hierarchical stochastic production planning for flexible automation workshops

Hierarchical stochastic production planning for ¯exibleautomation workshops

Hong-Sen Yan

Research Institute of Automation, Southeast University, Nanjing, Jiangsu 210096, People's Republic of China

Accepted 24 August 2000

Abstract

This paper presents the hierarchical stochastic production planning (HSPP) problem for ¯exible automation

workshops (FAWs) in agile manufacturing environments, which is a multiple-period multiple-product problem

with random material supply, demands, capacities, processing times, rework and waste products. To solve the

HSPP problem, a mathematical model is built up ®rst. Then, an algorithm for HSPP is deduced in detail by using a

stochastic interaction/prediction approach. The corresponding software package named as stochastic interaction/

prediction algorithm (SIPA) has been developed and is presented in this paper, through which examples of HSPP

have been studied, and which show that the algorithm can optimally decompose medium-term random product

demand plans of an FAW into short-term stochastic production plans to be executed by FMSs in the FAW. Finally,

the application of the algorithm is presented in detail through one of those examples. q 2000 Elsevier Science Ltd.

All rights reserved.

Keywords: Hierarchical stochastic production planning; Flexible automation workshop; Stochastic interaction/prediction

algorithm

1. Introduction

A manufacturing department of a manufacturer in China is usually composed of several workshops orsub-factories, each of which is usually composed of some workshop-sections. Thus, the manufacturingdepartment of the computer integrated manufacturing system (CIMS) in such a manufacturer is alsocomposed of several shops, each of which consists of some manufacturing cells or ¯exible manufactur-ing systems (FMSs). Secondly, in theory, the workpiece-transport time and charges can be decreased byrecon®guration of the manufacturing cells (Chung & Fang, 1993; Rheault, Drolet & Abdulnour, 1995).

Computers & Industrial Engineering 38 (2000) 435±455

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PII: S0360-8352(00)00056-5

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E-mail address: [email protected] (H.-S. Yan).

However, because main machines in cells (or FMSs) are NC machines and machining centers thatcannot be arbitrarily moved, the physical recon®guration of cells cannot be realized and the logicalrecon®guration of cells cannot obviously decrease the workpiece-transport time and charges. Thirdly,although the commercial software of manufacturing resource planning (MRP II) can directly assignmanufacturing orders to manufacturing cells, these orders are usually not optimal (Zou & Su, 1994).This is because the manufacturing orders are made according to a material requirement plan that,without workcentre capacities being taken into consideration, is generated only from a master produc-tion schedule, bill and lead time of a material. After the manufacturing orders are obtained, a capacityrequirement plan is developed by simulation. If the capacity requirement plan does not match work-centre capacities, these manufacturing orders are revised manually after the amount of material to beprocured and/or the amount of work to be subcontracted have been changed manually, or other manu-facturing orders are remade after the master production schedule has been modi®ed manually until theymatch each other. Thus, those manufacturing orders are not optimal. It is better for MRP II to assign ashop the manufacturing orders that are to be optimally decomposed into short-term plans to be executedby manufacturing cells or FMSs in the shop, about which little has been written in the literature onproduction planning (PP). Fourthly, the solutions of PP problems in workshops will provide a theoreticalbasis for implementing manufacturing execution systems (Baliga, 1997) similar to shop ¯oor controllers.Fifthly, with the increasingly keen competition for markets, there will be no manufacturer that has all theresources to win a victory. Thus, several manufacturers with complementary resources will temporarilyform themselves into an agile virtual enterprise (Goranson, 1995) to take advantage of a transient marketopportunity and to win a victory in the competition. In such an agile manufacturing environment, a shopis empowered to organize production autonomously according to manufacturing orders (product demandplans) not only from the manufacturer (which it belongs to) but also from the agile virtual enterprise(which it belongs to), which generates more uncertainty in the product demand than the traditionalproduction. Besides, the equipment capacity in the shop is uncertain because of unplanned maintenanceand the material supply for the shop is also uncertain because of the supplier's capacity and the materialquality. Therefore, the shop can be taken as a stochastic manufacturing system to some extent. On theother hand, the production planning in a manufacturing setting is essential to achieve ef®cient resourceallocation over time while meeting demands for ®nished products. It is thus clear that the study on thestochastic PP problem for a ¯exible automation workshop (FAW) consisting of FMSs (or cells) is ofgreat signi®cance.

Since the scope of PP problems generally prohibits a monolithic modeling approach, a hierarchicalproduction planning (HPP) approach has been widely advocated in the PP literature (Davis & Thomp-son, 1993). To model PP problems, the existing hierarchical approaches usually employ the followingconcepts: (1) product disaggregation (Davis & Thompson, 1993; Bitran, Haaas & Hax, 1981; Graves,1982; Simpson & Erenguc, 1998; Simpson, 1999; Kira, Kusy & Rakita, 1997), (2) temporal decom-position (Malakooti, 1989; Nguyen & Dupont, 1993; Carravilla & Sousa, 1995; Qiu & Burch 1997;Bassok & Akella, 1991), (3) process decomposition (Villa, 1989; Yan, 1997; Yan & Jiang, 1998), and(4) event-frequency decomposition (Gershwin, 1988). However, these articles focus on deterministicHPP problems except (Davis & Thompson, 1993; Gfrerer & Zapfel, 1995; Kira et al., 1997) that are onthe HSPP problems.

The existent articles on the PP problems with uncertainties mostly focus on uncertainties of thedemand, capacity and material supply in the single-period or in®nite-horizon setting (Bassok & Akella,1991; Ciarallo, Akella & Morton, 1994; Ishikura, 1994; Kasilingam, 1995; Metters, 1997; Hwang &

H.-S. Yan / Computers & Industrial Engineering 38 (2000) 435±455436

Medini, 1998). However, Davis and Thompson propose an integrated stochastic decision-makingapproach of integrating the techniques of mathematical programming and Monte Carlo simula-tion, which is employed within each implemented generic controller to rigorously address theuncertainties that are inherent to multi-period PP problems (Davis & Thompson, 1993). Gfrererand Zapfel address hierarchical production planning for a multi-period model consisting of anaggregate planning level and a detailed planning level in the case of uncertain demand (Gfrerer& Zapfel, 1995). Kira et al. propose a stochastic linear programming approach to solve hier-archical production planning problems under uncertain demand (Kira et al., 1997). Bitran andLeong examine deterministic approximations to multi-period multi-item production planningproblems in environments with stochastic process yields and substitutable demands (Bitran &Leong, 1992). Schmidt presents a Markov decision process model that combines features ofengineering design models and aggregate production planning models to obtain a hybrid modelthat links biological and engineering parameters to optimize operations performance in biophar-maceutical production processes (Schmidt, 1996). Stecke and Raman use an open queueingnetwork model of a ¯exible manufacturing system to determine the optimal con®gurations andmachine workload assignments for the no grouping and total grouping cases (Stecke & Raman,1994). Bonissone et al. propose a new approach to planning in the ®nancial domain of mergersand acquisitions and in dynamic and uncertain environments (Bonissone, Dutta, & Wood, 1994).The planning is viewed as a process in which an agent's long-term goals are transformed intoshort-term tasks and objectives, given the agent's strategy and the context of planning (Bonissoneet al., 1994).

In contrast, the HSPP problem in this paper involves not only the three kinds of uncertainties(demands, capacities and material supply) in the existent PP literature, but also the uncertainties ofprocessing times, rework and waste products. Besides, the system under consideration is also verycomplicated (an FAW consists of some FMSs, each of which consists of several machines) and theproblem is also a multi-period multi-product one. It is thus clear that the problem gives a challenge.Because most stochastic programming problems in the literature are solved for the single-period orin®nite-horizon (Bitran & Leong, 1992; Higle, Lowe & Odio, 1994) and the stochastic linear program-ming approach in Kira et al. (1997) is suitable for solving the hierarchical PP problem only underuncertain demand and because the solution of multistage stochastic optimization problems requiresthe tree-like decision-making structure (Mulvey & Ruszczynski, 1995) that is not suitable for thesolution of the HSPP problem, it is necessary to ®nd a new approach to solve the problem. Thus, thepaper proposes a new stochastic interaction/prediction approach that can effectively solve the HSPPproblem.

H.-S. Yan / Computers & Industrial Engineering 38 (2000) 435±455 437

Fig. 1. Topologic structure of FAW's LAN (Yan and Jiang, 1998).

2. Stochastic production modeling of FAWS

An FAW under consideration in this paper is generalized from the ¯exible automation workshop ofthe CIMS in the Chengdu Aircraft Industry Company that consists of the two FMSs and two ¯exibledirect numerical control systems. To solve the general problem of HSPP, we suppose that an FAWcomprises a shop computer, a material handling system (MHS), a tool handling system (THS), a shoptesting and monitoring system (STMS), and FMS1, FMS2,¼,FMSM (M is the number of FMSs in theFAW). They are connected by a local area network (LAN), as shown in Fig. 1. The shop computermanages the production of the FAW. Its main functions are: (1) receiving medium-term random manu-facturing orders from the manufacturer or the agile virtual enterprise and decomposing them into short-term stochastic production plans (to be sent to the FMSs in the FAW); (2) from short-term stochasticproduction plans, developing stochastic tool requirement plans (to be sent to the THS) and stochasticmaterial requirement plans (to be sent to the MHS); and (3) reporting, in time, to the manufacturer or theagile virtual enterprise on error information, production situations and so on. Each of the FMSs in theFAW can produce ®nished products or semi-®nished products (which need sending to the successiveFMSs for further processing). The main functions of the FMSs are: (1) executing short-term stochasticproduction plans coming from the shop computer, and (2) reporting to the shop computer on the errorinformation and production situations in time (Yan, Wang, Cui & Zhang, 1997; Yan, Wang, Zhang &Cui, 1998).

An FAW in agile manufacturing is empowered to organize production autonomously according to themedium-term random manufacturing orders from the manufacturer or the agile virtual enterprise ordirectly from customers, that is, to make a medium-term random product demand plan from the manu-facturing orders and decompose it into short-term stochastic production plans to be executed by FMSs inthe FAW. Thus, the HSPP problem in this paper is a problem of how to optimally decompose themedium-term random product demand plans of an FAW into short-term stochastic production plans tobe executed by FMSs in the FAW. In fact, this problem also originates from the operating manner of theworkshop controller of the CIMS in Chengdu Aircraft Industry Company. In operation, the workshopcontroller decomposes a one-month production plan from MRP II into ten-day plans to be executed bycell controllers that schedule and control the production of the two FMSs and the two ¯exible directnumerical control systems.

The basic principle of production in a workshop is supposed to minimize production cost under thecondition of meeting customers' requirements as far as possible. Thus, the objectives of hierarchicalstochastic production planning in the FAW are: (1) to satisfy the requirement of work-in-process inFMSs; (2) to maximize the machine utilization and the balance of the loads on the machines; and (3) tosatisfy the stochastic demands for products. Since all of the objectives may not be attained simulta-neously, it is necessary to reach a compromise among them, i.e. to arrive at their combinatorial opti-mization. Therefore, the objective function of the HSPP problem in the FAW can be described asfollows:

J � E

(XMi�1

"1

2ixi�N 1 1�2 ai�N 1 1�i2

Qi�N11� 1XNk�1

1

2�ixi�k�2 ai�k�i2

Qi�k�

1iTi�k�ui�k�2 bi�k�i2Ri�k� 1 iCi�k�ui�k�2 di�k�i2

Ki�k��#) �1�


where

N: number of planning periods in a planning horizon (H). H is generally a month or a week. Aplanning period is shorter than H. A period is usually a week, a day or a shift.xi(k): work-in-process of FMSi at the beginning of period k for k � 2;¼;N 1 1: It is an ni-dimensionalstochastic column vector and ni is number of types of workpieces (which are manufactured by FMSi

over the planning horizon). Its lth component is the amount of work-in-process of the lth type ofworkpiece manufactured by FMSi for l � 1; 2;¼; ni:

ui(k): workpieces (®nished or semi-®nished products) of a planned production by FMSi in period k fork � 1; 2;¼;N: It is an ni-dimensional stochastic column vector whose lth component is the number ofworkpieces of the lth type of planned production by FMSi for l � 1; 2;¼; ni:

a i(k): given work-in-process level of FMSi in period k for k � 1; 2;¼;N 1 1: It is an ni-dimensionalcolumn vector whose lth component is the given work-in-process level of the lth type of workpiecemanufactured by FMSi for l � 1; 2;¼; ni: If a i(k) is a zero vector, zero work-in-process is required forFMSi in period k.b i(k): processing time available to machines (NC machines, machining centers, etc.) of FMSi inperiod k. Not including the machine repairing time and so forth, it is shorter than a planning period.It is an mi-dimensional stochastic column vector and mi is number of machines in FMSi. Its sthcomponent is the processing time available to the sth machine of FMSi for s � 1; 2;¼;mi:

di(k): demands for products out of FMSi in period k. It is an n-dimensional stochastic column vectorand n is number of types of ®nished products (which are processed by the FAW over the planninghorizon). Its jth component is the demand for the jth type of product out of FMSi for j � 1; 2;¼; n:

Ti(k): an mi £ ni dimensional stochastic variable matrix independent of ui(k) and b i(k). It representsthe processing time (which is taken by the machines in FMSi to process ni types of workpieces inperiod k). The element in the sth row and the lth column of Ti(k) is the processing time it takes the sthmachine in FMSi to process the sth type of workpiece for s � 1; 2;¼;mi; l � 1; 2;¼; ni: Refer to Yanet al. (1997) and Yan and Jiang (1998) about its computation.Ci(k): an n £ ni dimensional Boolean matrix. It is an output matrix of FMSi in period k. If the elementin the jth row and the lth column of Ci(k) is 1, the lth type of workpiece of planned production by FMSi

is the jth type of ®nished product for j � 1; 2;¼; n; l � 1; 2;¼; ni: And if the element is zero, the lthtype of workpiece is a kind of semi-®nished product.Qi(k): an ni £ ni dimensional symmetrical (semi-)positive-de®nite weight matrix of FMSi in period kfor k � 1; 2;¼;N 1 1: If it is taken as a diagonal matrix, the square root of the lth element on itsleading diagonal is the additional cost per unit, with the work-in-process of the lth type of workpiecemanufactured by FMSi higher or lower than the corresponding given work-in-process level for l �1; 2;¼; ni:

Ri(k): an mi £ mi dimensional symmetrical positive-de®nite weight matrix. If it is taken as a diagonalmatrix, the square root of the sth element on its leading diagonal is the per-unit cost, with the workload(the required processing time) of the sth machine of FMSi more than or less than the correspondingavailable processing time, that is, a per-unit over-time cost or a per-unit resource idle cost, for s �1; 2;¼;mi:

Ki(k): an n £ n dimensional symmetrical positive-de®nite weight matrix. If it is taken as a diagonalmatrix, the square root of the jth element on its leading diagonal is the per-unit cost, with the numberof ®nished products of the jth type of planned production by FMSi greater or smaller than the


corresponding demand, i.e. a per-unit holding cost for the jth type of ®nished product or a per-unitbacklogging cost, for j � 1; 2;¼; n:iai2

B : aTBa where a is a vector and B a matrix.

From the above-mentioned objective function, we can see that the result of optimization will makereal work-in-process approximate to the given work-in-process level, machine workload approximate toavailable processing time (capacity) and ®nished products approximate to demands as far as possible. Itcan also be seen that the per-unit overtime cost is the same as the per-unit resource idle cost and that theper-unit holding cost is the same as the per-unit backlogging cost.

The FAW's stochastic dynamic equation can be described by:

xi�k 1 1� � xi�k�2 ui�k�1 zi�k�1 Gi�k�r�k�1 vi�k�initial condition xi�1� � xi1

i � 1; 2;¼;M; k � 1; 2;¼;N

�2�

where

xi1: initial work-in-process of FMSi. It is an ni-dimensional column vector.zi(k): number of semi-®nished products that are sent from the other FMSs to FMSi in period k. It is alsoknown as interaction input and it is an ni-dimensional stochastic column vector whose lth componentis the number of semi-®nished products of the lth type sent from the other FMSs to FMSi for l �1; 2;¼; ni:

r(k): input of blanks into the FAW in period k. It is an n-dimensional stochastic column vector whosejth component is the number of blanks of the jth type of product for j � 1; 2;¼; n:

vi(k): difference between the number of the quali®ed products of reprocessed products and the sum ofthe waste products and the products whose rework begins in period k. It is an ni-dimensional stochasticcolumn vector whose lth component corresponds to the lth type of product for l � 1; 2;¼; ni:

Gi(k): an ni £ n dimensional Boolean matrix. It is an input matrix of FMSi in period k. If the element atthe lth row and the jth column of Gi(k) is 1 (or 0), blanks of the jth type are (or not) supplied forplanned production of the lth type of workpiece by FMSi for l � 1; 2;¼; ni; j � 1; 2;¼; n:

The output equations of products from the FAW can be described by

y�k� �XMi�1

yi�k� �3�

yi�k� � Ci�k�ui�k� �4�where

y(k): ®nished products of the planned production by the FAW in period k. It is an n-dimensionalstochastic column vector whose jth component is the yield of products of the jth type for j � 1; 2;¼; n:yi(k): ®nished products of the planned production by FMSi in period k. It is a part of y(k), and is also ann-dimensional stochastic column vector whose jth component is the yield of ®nished products of thejth type of planned production by FMSi for j � 1; 2;¼; n:


Suppose that time for transporting a workpiece between FMSs can be omitted. That is, as soonas a semi-®nished product has been processed by an FMS, it will be sent to another FMS. And ithardly takes time to transport the semi-®nished product. Then the interaction equation can bedenoted by

zi�k� �XMj�1

Lij�k�uj�k� �5�

where

Lij(k): an ni £ nj dimensional Boolean interaction matrix. It re¯ects the interaction betweenoutput of the semi-®nished products from FMSj and input of semi-®nished products intoFMSi in period k. If the element in the lth row and the bth column of Lij(k) is 1 (or 0),the semi-®nished products of the bth type from FMSj are (or not) supplied for plannedproduction of the lth type of workpiece by FMSi for l � 1; 2;¼; ni; b � 1; 2;¼; nj:

Now, we can formulate the problem of optimal hierarchical stochastic production planning inthe FAW as follows:

minu

J � E

(XMi�1

"1

2ixi�N 1 1�2 ai�N 1 1�i2

Qi�N11� 1XNk�1

1


Qi�k�

1iTi�k�ui�k�2 bi�k�i2Ri�k� 1 iCi�k�ui�k�2 di�k�i2

Ki�k��#)

s:t: �2�; �5�

�6�

3. Stochastic interaction/prediction algorithm

Solving the HSPP problem (6) by the stochastic interaction/prediction approach, we can obtainthe algorithm to optimally decompose the random product demand plans into the short-termstochastic production plans. The basic ideas of the stochastic interaction/prediction approachare: (1) by predicting interactions among FMSs, the FAW's optimal HSPP problem (6) is dividedinto M FMSs' optimal SPP (stochastic production planning) sub-problems; (2) then solving eachof the sub-problems; and (3) improving the predictions and continuing the ®rst two steps until thesolution is obtained. The interaction/prediction approach has been applied to solve the problem ofhierarchical control of large-scale deterministic discrete-time systems (Li & Shao, 1987). In thefollowing, we shall deduce the stochastic interaction/prediction algorithm of HSPP in the FAW.For this end, by the method similar to that in Li and Shao (1987), we take the following


Lagrangian function into consideration:

L�x; u; z;l; p� �XMi�1

Li�xi; ui; zi; li; pi��XMi�1

E

(1

2ixi�N 1 1�2 ai�N 1 1�i2

Qi�N11�

1XNk�1

"1


Qi�k� 1 iTi�k�ui�k�2 bi�k�i2Ri�k� 1 iCi�k�ui�k�2 di�k�i2

Ki�k��

1lTi �k�zi�k�2

XMj�1

lTj �k�Lji�k�ui�k�1 pT

i �k 1 1��xi�k�2 ui�k�

1zi�k�1 Gi�k�r�k�1 vi�k�2 xi�k 1 1��#)

(7)

For any given z � zp and l � lp; Eq. (7) can be changed into M FMSs' optimal SPP sub-problems.

And by the method similar to that in Li and Shao (1987), we take the following Hamiltonian of FMSi intoaccount:

Hi�k� � 12

ixi�k�2 ai�k�i2Qi�k� 1 1

2iTi�k�ui�k�2 bi�k�i2

Ri�k� 1 12

iCi�k�ui�k�2 di�k�i2Ki�k�

1lpTi �k�zp

i �k�2XMj�1

lpTj �k�Lji�k�ui�k�1 pT

i �k 1 1��xi�k�2 ui�k�1 zpi �k�1 Gi�k�r�k�1 vi�k��

�8�Then we have

Li�xi; ui; zpi ; l

p; pi� � E

(1

2ixi�N 1 1�2 ai�N 1 1�i2

Qi�N11� 2 pTi �N 1 1�xi�N 1 1�

1XNk�1

�Hi�k�2 pTi �k�xi�k��1 pT

i �1�xi�1�)

�9�

Its variation of the ®rst order becomes

dLi � E�Qi�N 1 1��xi�N 1 1�2 ai�N 1 1��2 pi�N 1 1��Tdxi�N 1 1�

1XNk�1

E2Hi�k�2xi�k� 2 pi�k�

� �T

dxi�k�1XNk�1

E2Hi�k�2ui�k�

� �T

dui�k�

1XNk�1

E2Hi�k�

2pi�k 1 1� 2 xi�k 1 1�� T

dpi�k 1 1�1E�pi�1��Tdxi�1� � 0 �10�

Because of the arbitrariness of dxi(N 1 1), dxi(k), dui(k) and dpi(k 1 1), because of dxi�1� � 0 and


because of the independence of Ti(k) from ui(k) and b i(k), we have that

E�xi�k 1 1�� E2Hi�k�

2pi�k 1 1��

� E�xi�k��2 E�ui�k��1 zpi �k�1 Gi�k�E�r�k��1 E�vi�k�� 11�

E�pi�k�� E2Hi�k�2xi�k�

� �� Qi�k��E�xi�k��2 ai�k��1 E�pi�k 1 1�� 12�

E�pi�N 1 1�� Qi�N 1 1� �E�xi�N 1 1��2 ai�N 1 1�� 13�

E

2Hi�k�2ui�k�

!� E

(�TT

i �k�Ri�k�Ti�k�1 CTi �k�Ki�k�Ci�k��ui�k�2 TT

i �k�Ri�k�bi�k�

2CTi �k�Ki�k�di�k�2

XMj�1

LTji�k�lp

j �k�2 pi�k 1 1�)

� �E�TTi �k�Ri�k�Ti�k��1 CT

i �k�Ki�k�Ci�k��E�ui�k��2 E�TTi �k��Ri�k�E�bi�k��

2CTi �k�Ki�k�E�di�k��2

XMj�1

LTji�k�lp

j �k�2 E�pi�k 1 1�� 0 �14�

Setting �xi�k� � E�xi�k��; �ui�k� � E�ui�k��; �r�k� � E�r�k��; �vi�k� � E�vi�k��; �b i�k� � E�bi�k��; �di�k� �E�di�k��; �pi�k� � E�pi�k�� and �Ti�k� � E�Ti�k��; we obtain

�xi�k 1 1� � �xi�k�2 �ui�k�1 zpi �k�1 Gi�k��r�k�1 �vi�k�

s:t: �xi�1� � xi1

�15�

�pi�k� � Qi�k�� xi�k�2 ai�k��1 �pi�k 1 1� �16�

�pi�N 1 1� � Qi�N 1 1�� xi�N 1 1�2 ai�N 1 1�� 17�

E2Hi�k�2ui�k�

� �� E�TT

i �k�Ri�k�Ti�k��1 CTi �k�Ki�k�Ci�k�� ui�k�2 �TT

i �k�Ri�k� �b i�k�2 CTi �k�Ki�k� �di�k�

2XMj�1

LTji�k�lp

j �k�2 �pi�k 1 1� � 0 (18)

Solve for

�ui�k� � �E�TTi �k�Ri�k�Ti�k��1 CT

i �k�Ki�k�Ci�k��21� �TTi �k�Ri�k� �b i�k�1 CT

i �k�Ki�k� �di�k�

1XMj�1

LTji�k�lp

j �k�1 �pi�k 1 1�� 19�


Suppose that the relationship between �pi�k� and �xi�k� can be described by a linear vector equation

�pi�k� � Si�k� �xi�k�1 gi�k� �20�From Eqs. (20), (15) and (19), we can get the representation of �pi�k 1 1�: By comparing Eq. (16) (in

which the representation of �pi�k 1 1� has been substituted) with Eq. (20), we obtain the following Riccatiand adjoint equations:

Si�k� � Qi�k�1 B21i �k�Si�k 1 1�

s:t: Si�N 1 1� � Qi�N 1 1��21�

gi�k� � B21i �k�gi�k 1 1�1 Fi�k�2 Qi�k�ai�k�

s:t: gi�N 1 1� � 2Qi�N 1 1�ai�N 1 1��22�

where

Bi�k� � I 1 Si�k 1 1�A21i �k� �23�

Ai�k� � E�TTi �k�Ri�k�Ti�k��1 CT

i �k�Ki�k�Ci�k� �24�

Fi�k� � B21i �k�Si�k 1 1��zp

i �k�1 Gi�k��r�k�1 �vi�k�2 A21i �k�Ei�k�� 25�

Ei�k� � �TTi �k�Ri�k� �b i�k�1 CT

i �k�Ki�k� �di�k�1XMj�1

LTji�k�lp

j �k� �26�

Substituting �pi�k 1 1� representation into Eq. (19) and Eq. (27) into Eq. (15), we have

�ui�k� � A21i �k�B21

i �k�Si�k 1 1� �xi�k�1 A21i �k��Ei�k�1 Fi�k�1 B21

i �k�gi�k 1 1�� Mi�k� �xi�k�1 Ni�k� �27�

�xi�k 1 1� � �I 2 Mi�k�� xi�k�1 zpi �k�1 Gi�k��r�k�1 �vi�k�2 Ni�k� � Vi�k� �xi�k�1 Wi�k�

s:t: �xi�1� � xi1

�28�

where

Mi�k� � A21i �k�B21

i �k�Si�k 1 1� �29�

Ni�k� � A21i �k��Ei�k�1 Fi�k�1 B21

i �k�gi�k 1 1�� 30�

Wi�k� � zpi �k�1 Gi�k��r�k�1 �vi�k�2 Ni�k� �31�

Vi�k� � I 2 Mi�k� �32�The task of the coordination level is considered in the following (i.e. zp

i and lpi are improved). By the


necessary condition of optimization, from Eq. (7) we obtain

2L

2zi

li � lpi

zi � zpi

�� E�lpi �k�1 pi�k 1 1�� lp

i �k�1 �pi�k 1 1� � 0 �33�

2L

2li

li � lpi

zi � zpi

�� E zpi �k�2

XMj�1

Lij�k�uj�k�0@ 1A � zp

i �k�2XMj�1

Lij�k� �uj�k� � 0 �34�

Solve for

lpi �k� � 2 �pi�k 1 1� � 2Si�k 1 1� �xi�k 1 1�2 gi�k 1 1� �35�

zpi �k� �

XMj�1

Lij�k� �uj�k� �36�

Therefore, the coordination task at the second level is

lpi �k�

zpi �k�

" #l11

�2Si�k 1 1� �xi�k 1 1�2 gi�k 1 1�XM

j�1

Lij�k� �uj�k�

26643775

l

�37�

where l is the number of iterations.From Eqs. (21) and (23), we know that the problem of optimal hierarchical stochastic production

planning (6) can be solved if Bi(k) and Ai(k) are of full rank. Thus, the stochastic interaction/predictionalgorithm is described as follows.

Algorithm 1. Stochastic interaction/prediction algorithm of optimal hierarchical stochastic produc-tion planning in the FAW:

Step 1. Input or compute expectations of stochastic column vectors di(k), r(k), vi(k) and b i(k) and ofstochastic variable matrixes Ti(k) and TT

i �k� Ri(k) Ti(k) for i � 1; 2;¼;M; k � 1; 2;¼;N:

Step 2. At the coordination level, let l � 1 and make conjectures upon initial values zi�k� � zpi �k� and

li�k� � lpi �k� for i � 1; 2;¼;M; k � 1; 2;¼;N: Then send them to the ®rst level.

Step 3. Using Eqs. (23) and (24), solve M Riccati Eq. (21) and memorize their solutions Si(k) at the®rst level.

Step 4. Using Eqs. (23)±(26), solve M adjoint Eq. (22) and memorize their solutions gi(k) at the ®rstlevel.

Step 5. Using Si(k), gi(k) and Eqs. (27)±(32), calculate �ui�k� and �xi�k 1 1� for i � 1; 2;¼;M; k �1; 2;¼;N:

Step 6. Examine norms to the left-hand side of Eqs. (33) and (34) to see whether they are less than thelittle positive numeric d . If they are, stop iteration and go to step 7. Otherwise using Eq. (37), updatelp

i �k� and zpi �k�: Let l � l 1 1: Then go to step 4.

Step 7. Output �upi �k� that is the expectation of the short-term stochastic production plan to be executed

by FMSi in period k.


From Eq. (23), we know that Ai(k) is of full rank if Algorithm 1 is feasible. Refer to Yan (2000) aboutthe two necessary conditions for Ai(k) to be of full rank and the computation of E�TT

i �k�Ri�k�Ti�k��:Algorithm 1 only ®nds out the expectation number of workpieces of planned production by each FMS

in each period. To develop the real production plan to be executed by each FMS in each period, we mustadd such parameters as part no, part priority, available machine, NC program, processing time and toolto the number (Yan et al., 1997, 1998).

In addition, for the deterministic case, all of the above-mentioned stochastic variables will becomedeterministic variables and their expectations will become themselves so that the algorithm is stillsuitable in this case. Thus, a deterministic HPP problem is a special case of the HSPP problems inthis paper.

4. Development of SIPA

We have employed Algorithm 1 to develop the software package named as SIPA which is written inTURBO C2.0. Its main functions are: (1) receiving medium-term random manufacturing orders from themanufacturer or the agile virtual enterprise or directly from users and decomposing them into short-termstochastic production plans to be sent to, and to be executed by FMSs in the FAW; and (2) reporting, intime, to the manufacturer or the agile virtual enterprise or the users on error information and productionsituations from the FMSs. Its software structure is shown in Fig. 2. The arrows indicate the exchange ofinformation between the two modules. The user interface allows comfortable work with SIPA,especially for users who are not familiar with the underlying software. The order processing componentmakes a medium-term random product demand plan from the manufacturing orders. The data ®lesare used to store �di�k�; �r�k�; �vi�k�; �b i�k�; �Ti�k�; E�TT

i �k�Ri�k�Ti�k��; xi1, zpi �k�; lp

i �k�; a i(k), Gi(k),Lij(k), Ci(k), Qi(k), Ri(k), Ki�k�, �up

i �k�; �xpi �k�; �yp

i �k�; Jp; l (refer to Sections 2 and 3) and so on. The

calculation component is the implementation of Algorithm 1 whose program chart is shown in Fig. 3.


Fig. 2. Software structure.


Fig. 3. Program chart of Algorithm 1.

5. Application

With the help of SIPA, many examples of HSPP have been studied, which show that the algorithm canoptimally decompose the medium-term random product demand plans of an FAW into short-termstochastic production plans to be executed by FMSs in the FAW. In the following, the application ofthe algorithm and SIPA is presented through an example of HSPP in an FAW.

Without loss of generality, suppose that an FAW consists of four FMSs with complementary func-tions. Each of the FMSs comprises four machines whose functions are alike and/or complementary. Theweek random product demand plan is shown in Table 1, in which the lot of each type of parts is theexpectation and the processing times are deterministic. From Table 1, we know that there are four typesof parts, and each type must pass through each FMS. However, different types of parts pass throughFMSs in different sequences. Parts can pass through each machine or partial machines in an FMS. Theyare of the ¯exibility of paths. In other words, they can select one of the available machines for each


Table 1

Week random product demand plan

Part Sequence FMS Available machine Lot Processing time (h)

P1 5 4 M41,M43 50 0.98

10 M42,M44 1.20

15 2 M21,M23,M24 0.72

20 3 M34 0.40

25 M32 1.30

30 1 M12,M14 0.62

P2 5 3 M33 40 1.62

10 M31,M34 0.50

15 1 M14 1.05

20 4 M43 0.62

25 M42 0.50

30 2 M23 1.45

35 M21,M24 1.62

40 M22 1.06

P3 5 1 M11,M12,M14 60 0.51

10 M13 0.33

15 2 M22 0.44

20 M21 0.42

25 3 M31,M34 1.32

30 4 M41 0.41

35 M44 0.33

P4 5 1 M11 70 0.83

10 M12 0.59

15 M13 0.71

20 3 M31 0.28

25 2 M24 0.36

30 4 M41,M43 0.56

35 M42,M44 0.56

sequence. Besides, it is apparent that the FMS blocks in Table 1 are of varying sizes because FMSs arethe ¯exibility of job shops. It is a ®ve-day week. A workday is divided into two shifts. Time available toeach machine every shift is a stochastic variable whose expectation is seven hours (excluding machinerepairing time, etc.). The demand for ®nished products of part P1 each shift is a stochastic variable whoseexpectation is ®ve; P2, four; P3, six; and P4, seven.

From Table 1 and Eq. (1), we have (refer to Yan and Jiang (1998) about the computation of �Ti�k��

�T1�k� �

0 0 0:17 0:83

0:31 0 0:17 0:59

0 0 0:33 0:71

0:31 1:05 0:17 0

26666664

37777775 �T2�k� �

0:24 0:81 0:42 0

0 1:06 0:44 0

0:24 1:45 0 0

0:24 0:81 0 0:36

26666664

37777775

�T3�k� �

0 0:25 0:66 0:28

1:30 0 0 0

0 1:62 0 0

0:40 0:25 0:66 0

26666664

37777775 �T4�k� �

0:49 0 0:41 0:28

0:60 0:50 0 0:28

0:49 0:62 0 0:28

0:60 0 0:33 0:28

26666664

37777775for k � 1; 2;¼; 10:

From Table 1, we know that sequence 5 of P3 or P4 is the ®rst of the seven sequences in the FAW andpasses through FMS1, which means the necessary supply of the corresponding blanks for the processingof sequence 5 of P3 or P4. For simplicity, the third kind of workpiece processed by FMS1 is correspond-ing to P3 and the fourth to P4. Thus, from the de®nition of Gi(k) in Eq. (2), we get

G1�k� �

0 0 0 0

0 0 0 0

0 0 1 0

0 0 0 1

26666664

37777775 � diag�0; 0; 1; 1�

Similarly, from Table 1 and the de®nitions of Gi(k), Lij(k) and Ci(k) in Eqs. (2), (5) and (1), respectively,we have

G2�k� � diag�0; 0; 0; 0� L13�k� � diag�1; 1; 0; 0�

G3�k� � diag�0; 1; 0; 0� L21�k� � diag�0; 0; 1; 0�

G4�k� � diag�1; 0; 0; 0� L23�k� � diag�0; 0; 0; 1�

C1�k� � diag�1; 0; 0; 0� L24�k� � diag�1; 1; 0; 0�

C2�k� � diag�0; 1; 0; 0� L31�k� � diag�0; 0; 0; 1�

C3�k� � diag�0; 0; 0; 0� L32�k� � diag�1; 0; 1; 0�


C4�k� � diag�0; 0; 1; 1� L41�k� � diag�0; 1; 0; 0�

L42�k� � diag�0; 0; 0; 1� L43�k� � diag�0; 0; 1; 0�

Lij�k� � diag�0; 0; 0; 0� for all other ij2values

for k � 1; 2;¼; 10:From the assumption of the time available to each machine every shift and Eq. (1), respectively, we

have

�b i�k� � � 7 7 7 7 �T

for i � 1; 2; 3; 4; k � 1; 2;¼; 10:Suppose that blanks into the FAW are stochastic variables and their expectations can be described by

�r�1� � � 6:2 5:0 7:4 8:6 �T �r�9� � � 6:0 5:4 7:6 9:0 �T

�r�7� � � 6:0 4:8 7:2 8:6 �T �r�10� � � 6:0 5:0 8:0 9:2 �T

�r�8� � � 6:0 5:0 7:4 8:6 �T �r�k� � � 6:0 4:8 7:2 8:4 �T for the others

Suppose that there is no rework and that waste products are stochastic variables and their expectationscan be represented by

�vi�k� � �20:25 20:2 20:3 20:35 �T

for i � 1; 2; 3; 4; k � 1; 2;¼; 10:From Table 1 and Eq. (1), we obtain

�d1�k� � � 5 0 0 0 �T

�d2�k� � � 0 4 0 0 �T

�d3�k� � � 0 0 0 0 �T

�d4�k� � � 0 0 6 7 �T

�d�k� �X4

i�1

�di�k� � � 5 4 6 7 �T

for k � 1; 2;¼; 10:Let initial states

xi1 � � 0 0 0 0 �T

for i � 1; 2; 3; 4:


Let given work-in-process level

a1�5� � � 0 0 0 0:02 �T a3�11� � � 0:20 0 0 0 �T

a1�6� � � 0 0 0 0:10 �T a4�3� � � 0 0:02 0 0 �T

a1�7� � � 0 0 0 0:20 �T a4�4� � � 0 0:01 0 0 �T

a2�11� � � 0:10 0 0 0 �T ai�k� � � 0 0 0 0 �T for the others

Let weight matrixes

Qi�11� � diag�1; 1; 1; 1�

Qi�k� � Ri�k� � Ki�k� � Qi�11�for i � 1; 2; 3; 4; k � 1; 2;¼; 10:

Then, we make conjectures upon initial values

zpi �k� � lp

i �k� � � 1 1 1 1 �T

for i � 1; 2; 3; 4; k � 1; 2;¼; 10:From Step 6 of Algorithm 1, the condition to stop iteration is that the norm ilp

i �k�1 �pi�k 1 1�i2 tothe left-hand side of Eq. (33) is less than d (equal to 0.3162), so is the norm izp

i �k�2PMj�1 Lij�k� �uj�k� i2 to the left-hand side of Eq. (34).Having run SIPA on Pentium 133 for 1.5 s, we obtain FMSs' optimal shift stochastic production plans,


Table 2

FMS1's shift stochastic production plans

Part Period

1 2 3 4 5 6 7 8 9 10

P1 5.00 5.00 5.00 5.00 4.99 4.98 4.95 4.92 4.90 4.92

P2 4.44 4.44 4.41 4.38 4.36 4.35 4.37 4.37 4.34 4.22

P3 7.04 6.93 6.90 6.89 6.90 6.93 6.88 7.00 7.02 7.11

P4 8.13 8.10 8.08 8.06 8.04 8.06 8.19 8.17 8.17 7.85

Table 3


Part Period

1 2 3 4 5 6 7 8 9 10

P1 5.56 5.52 5.51 5.51 5.50 5.48 5.44 5.42 5.41 5.45

P2 3.96 3.97 3.97 3.97 3.95 3.94 3.94 3.92 3.90 3.89

P3 6.68 6.63 6.61 6.60 6.61 6.62 6.59 6.61 6.60 6.58

P4 7.39 7.39 7.39 7.38 7.36 7.35 7.35 7.35 7.35 7.20

as shown in Tables 2±5. For simplicity, each plan only shows the expectation number of workpieces(®nished or semi-®nished products) of planned production by the corresponding FMS each shift, that is,�up

i �k� for i � 1; 2; 3; 4; k � 1; 2;¼; 10:By rounding data in Tables 2±5, we get

�up1�k� � � 5 4 7 8 �T for k � 1; 2;¼; 10

�up2�k� � � 6 4 7 7 �T for k � 1; 2;¼; 5

�up2�k� � � 5 4 7 8 �T for k � 6; 7;¼; 10

�up3�k� � � 5 5 6 8 �T for k � 1; 2;¼; 10

�up4�k� � � 6 4 6 7 �T for k � 1; 2;¼; 10

Thus, the expectations of ®nished products of planned production by FMSs become

�yp1�k� � C1�k� �up

1�k� � � 5 0 0 0 �T

�yp2�k� � C2�k� �up

2�k� � � 0 4 0 0 �T


Table 4


Part Period

1 2 3 4 5 6 7 8 9 10

P1 5.28 5.27 5.26 5.25 5.24 5.22 5.19 5.16 5.16 5.19

P2 4.66 4.63 4.61 4.60 4.60 4.60 4.61 4.64 4.66 4.51

P3 6.36 6.34 6.32 6.29 6.28 6.29 6.28 6.27 6.21 6.13

P4 7.76 7.74 7.73 7.71 7.70 7.71 7.79 7.77 7.74 7.51

Table 5


Part Period

1 2 3 4 5 6 7 8 9 10

P1 5.86 5.79 5.76 5.75 5.74 5.72 5.70 5.68 5.66 5.63

P2 4.17 4.19 4.19 4.18 4.16 4.15 4.15 4.14 4.09 3.98

P3 6.03 6.03 6.01 6.00 5.99 5.98 5.98 5.96 5.92 5.89

P4 7.01 7.03 7.02 7.02 7.02 7.02 7.01 6.98 6.94 6.88

�yp3�k� � C3�k� �up

3�k� � � 0 0 0 0 �T

�yp4�k� � C4�k� �up

4�k� � � 0 0 6 7 �T

for k � 1; 2;¼; 10: Note that �ypi �k� � �di�k�: That is, demands for products are completely satis®ed.

Besides, the optimal objective value Jp is 21.75. The number of iterations is 20. Because of wasteproducts, expectations of work-in-process xp

i �k� are not equal to zero and their ranges are

� 0:011 0:018 0:003 0:027 �T # �xp1�k� # � 0:083 0:435 1:001 1:653 �T

� 0:023 0:004 0:029 0:007 �T # �xp2�k� # � 0:097 0:123 0:473 0:135 �T

� 0:025 0:090 0:021 0:002 �T # �xp3�k� # � 0:071 1:079 0:363 0:204 �T

� 0:034 0:003 0:007 0:014 �T # �xp4�k� # � 0:403 0:302 0:069 0:134 �T

for k � 2; 3;¼; 11:

6. Conclusions

Because the existing papers on the PP problems with uncertainties mainly focus on uncertainties of thedemand, capacity and material supply in the single-period or in®nite-horizon setting and because moststochastic programming problems in the literature are solved for the single-period or in®nite-horizon, we propose the new stochastic interaction/prediction approach to solve the HSPPproblem for FAWs in agile manufacturing that is a multiple-period multiple-product problemwith random material supply, demands, capacities, processing times, rework and waste products.In the paper, a mathematical model of stochastic production of an FAW is built up ®rst. Thestochastic interaction/prediction algorithm of HSPP for the FAW is then deduced. Based on thealgorithm, the software package named SIPA has been developed and is presented in this paper.By means of SIPA, many examples of HSPP have been studied. Through one of those examples,the application of the algorithm and SIPA is introduced.

As compared with the existing approaches to solve the PP and HPP problems with uncertainties, thepresent algorithm has the following features:

1. It solves the multiple-period multiple-product problem with more uncertainties and in a more compli-cated manufacturing setting (see Section 1).

2. It needs only expectations of r(k), di(k), vi(k), Ti(k) and TTi �k� Ri(k) Ti�k� and independence of Ti�k�

from ui�k� and bi�k�, but does not require their exact distributions.3. On condition that expectation blanks and expectation demands for products in the FAW are given, the

combinatorial optimization of the objectives, such as the work-in-process in FMSs, machine utiliza-tion, balance of loads on machines, and satisfaction of demands for products, is reached.

4. Since the FAW's optimal HSPP problem is decomposed into M FMSs' optimal SPP sub-problems,the complexity of the problem is greatly reduced, thus speeding up the process of solving the problem.

5. The interactions zpi �k� among FMSs in the FAW can be obtained.


Weight matrixes Qi(k) �for k � 1; 2;¼;N 1 1� are symmetrical and (semi-) positive de®nite. Ri(k) andKi(k) are symmetrical and positive de®nite. For simplicity, they can be taken as diagonal matrixes. Eachelement in them must not be too great in case the loss of signi®cant digits in the matrix calculation makesthe problem ill-conditioned. For the decomposition of the ten-day or week random product demandplans, the hour is taken as a unit of the element in Ti(k) and b i(k) in order to decrease its numeric andincrease the accuracy of the matrix computation. To increase the satisfaction of the demands forproducts, the numeric of the element in Ki(k) can be properly increased. When some elements of�xpi �k� are negative, the corresponding elements (that can be decimal) of a i(k) can be increased.

SIPA has the friendly user interface. It is convenient for users to input and update both data andparameters of the algorithm. It can run by itself and also be taken as an algorithm module of algorithmbase of the FAW controller if it is properly modi®ed. It can be used not only to optimally decomposerandom product demand plans of FAWs but also to examine the rationality of both the expectation blanksupply and the expectation product demand.

If the expectation supply of blanks is not suitable to the given expectation demand for products, theexpectation demand can not be satis®ed, as may be imagined. Therefore, we will probe further into theproblem of how to determine the expectation blank supply appropriate for the given expectation demandso as to meet it.

Acknowledgements

This research is supported by the National High Tech R&D Program, the People's Republic of China,under grants 863-511-943-005 and 863-511-708-008. We thank the anonymous referees for valuablecomments and suggestions.

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