Flow shop scheduling by a Lagrangian relaxation and network flow approach

WAS

=

11:20

FLOW S H O P SCHEDULING BY A LAGRANGLAN RELAXATION

AND

NETWORK FLOW APPROACH*

Pmceedlngr 01 the 29th Confareffie on D d s l o n md Conlrol Honolulu, Hawall Decanbsr 1WO

Shi-Chung Chang, Da-Yin Liao, Fu-Shiung Hsieh Dept. of Electrical Engineering

National Taiwan University Taipei, Taiwan, R. 0. C.

ABSTRACT

This paper develops, by using advanced optimization methods, a production scheduling algorithm for discrete-part, make-twrder type of flexible flow shops. The goal of

scheduling is to meet due dates and the problem is formulated

as

a large-scale integer programming problem. Our scheduling algorithm includes four parts : decomposition by Lagrangian relaxation into subproblems, the minimum cost linear network flow algorithm for solving subproblems, a subgradient algorithm for solving the dual problem, and a heuristic to obtain a near-optimal and feasible schedule. The algorithm is then compared with a heuristic rule that is considered effective by some local manufacturing firms. Preliminary results show that our scheduling algorithm not only is better in optimality but also provides more insights into scheduling complicated operations.

I. INTRODUCTION

Production scheduling is one of the most important issues in shop floor control of a manufacturing firm. Although there have been numerous researches on this topic in the literature [4], there are needs for new concepts and advanced techniques to schedule manufacturing operations as many new technologies have been developed and installed in manufacturing firms today [7]. In this paper, we consider a make-twrder flow shop which manufactures discrete parts of different types. Each type of parts have its own due date and desired quantity. Machine groups of limited processing capacities with infinite intermediate buffer sizes are used to manufacture these parts. Each machine group consists of homogeneous machines. There is one production process associated with each part type which consists of a sequence of operations requiring different machining facilities. However, different part types may have processin operations that need a same machine group. Given a set of orders, we want t o find a schedule that minimizes weighted production tardiness subject to constraints of (1) machine capacity and availability, 2) end product demand, and (3) precedence We first formulate the above scheduling problem as an integer programming problem in Section 11. In Section 111, we describe our solution algorithm which is developed based on a La rangian relaxation and network flow approach. Some prefiminary scheduling results of our algorithm are presented in Section IV. Its comparisons to the "Priority Factor" (PF) heuristic 31 are also given. Section V then relationship

o

!

manufacturing processes.

briefly concludes our stu

1

y.

II. PROBLEM FORMULATION

To convey our main idea and simplify the discussion, we assume that all types of parts require the same sequence

of

processing by machine groups in the flow shop, i.e., the

M

machine groups and the associated buffers can be organized

as

a line of production shown in Figure 2.1.

STOCK

b l m i b2 m 2 'M m M bM+l Figure 2.1

*

This work was supported in part by the National Science Council of the Republic

of

China under Grants NSC-78-0422-E00249 and NSC-79-0422-E002-05,

Chi-Tsai Yang

Information and Administration Dept. Fu Sheng Industrial Co.

San-Chung, Taiwan, R. 0. C.

Let us now define some notations for describing the above flow shop.

Notations I i Di di M m 'm 'im t b 'ibt 'imt 'it

:total number of part types; :part type index, i=l, e . . , I;

:demand of type i parts; :due date of type i parts; :total number of machine groups; :machine group index, m = 1,

-

*,

M; :capacity of machine group m;

:processing time of unit type i part on machine group m;

:time index;

:buffer index, where b=m for the buffer before machine roup m, and b=M+1 for the stock of finishe! parts;

:number

of

type i parts in buffer b at the beginning of time period t;

:number of type i parts loaded onto machine group m for processing at the beginning of period t;

:number of type i parts arriving at the stock

at t i m e period t, where Zit = u ~ ~ ( ~ - ~ ~ ~ ) . Assume that it takes the same amount of processing

time as long as the number of type i parts being processed is within the capacity of machine group m. Also assume that all the production demands IDi} are released at the beginning of scheduling. In such a shop, a batch of type i parts loaded onto machine (m-1) for processing at period t-Pi(m-l) go into buffer m after Pi(m-l) periods of processing

at

machine group (m-1). The stock buffer serves

as

a sink and accumulates finished parts. The flows of parts therefore must satisfy the following flow balance equations. Flow Balance Equations

Xilo = Di; (2.1 .a)

'il(t+l) 'ilt

-

'ilt; (2.1. b)

'im(t+l) = 'imt - 'imt + ui(m-1)(t-Pi(m-l))7 'i(M+l)(t+l)' 'i(M+l)t -I- 'iM(t-PiM);

Since a batch of uimt parts loaded onto machine group m needs P i m periods to complete the processing, the

[E::\

V m#l, M+1;

for i = 1, e . . , I.

I t

quantity

C

C

uimr is the total number of parts i = l r=t-P. +1

im

which is being processed on machine m during time period t. This quantity must not exceed the processicg capacity of machine group m, i.e.,

Machine Capacity Constraints

I t

c

c

Uimr 5

cm,

V t , m. (2.2) i=l T=t-P. $1

uimt and Xibt are nonnegative integers, V i, m, b, t. im

Obviously, we should have the constraints that

(2.3)

Our objective of production scheduling is to meet due dates if possible. A pendty cost is incurred by an overdue production, which increases with production tardiness. We then formulate the production scheduling problem as follows:

structure of flow balance equations (2.1). The network, which is a graph consisting of nodes and arcs, is constructed as follows. Let

a

node nmt represent buffer m at time t , m=l, . . a , M, M+l and t = l , . . . , T. Since it takes Pim

periods of processin time for

a

part to go from buffer m to buffer m + l , we form an arc (nmt,n(m+l)(t+p. )) to

U i = l t = l im

subject to constraints (2.1-3), represent a flow path between the two buffers. The remaining parts in buffer m after loading uimt for processing at t is carried by buffer m over to time period t + l , i.e., they flow through arc (nmt,nm(t+l)). In addition, we let nodes S and T. be the source and sink nodes of the network respectively. The source node

S

is connected by an arc

to

node nll with a constant flow equal to the demand Di. The The production scheduling problem (P) formulated terminal node T serves. as a sink of flows. The flow on arc above is an integer pro ramming problem of NP-hard (nmt,n(m+l)(t+pim)) 1s "imt 2nd Zit On arc ( n ( ~ + l ) t , T ) . computational complexity $61. Here, we develop for it a

computationally feasible, near-optimal solution algorithm. It It can be easily checked out that the flow conservation at consists of four parts : each node represents one of the flow balance equations. We

(1) the dual problem formulation and decomposition into can find the associated arc cost for

arc

production scheduling subproblems according to part (nmt,n(m+l)(t+Pim)) by aPDroPriatelY rearranging terms in types by applying Lagrangian relaxation to the machine

capacity constraint;

V t and m. The cost of (2) application of

a

minimum cost network flow algorithm subproblem (P-i) as 'mr'

(3) application of a simple nondifferentiable optimization arc (n(M+1It,T) is '$'it. There is no cost for flows On arc scheme to the dual problem, and (nmt,nm(t+l)). From the above discussion, subproblem (4) development of a heuristic algorithm that finds a

near-optimal, feasible solution based on the solution of (P-i) is now clearly formulated as a minimum cost linear the relaxed problem and the network structure. network flow (MCLNF) problem, which has an integer optimal solution. We adopt the RELAX code of Bertsekas and Tseng [I] to solve it.

111.3 Solving the Dual Problem

I T m i n E $+,Zit (P) 0, t

5

di,

[

Ai(t-di), t

>

di, where Qit =

and Ai is the overdue penalty cost coefficient.

t +Pim-l to solve each production scheduling subproblem; r = t

111.1 L w r n

In problem (P), we observe that coupling among flows of different part types is through the machine capacity constraints. Applying Lagran e Relaxation to constraint (2.21, we form the Lagrangian function as

*

Let (ui} be the optimal solution to subproblems for the set of Lagrange multipliers

Ak

at kth iteration. Since

c

c * . z . + c

c

'imr-'m) @(A) is nondifferentiable with respect to A, we adopt

a

subgradient scheme of [2] to solve (D). The scheme is

I T M t

T T M I t

i = l t = l it 1' t = l m Z l X m t ( i 2 1 r=t-p. $1 im

(3.l.a) summarized as follows :

The gradient of @(A) with respect to Xmt is

I t 'imr)

*

a

=

c c

(*itzit

+

c

_*

'imr

-

'm, i=l t = l m21Xmtr=t-P. im +1 gmt(u ) -JJ-@(A) =

E

c

i=l r=t-P. +I and the Lagrange multipliers are updated by

Im

T M m t

-

where Xmt is the associated Lagrange multiplier. A (3.1.b) m = l 'TI' m Since the function above is additively separable in ui for a

given set of Lagrange multipliers A, we define

Xi;'=

k

T M t if Xmt=O and gmt(u*)<O,

i s the step size at the kth and the dual function

@(A)s m i n Li(ui,A)

-

A C (3.3) iteration, 0

<

y <: 2, and m(A ) is an estimate of the The iteration step terminates if cr is smaller than

a

threshold. Polyak proved that this method has

a

linear

(D)

y:t

@(A), convergence rate [8].

111.4 Construction of a Good Feasible Schedule and the scheduling subproblem for type i parts is

The schedule derived from the dual problem is (p-i) min [L.(u- A)s ($itZit+m~lXmt

generally infeasible for the primal problem (P because (P) subject to (2.1) and (2.3) for type i only, which is not a convex programming problem [5] an the machine is independent of the scheduling for other types. capacity constraints (2.2) may not be satisfied after the relaxation. We develop a heuristic algorithm to adjust the 1n.2 A Network Model schedule to a feasible one. We check capacity constraints in Each scheduling subproblem (P-i) can be formulated as an ascending sequence of indices m and t. For each violated

a

minimum cost network flow problem due to the network capacity constraint, we

k

r[W*)-r(A"l

L.(u.,A)z

c

( 9

z.

+

c

Amt U. (3.2) It 1' m = l 7zt-p. +1 1 m ~ ) where cr = im 1 1 t = l I T M

t = l m = l mt optimal dual cost i

i=l U

k

subject to (2.1) and (2.3). The dual problem is then

T M t Uimr)]

d

',

t = l r=t-PimS1 'i 123

residual rocessin time

124

Case Optimal Cost Dual Cost Our Algorithm Heuristic LB = t o t a requireP processfng time 1

by using the CurrAnt schedje; the lower the P F value, the higher the priority, and

reroute excessive production flows starting from the .

2 (2)

60000.0 59998.7 60000.0 60000.0 160000.0 143026.0 160000.0 220000.0 t pe of the largest priority. 3 200000.0 147205.0 240000.0 440000.0

subnetwork and then reroute the flow by solving another MCLNF problem.

IV.

NUMERICAL RESULTS

In this section, we present very preliminary testing

, Case Our Algorithm Heuristic

1 1.05 0.16

2 1.29 0.19