliest available machine, has the worst error to optimum ratio 3/4-1/(3m), where m is the number of parallel machines. It complicates the situation that there are precedence constraints and tardiness is applied as the objective function.Recently, effort has been made in developing alternative approach which is deroonstrated efficient and near-optimal. The approach is based on Lagrangian relaxation method. The dual solution provides an ordered list for the list scheduling method. P.B. Luh, et al(1988,1989) and Debra, et al(1989) have studied the non-preemptive schedul ing of jobs with due dates on identical, parallel machines. Each job is made up a small number of operations which must be processed in a particular order. The simulation provided shows that the near-optimal solution are within 1X of the lower bound of the tested data. There are three factors may limit the application of the methods: 1). The discretetime domain in which the problem is formulated; 2). the limitation that is set on the number of operations; 3). The simple precedence constraints wh i ch can not acconmodate the compl i cat ions of production environment.An extension of the work mentioned above is the schedul ing of jobs on jobshops. Each job consists a number of operations which are processed in a particular order named precedence constraints. Assembly and disassembly may be acconmodated in the structure. The problem is formulated in the continuous-time domain and the algorithm is simpl ified. Similar work has been simultaneously done by P.B. Hoitomt(1990). However, our approach is different from that of P.B. Luh's.

Since t i j and Sij are given, the decision variable becomes
is to be processed before operat i on (k, n k ) The objective function is defined as the conbined quadratic tardiness and job's makespan:

170

I' j -

(bjj-bil-til-Sil)

)'

(j:n"EPJ

3.1

N

J-)' WjU( Cj-Dj) ~1

U(bjj-b il - t il- S ill ,

(Cj-D j ) 2

(i-l, . . . ,N) .

2.1

-
H j ( t)

Here J is convex or concave to

b jj

•

U(E

scheduling problem can be formulated as follows: P:

1j

(t) -Hj (t)} ·

1

Then the

3.2

a 1j (t) -Hj (t) ) ,

1

2.2

HinJ b'j

N j

subject to precedence constraints: b l1 -b jj -s j j - t jj~O, if(j, 1) EPj ,

(t) -~'3" jj (t) -i)'

3.3

U(~'3" jj (t) -1)

2.3

and

Nad ll

h

1.k. [(i.n, • (k ,nkIEP...J

3.4

(b1n,-bJcnk-tJcnk-SJcn) .

b jn ,- tJcn k-sJcn k-bJcnk~O'

U(b jnj -bJcnk - tJcn k-SJcn k) •

i f ( i n 1 ,kn k ) EP ad ,

2.4 and capacity constraints: NJ

~

a jj (t) -Hj ( t) SO,

To decompose the scheduling problem P with respect to operations, we relax capacity constraint (2.5) and (2.6), and precedence constraint (2.3) and (2.4) by introducing

2.5

l,

Lagrangian multipliers

8,

and y

to

form the relaxed optimization problem:

N,

~ '3" jj (t)

It,

-ls0

2.6 Hin)' w j (C j -D j )2U(Cj -DJ + b'j

Note that the problems are not only coupled by the machine capacity constraints, but also by the job capacity constraints as well as precedence constraints of different jobs as the result of operation of assembly and disassembly.

fj

R:

AI' j+8H j (t) +

3.5

ltN j (t) +yN"d'

Then, the dual problem becomes

SOlUTION METHCOOl.OGY

Lagarangian Relaxation Approach to the ScheclJl ing Proble.

Hax L, l , .. ,y.8

Lagrangian relaxation is a useful approach to decompose constrained optimization problem into some subproblems which are easier to be solved. Debra, et al,(1989) has briefly discussed the development of the approach and its application in the schedul ing of related problems. For convenience, we define

0:

with

L-lEI' j+8EH j (t) + 1

]

3.6

ltE N j (t) +yN"d' 1

The above problems can be further decomposed into operation (i,j) level. The derivation given above presents a basic framework for the solution of the stated problems. Our formulation is different from that of Debra, et al,(1989) in: 1). the lagrangian .ultipliers are given to be constant over the tiE interval to be considered; which

171

silllPl ifies the cClIIpItation; 2). ti_ ctc..in appl ied since the objective fWlCtion is piece wise convex or concave and the ....mer of concave or convex ti_ intervals is 0(11); 3). and the precedence structure which can cover wide variety of practical applications. By introducing step function U(*), it is valid to assume the Lagrangian rultipl iers to be time-invariable. The next section will briefly discuss the solution methods.

A TWO LAYER axJID IlIATI~

It has been mentioned earlier in the paper that a large jobshop can be divided into some small jobshops with the help of group technology. The decomposition of such a jobshop, however, is incomplete. There may exist coupl ing between some small jobshops, but the coupl ing is weak. Only a few jobs are processed in more than one jobshops. Here a two layer hierarchical framework is proposed for schedul ing individual jobshops and coordinating jobshops in a global sense. The upper layer is the coordinator performing the coordination and the lower layer consists of the jobshop schedulers. It is shown in Fig. 1.

The Solution Observing the forrulation given in the previous subsection, we find that both precedence and capacity constraints are relaxed. Here we would like to point out that operation (i,j) be processed

at

machine

j.

Given

HIERARCHICAL FRME\DUC FCIt SHOP

and

).,8,y, and It , the related items in the

objective function is calculated as the cost of scheduling operation (i,j) beginning at

b Jj

•

Si nee Lis pi ece wi se concave or convex wi th respect to

b Jj

,

we can find

b Jj

which

Fig. 1. The framework for jobshop coordination.

minimizes L. The time complexity of the problem Here JSi performs the schedul ing for jobshop i. The scheduling of jobshops has been discussed in previous sections. It has been stated earl ier that an additional item should be added to the objective functions which is minimized in the process of jobshop scheduling in order to keep the operations of the job which is processed in more than one jobshops closer in time, so that the coordination of jobshops becomes easier. It should be clear that an operation rust be scheduled in a permissible interval. Let oZJ=:{operations of job I processed in jobshop J and J is not the jobshop where the last operation of job I is processed}; Iu-' [Bu'Cu ] the time interval within which 0u is performed; Tu=: the least possible time in which 0u; ou.=:{operations of job I processed in jobshop JL in which job I completes}; ou.-' [B u .,-], the time interval. It is clear that

is O( ~NJ ).

Solvi~

the Dual Problell

To solve the dual problem 0, several methods for rultiplier generating the Lagrangian )., 8, y, and It have been deve l oped recent l y. See Held, et al,(1971), Fisher(1981), Luh, et al,(1989). The later one is adopted here.

3.7 where n is the iteration index, an is the step size and g().n) is the subgradient of L with respect to ).. ex changes according to 3.8

4.1

Then the following algorithm has been proposed for the coordination:

where L n is the value of L at nth iteration. It,y, and 8 can be obtained similarly.An al ternative approach may be appl ied. At each iteration, increase the rultipliers by a small stepsize l.I'Iti l the feasible solution has been obtained or almost obtained since there may be some violations of precedence and capacity constraints and the corrections of such violations only results in little change in the objective function.

Algorithm co: 1. Assign permissible intervals to sets of operat i ons of those jobs processed in more than one jobshops; 2. Perform jobshop scheduling for each jobshops;

J1-b. RN

3.

The feasible sched.Ile

4.

The feasible schedule can be obtained by some small adjustment in the schedule with little increase in the objective function.

5.

172

Calculate

(C):-D):)2U(C):-D):)

for

jobshop l(l=1,2,L); Do 1 in accordance with the result obtained in 3; Stop if desired result has been achieved; Otherwise repeat 2-5.

Comments have to be made for 3 in the algorithm. For jObshops with larger values obtained in 2, relatively looser time intervals should be assigned. For jObshops with smaller values, relatively tighter time intervals should be assigned. Of course, these are subjected to the constraints given in previous sections. At step 4 of the algorithm, the time intervals are adjusted as follows:

Dempster,J.K. Lenstra, and A.H.G. Rinnooy Kan, eds,Boston; D. Reidel Publishing Co •• Lenstra, J.K. and A.H.G. Rinnooy Kn, (1978), "cOft1)lexi ty of schedul ing under precedence constraints," Operation Research, vol. 26, no.1,pp22-35. Fisher,

4.2

M.L. (1981), "Lagrangian relaxation method for solving integer programming prob l ems , " Management Sc i. , vo l. 27, pp.1-1

where n is the i terat i on index, g n is the stepsize at iteration n, is the average cost calculated at step 3, K is the successive jObshop where some operations of job I are performed, and J;" and J!z are the costs of jObshop I and K, respectively. Once e;" are decided, B;" can be decided accordingly. Simulations on jObshop schedul ing and coordination have been partially performed. The resul ts show that the methodology proposed is sub-optimal. As result of limited scope of the paper, the simulation is not presented here.

Held, M. and R.M. Karp(1971), "The travel ing salesman problem and minimum spaming trees: Part 11," Mathematical Programming Study, no. 1, pp.6-25.

Hoitmt,

D., Luh, P., Pattipati, K.,(199O) "JObshop scheduling," Proc. of First Int. Conf. on Automatic Tech., Taipei, Taiwan, pp.565-574.

CCNCLUSHII

Lawler,

The paper presents a systemat i c approach for schedul ing jobs on jobshops. Some practical factors such as operat i on of di sassembl y and assembly are considered. Lagrangian relaxation based on integer programming formulation in continuous time domain is appl ied. A framework for coordination of jObshops is also included. There exists similarity between the method proposed and neural network method in solving combinatorial problems.

E.L.,J.K. Lensra, and A.H.G. Rinnooy in Kan(1982), "Recent developments deterministic sequencing and schedul ing, A survey on deterministic and stochastic Scheduling, M.A.H.

Luh, P.B., D. Hoitmt, K. Pattipati, an E. Max(1988) "Parallel Machine schedule using Lagrangian relaxation," Proc. Int. Conf. COft1). Integr. Manuf., pp.244-248, Troy, NY, May.

REFERENCES

Luh,

J:

Debra J. Hoitmt, P.B. Luh, E. Max and Krishna R.Pattipati(1990), "Schedul ing jobs with simple precedence constraints on parallel machines," IEEE Control Systems Magazine, February, 1990.

173

P.B., D. Hoitmt, K. Pattipati, and E. Max(1989), Schedule generation and reconfiguration parallel machines," Proc. of 1989 IEEE Int. Conf. Robot and Automation, pp.528-533.