a multistage benders decomposition method for production planning problems with approved vendor...

Journal of ManuJacturing Systems Vol. 24/No. 4

2005

A Multistage Benders Decomposition Method for Production Planning Problems with Approved Vendor Matrices

L.H. Lee, E.P. Chew, and T.S. Ng, Dept. of Industrial and Systems Engineering, National University of Singapore, Singapore

Abstract A production planning problem is studied based on an

actual manufacturer of hard-disk drives that offers the approved vendor matrix as a competitive advantage. An approved vendor matrix allows customers to pick and choose the component suppliers for individual components or pairs of components constituting their product. The problem is to develop production plans that minimize the total tardiness in fulfilling customer orders while observing the matrix restrictions and limited component supplies. It is first shown that this problem has an equivalent multicommodity network flow representation. A solution procedure using multistage Benders decomposition is then developed. The computational efficiency of the approach is compared with the column-generation method and the CPLEX general-purpose LP solver under different scenarios of matrix restrictions.

Keywords: Production Planning, Approved Vendor Matrix, Network Flows

1. Introduction A production planning problem is considered

based on an actual hard-disk drive manufacturer. The hard-disk drive is basically an assembly of a number of important components, and for this manufacturer there are several vendors supplying each component. As a competitive advantage, the manufacturer allows customers to specify their own approved vendor matrix (AVM). The AVM lets the customers pick and choose preferred vendors for individual components or combinations of components in their product. In this work, an end-product is defined as the final assembly of components. A build type is the set of all end-products that uses the same combination of component vendors. Custom- ers specify their demand in the form of order types. A build type can be assigned as a customer's order type only if it complies to the AVM specified by the customer. A summary of the problem is as follows. At the beginning of each planning horizon, there is

a set of customer demands due in any period. De- mand that is unfulfilled by the due date is backlogged and charged with a tardiness penalty until it is fulfilled. Demand that is unfulfilled by the end of the planning horizon is penalized as a shortage. Produc- tion resources like manpower availability and component supply schedules cannot be changed within the planning horizon. The decisions to be made are which build types to produce in each period and their corresponding levels, and how to pack them for different customers, that is, how much of each build type to assign for each order type in order to achieve the minimum total backlog and shortage costs. This is the BuiM-Pack Problem first introduced by Lee, Chew, and Ng (2005).

The chief characteristic of the build-pack problem is the high proliferation of build types and the com- plexity of production planning imposed by the AVM, A hierarchical planning approach (Hax and Meal 1975; Graves 1982) is unsuitable because the set of order types overlaps the set served by each build type. Chu (1995) developed a myopic decomposition heu- ristic for a production problem that allows customers to specify preferred suppliers for individual components, but still does not explicitly address the problem of high build type proliferation. Lee, Chew, and Ng (2005) formulated the build-pack problem as a total tardiness problem (Wang 1995). The column- generation technique is then used to solve the model for an optimal build-and-pack schedule.

This work first shows that the column-generation approach can be an-ived at by considering the build- pack problem as a multiconunodity network flow (MCNF) problem. Furthermore, the MCNF model for the build-pack problem has a special multistage network structure. The primary motivation of this work is to study the performance of a solution ap-

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proach that is based on this multistage network structure. Secondly, the motivation of presenting this solution and modeling approach is practical one, based on the sensible question of whether, in a situation where there are very little AVM restrictions, it is possible to avoid a formulation such as that in Lee, Chew, and Ng (2005), which explicitly identifies the build types. In the case when there are no AVM restrictions, the build-pack problem collapses into a very. simple total tardiness planning problem that can be solved efficiently. The solution, which can be viewed as a master schedule of the production levels for each customer order, would then be sufficient because it would not be necessary to explicitly identify the different build types. It would, hence, be of value if this basic result can be used in some way when AVM restrictions are present. Some scheme is then required to disaggregate the master schedule solution into build schedules, and then to repair the solution if AVM restrictions are violated. The issue of interest would then be comparing the economy of the effort required to perform the repairing versus the column- generation approach.

The following notation is used for the inputs and parameters of the build-pack planning problem in this work:

t

P k v

¢ g~

4 FFI~: t

P

v;

K K~

Fk,,,,,,' =

production period, t = 1. • -T product component order type component vendor

demand level for k due in t

tardiness cost per unit of k in period t

units of p to build per unit of k units of v arriving in period t set of all components p

set of all vendors of component p E P

set of all vendors of p ~ P that is acceptable in the AVM of k set of all k set of all k E K that cma use vendor v E Vf, for component p E P to make the final product

-1 if for k, vendor v of component p cannot

be used together with vendor v' of

component p', where v ~ Vp, v' ~ Vp.Vp,

p '~ P.

0 otherwise.

The next section is a case study of a manufacturer of hard-disk drives, which provides the background for this work. Section 3 presents a MCNF representation of the build-pack planning problem. Section 4 gives a multistage formulation of the problem and develops a solution procedure that uses the Benders decomposition approach. Computational results are presented in section 5 to draw comparisons with the well-known column-generation method for solving MCNF problems. Conclusions are given in section 6.

2. Case: Production Planning of Hard-Disk Drives

2.1 Background This work is motivated by a problem faced by an

actual manufacturer of hard-disk drives, whose customers are largely OEMs and reputable PC makers. This manufacturer purchases key components from multiple vendors on a long-term contract basis. It then assembles, tests, and packs the drives for the customers. The hard-disk drive company competes in a thin-margin, commoditized market, and to de- fend profit margins the company offers customer flex- ibility and on-time delivery as its compet i t ive advantage. One scheme to implement customer flex- ibility is to enable customers to pick and choose preferred vendors for individual or combinations of components constituting their hard-disk drive.

The problem addressed starts with the release of the Master Production Schedule (MPS), which is a schedule of order types (by demand quantity and due date) to be fulfilled in the current week. The MPS is used to drive even more detailed schedules, which are implemented at the assembly plant. A buiM plan schedules the run quantities of build types in each period, while a pack plan assigns the build types to order types to fulfill the MPS.

Once the build and pack plans are generated, the rest of the production process is relatively straight- forward. At the beginning of each production period, production supervisors refer to the build plans to draw components from the parts store, and these components are fed into the manufacturing cells. In the workcells, the components are assembled into the build types mad are then passed to the test cells for software coding and power-up tests. Finally, the drives are labeled and packed for the customers as specified in the pack schedule and are shipped out

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of the factory every day to regional distribution centers. The distribution centers are maintained by a third-party logistics finn, which charges customers a flat rate for some fixed amount of holding space.

In the current system, a team of human planners manually drafts the build and pack plans each week. During seasons of high demand, product proliferation is high and manufacturing resources are tight, and the manual planning process becomes time con- suming and inefficient. This work takes the scheduling model of the MPS and translates it into optimal daily build-and-pack production schedules. The problem considers limited manpower and component availability. It is not uncommon that although the resources meet the MPS requirements in an ag- gregate sense, daily availability of resources may not be fully synchronized with the build schedules. This result in underpacks, which are costly as they contribute to the failure to fulfill committed delivery to customers on time. An underpack of an order is the number of units short of the demanded quantity that is due. Underpacks are accumulated into subsequent periods as backlogs and are penalized as tardy orders until they are fulfilled. The objective is to schedule production in a manner so as to minimize the total daily production backlogs and shortages within the plamfing horizon.

2 . 2 A p p r o v e d V e n d o r M a t r i c e s

As a competitive advantage, the hard-disk drive manufacturer allows its customers to choose component vendors for their order type in the form of the AVM. In the hard-disk drive, there are three criti- cal components, the headstack assembly (HSA), disc, mad printed ch'cuit board (PCB). The performance of the hard-disk drive is well known in magnetic recording technology to be highly dependent on the interaction between the HSA and disc components, and customers have their own engineering evalua- tions on which HSA works well or not with which disc components. Tables 1 and 2 shows a typical AVM of one customer for the hard-disk drive company. H1-H3, DI-D3, and P1-P4 here denote the different suppliers of the HSA, disc, and PCBA components, respectively. In Table 1, a value of zero (one) is assigned to a combination of HSA and disc model if this combination of models cannot (can) be used to build the product for the customer. Table 2 indicates whether a PCB model can (value one) or

Table 1 AVM for Head-Disc Combinat ion for a Cus tomer

DISC/HSA H I H~ H 3 DI 0 1 l D, 1 1 0 D 3 0 1 1

Table 2 AVM fl~r Head-PCB Combination for a Customer

PCB/HSA HI H~ H 3 Pj 1 0 1 Pz 1 0 0 P3 1 0 1

cannot (value zero) be used to build the final product for the customer. All build types that do not vio- late the specifications of Tables 1 and 2 can be assigned as the order type of this customer. For example, the build type comprising components H I, D2, and P2 can be used to fulfill the demand of the customer, while the build type comprising H 1, D j, and P2 is not allowed to.

3. MCNF Representation of the Build-Pack Problem

Multicommodity network flow (MCNF) problems arise when several items (commodities) share arcs in a capacitated network. They have been studied extensively because of their numerous applications and their intriguing network structure exhibited. Ahuja, Magnanti, and Orlin (1993) and Kennington (1978) provide comprehensive surveys of the MCNF problem formulations and solution approaches.

This section develops a MCNF representation of the problem, where the objective is to find the minimum cost paths to ship commodit ies f rom the origin(s) to the destination(s) nodes through the given capacitated network. Let the network of con- cern be G = (N,A), where Nis the node set consisting of a supply node, a number of demand nodes, manpower nodes, and component vendor nodes. A C N × Nis the set of arcs connecting N in a special 'layer' structure to form the required network. The MCNF equivalent of a customer demand, k, is a commodity to be transported from the supply node to some demand node(s). Figure 1 shows an example of the multicommodity network in the hard-disk drive production planning context.

Let o be the supply node. The outflow of each commodity, k, at o is equal to the total demand for k

T

over the entire planning horizon, that is, Y~ d, k . t=l

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1 ............... -.-..;'% .... --- :j \ !

/ i / ....~ i ...................... ~--2 !

j >I . . . . . . . . . . . . . . . "-,..~.+.y "*..'~'~ . . . . . = . . . . . ?

~;, ~ , . ............ , , < . - ; / -..~ .............. / " ' ...................................... .;:~i ?(. "]:.,(: ..:; g "

....... ,d" .............. :7~ ", ........... ii

qa

Figure 1 Multicommodity Network for Hard-Disk Drive

P r o d u c t i o n Planning

Let q, be the demand nodes, t = 1. • .T. The net requirements of each commodity, k, at q, is the demand for k due in t, that is, d,*. Also defined is a shortages node, qr+l, with zero net requirements, qr+J is not a demand node by definition and is only used for transfer requirements f rom o into the demand nodes when shortages occur.

Let h,, t = I. • .T, denote the manpower nodes. Commodity k flows from o to h r when production of k in period t happens. The capacity at node h, is equivalent to the total manpower or labor resource available in the period t, that is, c,.

Let n; denote a component vendor node, where v E V,,, p E P, and t = 1- - -T. The system of the component vendor nodes is structured in the form of layers. Each layer is composed of all nodes for each p. Each commodity must flow through at least one node per layer to constitute a complete build type. If some level of a commodi ty k flows through n'~, then this level of k is produced using vendor v for component p in period t. The nodal capacity of n'~ is the net available supply of Vp in period t. Note that these capacities are dynamic because they de- pend on the flow and capacities of previous periods.

The cost of shipping on arc (q,+~, qt) per unit of k is the tardiness cost g,~. Note that the arc cost is unidirectional on the arc. There is no cost in shipping f rom q, to q,+l because holding costs are not consid-

ered in the problem. This is to be consistent with the problem in Lee, Chew, and Ng (2005). Consider- ation of holding costs, however, does not affect the validity of the model and solution approach.

The MCNF problem is to find a set of feasible flow paths and levels to ship the commodities from o to q, t = 1. • .T + 1, so as to minimize the total arc costs, subjected to the limited node capacities. De- fine F, k as the set of ,all feasible paths from the origin node o to node % t = 1. • .T. Similarly, F ~ is the set of all paths from origin o to all demand nodes % t = 1 . - -T , that is, F ~ = F~ k u ~ k . . 'wFr*. The path- flow variable, Xs~ for all p a t h s f E F~ * , is then equivalent to a scheduled production level of some build type assigned to a customer k in period t. Denote also the path-indicator parameters 8 f and 5[ so i , j '

that ~¢ = 1 if path f contains arc ( i , j ) , 5 j = 0 i , j i , j

otherwise. Similarly, 5[ = 1 if pa th fcon ta ins node i, 5[ = 0 otherwise. In this manner, the AVM restrictions can be explicitly modeled in the definition of F~ k ; that is, if customer k does not allow v ~ lip and v' E V ¢ to be used together for some pair of compo-

nents p and p ' , then 5~, v,,,v, = 0 for all the paths f E

F, * . The equivalent path-flow formulation can then be stated as:

Problem 7~: T

Minimize Z= Z Z Z g~'8~l;+,.q, "X, (1) kEK t = l f e F [ t

subject to:

~ XI= ¢ V k ~ K, t=I."T (2) f~v, k

~ 5,Y,, .Xi<c , t=l . . .T (3) k~ K ,f~ F ~

t ' = l t k~K,, t f ~F k

t

VveV ,peP, t=l...r t'= 1

(4)

XI~9I + V f r F 1~, V k ~ K (5)

The objective function in (1) minimizes the total costs of shipping all the commodities from the origin to destination nodes. Costs are only incurred when commodities ship on arcs (qt+l, qt), t = 1- • .T. Con-

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straints (2) are the flow requirements conditions at each demand node qt. Constraints (3) and (4) are the manpower capacity and the component supply limi- tations, respectively. Constraints (5) are unidirectional conditions on the path flows.

Although 7 9 is a linear program and can be solved directly by any general-purpose LP solver, large problem sizes render this approach inefficient. In particular, the number of columns (feasible paths) grows in exponential proportion with the number of nodes in the network. One popular approach for such problems is to apply the column-generation technique (Gilmore and Gomory 1961), whereby a restricted form of 79 is solved using only a subset of all possible paths in the solution process. The search for new improving paths is as per the simplex method, using the criteria of the reduced cost of nonbasic paths. Improving paths are found by solving a sepa- rate pricing problem, which can be shown to be a set of independent shortest path problems. The new paths found are augmented into the restricted version of 79, which is then re-solved. The procedure terminates when no more improving paths can be found, which is basically the optimality condition for the simplex method. The column-generation approach hence keeps 79 at a modest size, and search- ing for new entering paths is also efficient. Readers are referred to Ahuja, Magnanti, and Orlin (1993) for a detailed description of the application of the column-generation technique for MCNF problems. The implementation results for problem 79 are presented in section 5.

4. A Multistage Approach to Solve the Build-Pack Problem

This section presents an alternative solution approach to the problem that applies the outer linear- ization technique using Benders decomposition (Benders 1962). Benders decomposition is a popular technique in solving certain classes of difficult problems such as stochastic programming problems (Infanger 1994; Nielson and Zenios 1997) and mixed-integer nonlinear progranaming problems (Floudas, Aggarwal, and Ciric 1989; Geoffrion •972). Geoffrion and Graves (1974) demonstrated the solution of large-scale multicommodity distribution system models using Benders decomposition. Many other applications of Benders decomposition have also been proposed, including aircraft routing

(Richardson 1976), network design (Magnanti, Mirchandani, and Wong 1986), and vehicle assignment (Cordeau, Laporte, and Mercier 2000; Cordeau et al. 2001). Recent applications to stochastic programming include adapting Benders decomposition to the scenario version of robust network design problems (Kouvelis andYu 1997) and robust shortest path problems with interval data (Montemanni and Gambardella 2004). Multistage or nested Benders decomposition is largely applied to solve multistage stochastic programming problems [for example, capacity planning (Ahmed, Parija, and King 2003), scheduling hydroelectric generation (Jacobs et al. 1995)], These are basically extensions of the two- stage problems, where each stage is viewed as a Benders nv:tsterproblem and its successor stages the subproblem.

To exploit this approach, the problem is first cast as a multistage problem, where the first stage is simply a relaxation of the original problem formed by dropping all of the AVM requirements. The subsequent stages each consist of a transportation problem involving a pair of components. The solution of each transportation problem accounts for the AVM restrictions and yields a feasible component vendor assignment for the components involved. The transportation problems are solved in a sequential manner, from the first to the last component in the product. The solution for each stage sets up the transportation problem in the next stage. If the problem at any stage becomes ilffeasible, some AVM restrictions are then violated. A Benders feasibility cut is then generated and augmented to the previous stage, which is solved again for a new component vendor assignment. The procedure terminates when all the stages are feasible.

4.1 Multistage Formulation

Presented first is the multistage formulation to the build-pack problem. To begin, the product components are sequenced in the order p °, p~, • • • pU, where N + 1 = IPI, IPI being the cardinality of the set P, and p0 is the first component constituting the shortest path network presented in section 3.2 (or the MCNF network in section 3.4), pl the second component, and so on. This sequence is fixed throughout the solution procedure and determines the order in which the transportation problems are solved.

In the formulation, the variables Xk, t are defined as the production level for customer k in period t.

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G~,t indicates the backlog level for k in period t. Yf"

is the level of component j , vendor v (v ~ V1 ), that is

allocated to the production of k in t. For all components pi, where j = 1. • .N, Yv k'' is defined as

F k

yk.t= p~ S ~ yk,, V v6 V/ J = l ' ' ' x (6) v r k ~ v',v pJq i"~ V j_I

where yk,, is the disaggregation of yk,, which uses vl, v v '

~/ for the immediate predecessor component ff-~. Note that (6) holds only for pi f o r j --< 1 because p0 does not have a predecessor component by definition. The multiperiod build-pack scheduling problem can then be rewritten as:

Problem B: T

Minimize Z = ~ ~.&., .G~., (7) k E K t= l

subject to:

~ X , , , < c , t= l . . . T (8) k~K

t t

Z Z q . x a / < Z Z m , , , . V p ~ P , t = I . . . T (9) t '=l k ~ K t '=l ~ V p

t t

Gk,,>-~dk, , -~X~. c V k ~ K , t = I . . . T (10) t '= l t '=l

Gk. , -> ~ d k,, - X~, / Vk ~ K, t = 1..- T t '= l t '= l

ZY~k,'t'<Zmv,,. Vvc=V,o,t=l...T t '=l k ~ K t ' = l

(11)

(12)

r k

Z vZ= ' ' vk /c v'~V/_ l Fkpj ~'

j = I . . . N , t= l . . .T (13)

k F j y~,,_ p yk.,

v,v" - - " - ~ " v Vk e K, Vv ~ V;_,,

j =I . . .N, t= l . . .T (14)

Z£ Z Ye'/<Zm,,,/,,',v - V v e V / , v ' ~ V j _ 1 k E K t '=l t '= l

p

j = I . - . N , t = I . - . T (15)

y k , t ~ + X~,, ~9~+, Y,, ~'' ~ 9~+ . . . . ,,E (16)

The objective function (7) minimizes the total penalb, for tardiness of the orders [as has been men- tioned, inventory holding costs are not considered here for consistency with the problem in Lee, Chew, and Ng (2005)]. Holding costs can be included if required without any loss of generality in the model and solution development. Constraints (8) impose total manpower availability for each period, and Constraints (9) are the components availability ag- gregated over all vendors for each component and period. Constraints (10) are backlog equations. Con- straints (I I ) and (12) are the flow-balance and avail- abil i ty condi t ions for the first componen t , p0, respectively. Constraints (13) follows directly from the definition of y k., in expression (6). Constraints (14) together with (15) form the flow balance and availability conditions for the rest of the components f f , j = 1. • .N.

4.2 Solution Procedure

The solution procedure decomposes B into a master problem and a series of subproblems. Observe that for a fixed set of production levels Xk, ,, (11) and (12) can be replaced by a set of transportation feasibility conditions, with each Xk, , forming a demand point and each component supply mv,t VV E Vpo forming a supply point. Similarly, for a fixed set of pro-

duction levels Yv~"Vve V~_,, (14) and (15) can be replaced by transportation feasibility conditions with

each Y,~" forming a demand point and mv,, Vv ~ V~ forming the supply point, j = 1. • .N. In the following, the definitions of the decomposed problems are first formalized.

Define the master problem 7~B " {Z = Minimize (7), s.t. (8) - (10) }, which is just the basic total-tardiness problem. Note that when there are no AVM restrictions, the solution to ~ B forms an optimal production schedule.

The transportation problem associated with (11) and (12) is used to test for possible AVM violations for the first component, p0. The following defines the t ime-expanded transportation 'route' variable,

y k,,a a- = 1- • .t, where a" is interpreted as the arrival

period of the supply of v being used. Note that t

yk., = ~-- yk,,a v ~ ,, . The phase-one transportation problem

"~=1

can then be written as:

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Problem 77>o:

Minimize Z = k ~" ~-" ~-~ F~., • Y ~''a t=l I:=1 k ~ K ~ V o

subject to:

~ ~' y~,t,*-r ~ ~, - po'X~., V k e K , t = l . . . T "~=t v~V~ 0

(17)

(18)

T T T

Z Z Y L ' = Z Z m , , ~ - Z Z r e ' X k , v , pO ,

' t=l ~ V o $=1 v~Vo t=l k ~ K

(19)

T ZZ ZY,;'" t=x k ~ K v~V ~: pO

Vv~ V o, Z=I . . .T (20)

K?" So weK, v V,o, x=l...t, t=l.. .T

(21)

The objective function (17) minimizes the sum of all the shipment levels on inadmissible routes, or model assignment levels yk.,,~ which are 'illegal',

V

that is, in conflict with the AVM requirements. Con- straints (18) and (20) follow directly from (11) and (12), which are simply the demand and supply equations of the transportation problem. Constraint (19) captures the supply that is in excess of the demand in the formulation and is commonly termed as a 'dummy' demand point in unbalanced transportation problems, yfa is used to denote the assignment of the supply from v and a- to the dummy demand, /7:. Note that (9) ensures that there will be no unbal- ance in the other direction, that is, demand in excess of supply. Xk.r in (18) is fixed at the solution levels obtained from ~B. Constraint (21) requires

y~,,a to be long in the set So, def in ing So =

{Y *'"~ >_ O, s.t. Io} I0 being the set of cut constraints v

in 779o . Cut constraints I0 will be discussed later in this section. Note that I 0 is initialized as a null set in the first iteration of the solution procedure.

The transportation problem associated with (14) and (15) for each component/¢ where./-> 1 is used to test for possible AVM violations between component ff and its immediate predecessor component

t ~-~. Define the transportation route variable yk,,a t v,vt '

where "r is as defined previously, and Y~" = S 'Y k'''~ v, ]f ~ **',1 ~" . "c=l

Suppose we have a fixed set of vendor assignment

levels yk.,v ,gv~V,~_ , ,VkeK and t = 1- • . T f o r

component pi-~. The phase-one transportation problem associated component pJ can be defined as:

Problem 7~i: T

M i n i m i z e Z = ~ ~ ~ rk,,¢'Y k't'~ . . . . #" (22) t=~ ~=1 k~,v~,~vi ~_ v'~vj

subject to:

r k

~ Z yk,t,~= "pJ .yk,t v,v" r k v

~=1 v ' ~ V fi pj-I

Vk ~ K, Vv ~ V,j_,

t=l.. .T (23)

T

Y~"'~ Vv'e V~, "c=I...T (24) Z Z Z ,'.," =m¢,~ t='~ k ~ K v~Vpj_ 1

T T T r k pJ k, t

E Z YL':E,; E rn ," , , -ZZ~ 'Y~ ' ,%, ,=1 r ,_ l (25)

yk,ta E r,1/ S j

x= 1-.-t, t=l...T (26)

As in 779o, the objective function (122) of 77> i minimizes the sum of all the shipment levels on inadmissible routes, and (23) and (24) follow from (14) and (15), respectively. Constraint (25), as in (19), is the demand equation for the dummy demand point used to capture any supplies in excess of the demand.

The set Sj in (26) is defined as: S]: {Y~i~'~ > 0, s.t. I] }, where L. is the set of cut constraints in 779j.

In the initial iteration of the solution procedure, ~/3 is first solved :for a set of Xk,r If there are no AVM restrictions, a complete build and pack schedule can then be generated. However, if the AVM is present, a feasible pack schedule may not exist for the current solution of X~.,. The solution scheme is checked for possible AVM violations components- wise, starting with pO Given X~,,, a feasible component vendor assignment exists for p0 only if there are solutions to (11) and (12) with Xk. , held at the given levels. To check this, 7790 can be solved, fix- ing the demand quantities on the right-hand side of (18). If Z = 0 in (17), then a feasible assignment exists for the first component, p0 and the assignment for the next component can be checked. Otherwise,

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a Benders feasibil i ty cut can be genera ted us ing the dual solution of 779o . Benders feasibili ty cuts are based on locating extreme rays in the polyhe- dron of the dual Benders subproblem. The cut is written as:

T T

ZZgk.,'r~p~o'Xk,,+ZZv~,t'm,,.,+ k~K t=l ~ V 0 t=l

~t K • m.~ - ko "Xk..t + t=l k~K

Zb,

(27)

where Pk,t, 1)v.t, and g~ denote the optimal dual multipliers associated with (18), (20), and (191), respectively. "rr, and b~ refer to the dual mul t ip l ie r and right-hand side value of cut constraint i E I 0. Con- straint (27) is augmented in 7~B, which is then resolved for a new set of product ion levels, X~,,. 7720 is solved again, and if new violations are found, the corresponding cuts are then generated. The procedure is repeated iteratively until Z = 0 in (17).

Once a feasible assignment exists for p0, possible AVM violations are checked for the next compo-

nent. Given y k., Vv ~ Vj_,, where pm is the immediate predecessor component of pi by definit ion (] --> 1), a feasible assignment exists for component pi only i f t he r e are s o l u t i o n s to (14) and (15) w i t h

y,k., Vv~ V,j_~ fixed at the given levels. For j = 1,

Yv k'' Vv ~ Vj is obtained directly f rom the solution of 779o . F o r j > 1, this is obtained f rom the solution of 779j_1 and then applying (13). 779~ is then solved. As before, Z = 0 in (22) indicates that a feasible as- s ignment exists for component pJ, and the next component , / ;+1 is checked. Otherwise, a feasibility cut is generated, and this can be written as:

T

Z Z r=- k~Kv~Vpj_ 1 t=l pj-1 ~ V j t=l

r S ' S " P' Y~"

g~ " mca - z... ~ r ~ " ,, v j t=l kEK pj-I

y i~lj

r k T

pJ y k,~+ ZZVv, , . r nv , , +

+ (28)

where here Pg,v,t, Vv,,, and g i denote the optimal dual multipliers associated with (23), (24), and (25), respectively. As before, bi and 7ri are the dual price and right-hand side value of cut constraint i E ~ . Constraint (28) is augmented in ~-1, and Tp/_ 1 is then resolved. Note that y k,, in (28) is first replaced back with the transportation route variables of 7-5°/._1. If 77~i_ 1 remains feasible, a new set of model assign-

ment values, Yv k'' Vv e V/_,, is then generated. This

is used to solve ~ again, and if new violations are found, the corresponding feasibility cuts are then found. The procedure is repeated until Z = 0 in (22). I f T/gi_l becomes int'easible, then a cut (28) will have to be generated f rom 77~j_~ and passed back to the previous problem.

The solution scheme thus proceeds in a nested form, with ~ acting as the Benders subproblem for predecessor components and a master problem for success ive componen t s . The opt imal so lu t ion is found at the end of the solution procedure when Z = 0 in (22) Vp ' , where j = 0. • .N. The solution procedure is summarized below.

Multistage Algorithm:

1. Solve ~ B , yielding production levels Xk,,. Set the component index j = 0.

2. Fix Xk.r and solve 779o. If Z* = 0, increment j : j = j + 1 and proceed to Step 3. Otherwise, generate cut (27), update ~ B , and return to Step 1.

3. If j = N + 1 optimality is reached, and the procedure terminates. I f j = 0, go to Step 1. Other-

wise, solve 779j using the current Y~" Vv ~ V,~_,. 4. If Z* = 0, increment j : j = j + 1 and proceed to

Step 3. Otherwise, generate cut (28) and update !Jq- Decrement j : j = j - 1 and go to Step 3.

4 . 3 I m p l e m e n t i n g 7"Py

As the number of customers and component suppliers increases, the size of the problems 779/for j = 1- • -N may become considerably large because the number of transportation 'routes ' in 779j is in the order of IK1 x IV12 x T 2, where IK1 and IV] denote the cardinality of the sets K and V, respectively. This may render solution times to increase significantly even if ~ is a pure network problem (that is, ~. is empty). One way to ease the computat ional burden is to include in 779/only the demand points that have posi- tive demands and hence only the routes serving these points. Solving this compact version of ~ can help to reduce the problem size substantially. In the case

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when a cut is to be generated from 77~j, it is neces- s,'uy to obtain the dual prices associated with all the demand points. This can be accomplished as follows. First, note that the dual feasibility condition associated with each Y~'~ in ~ can be written as:

~k,,,',t + Vv,t + Z a i "~i <- rk,v,v" (29) i~ Ij

where gk,,',,, v,,,, 7rs, and i E /,. are as previously defined, ai denotes the coefficient of the route variable in cut constraint i, i E / / .

Because the omission of the zero demand points does not change the optimal solution, the solution obtained from the compact version of TPj is also optimal in the complete 77~/. It then follows from strong duality of linear programming that the corresponding dual solution, v , , and n~, is also feasible and optimal in the complete dual problem. Apply- ing (29), the dual prices associated with the zero demand points can then be recovered by simply s e t - / ' / t ing lak,,,,=miv.'n . In actual

implementation, oniy the set TM ~olf cut doefficients, a,,.

needs to be stored. Terms in the cuts are included into the model, only if the corresponding route variables are designated for nonzero demand points.

In implementing T]); with a nonempty set !i, the Dantzig-Wolfe decomposition algorithm (Dantzig 1963) is applied to solve Tp. i. Here the masterprob- lem consists of the cut constraints i E Ij, and the subproblem is the pure transportation problem. The master problem determines an optimal convex combination of corner point solutions from a subset of available transportation solutions. The dual nmlti- pliers associated with !i are then used to price out a new master variable by solving the transportation problem, where the objective is to minimize the reduced cost of the master variable. Optimality in 779j is achieved when no more negative reduced-cost variables can be priced out. The master problem is thus kept to a modest size, and the network structure of the subproblem is preserved. Readers are referred to Ho and Sundarraj (1989) for a comprehensive description of implementing the decomposition algorithm.

5. Computational Results and Discussion In the computational experiments, both the col-

umn-generation (section 3) and multistage decom-

position (section 4.2) algorithms are implemented using three industrial-strength problem sets. All problems were designed based on the hard-disk drive product structure. There are three components, the HSA, disc, and the PCB, hence IPI = 3. Each problem set consists of 10 instances with randomly generated demand and resource levels. Each instance is solved under different scenarios of AVM restrictions. The 'restriction level' t3 of an AVM between two components is defined as the proportion of illegal assignments among all possible vendor assignments, that is, for an order k and components/ ; and pi+l,

3-" , Fkv v,

13 = " - " " % , ," % , + , , ,

For example, 13 = 0.3 indicates that 30% of the component vendor assignments are not allowed in the AVM. In all of the computations, 13 is set to be the same for all orders and components in each scenario. Note that 1 > 13 --> 0, where a high 13 value indicates that there are many illegal component-vendor combinations. On the other hand, [3 = 0 refers to the case where there are no AVM restrictions.

The solution algorithms were all coded in C++. The CPLEX 7.0 LP and network solver libraries were invoked to solve the decomposed subproblems. All computations were performed on a Pentium IV 4.0 GHz PC with 512 MB RAM and 18 GB disk space. Tables 3 to 5 highlight the results of the computational study, where Cols and Rows indicate the total number of columns and rows in formulation 79 (see section 3), respectively. Note that the column count decreases with increasing 13 because the number of feasible pack types decreases. Results obtained via three solution approaches axe presented, that is, procedures GLP, CG, and 131). GLP refers to solving formulation 79 , with all feasible pack types enumerated directly using the CPLEX general-purpose LP solver. ('.(7, and BD refer to the column-generation procedure and the Benders decomposition procedure, respectively. CPU/s indicates the mean computation time (average of the 10 random instances) in sec- onds obtained at each 13 level. Itns indicate the mean number of pricing iterations used by procedure CG.

In the implementation of CG, the master problem initially consists of one column per customer order. For the pricing scheme, IK1 x T shortest path problems are solved during each pricing iteration. One

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Table 3 Problem /3 ,Set 1: IKI = 200, IV1 = 10, T = 7

CG BD GLP

[3 Cols Rows CPU Itns CPU/s Cuts CPU/s

0 1400000 1617 17.22 8 1.35 0 2614.7 0.1 1134000 1617 14.2 7 1.38 0 2171.0 0.2 896000 1617 13.4 6.7 1.38 0 1450.6 0.3 686000 1617 13.3 6.3 1.39 0 987.4 0.4 504000 1617 12.75 6.3 1.39 0 913.4 0.5 350000 1.617 12.65 6 1.4 0 652.0 0.6 224000 1617 12.4 5.9 1.4 0 448.3 0.7 126000 1617 12.2 5.7 1.39 0 340.7 0.8 56000 1617 11.9 5.1 1.74 0.2 240.2 0.9 14000 1617 l 1 4.7 4.1 2.2 128.5 0.98 560 1617 10.2 4.5 11.7 10.2 18.2

Table 4 Problem /3 Set 2: IKI = 100, IVI = 20, T = 7

CG BD GLP

[3 Cols Rows CPU Itns CPU/s Cuts CPU/s

0 5600000 1127 15.25 49 0.98 0 NA 0.1 4536000 1127 13.05 48.2 0.98 0 NA 0.2 3584000 1127 12.78 47 0.98 0.1 NA 0.3 2744000 1127 12.35 46.4 0.98 0.3 NA 0.4 2016000 1127 11.75 43.9 1.45 0.7 7184.25 0.5 1400000 1127 11.11 42.9 1.62 1.2 2402.1 0.6 896000 1127 10.44 41.7 2.82 2.2 1298.5 0.7 504000 1127 10 39.1 3.48 2.8 889.5 0.8 224000 1127 9.45 39.6 7.4 5.8 402.7 0.9 56000 1127 8.65 37.9 10.55 8.5 229.7 0.95 14000 1127 7.45 36.2 15.2 14.6 115.4

new column per shortest path problem is then augmented into the master problem if it has a negative reduced cost. A few other pricing schemes were also experimented with, but they generally gave inferior performance tbr the problem instances. Cuts refers to the mean number of cuts generated by procedure BD.

It should be noted that for all problems sets both the decomposit ion procedures, CG and BD, outper- form GLP in computation times even for the small- est problem instances. For large instances (over 2 million variables), CPLEX was unable to load the problem data (indicated by NA in Tables 4 and 5) due to limited storage resources, whereas procedures CG and BD solve the same problems using only modest computation times. For procedure CG, it is observed that the number of pricing iterations for problem set 2 (Table 4) is generally larger than in problem sets 1 and 3 (Tables 3 and 5). This may be because IK1 in problem set 2 is the lowest, and because each pricing iteration can add up to IK] x T new columns, less number of columns are being priced out per iteration. Note that the number of com-

Table 5 Problem /3 Set 3: IKI = 200, IV1 = 20, T = 7

CG BD GLP

[3 Cols Rows CPU Ires CPU/s Cuts CPU/s

0 11200000 1827 NA 56.18 18.6 2.5 0 0.1 9072000 1827 46.85 13 2.5 0 NA 0.2 7168000 1827 45.54 12.3 2.6 0 NA 0.3 5488000 1827 42.66 11.2 2.5 0 NA 0.4 4032000 1827 42.94 11.1 2.5 0 NA 0.5 2800000 1827 38.1 10.5 2.5 0.1 NA 0.6 1792000 1827 38.16 10.1 3.2 0.3 9835 0.7 1008000 1827 38.2 I0.1 5.4 1.1 4138 0.8 448000 1827 35.46 8.8 10,7 3.1 1060.8 0.9 112000 1827 34.36 8.3 24 7.1 352.2 0.95 28000 1827 33.7 8.2 100.8 17.6 283.5

ponent vendors, IV1, does not change this maximum number of new columns in the pricing algorithm.

The performance of CG and BD is now compared under different AVM restriction scenarios. £~gures 2, 3, and 4 show scatter plots of CPU time against the [3 level using the results in Tables 3, 4, and 5, respectively. In all problem sets, it is observed that for lower [3 values procedure BD outperforms CG, while for high [3 values procedure CG outperforms BD. Procedure CG is relatively insensitive to the [3 level (with slightly lower C PU times and fewer pricing iterations as [3 increases), whereas the computational effort required by BD increases significantly when [3 is high (in particular 13 > 0.9). The performance behavior exhibited by procedure BD is rea- sonably intuitive. When [3 is low, there are relatively few vendor assignments in the AVM that are disallowed. Hence for a given schedule generated in problem R13, the chance of finding a feasible vendor assignment is conceivably higher. For the example problems, solving the sequence of pure transportation problems ~ in the first iteration of the procedure proves to be sufficient for obtaining the complete solution when [3 is relatively low (in the range of 0 to 0.4 on average over all three problem sets). When [3 increases, more vendor assignments are illegal, and as a consequence an increasing number of feasibility cuts are generated before the solution converges to optimality. Despite this, BD still outperforms procedure CG consistently until [3 > 0.9 (average over all three problem sets) in the example problems.

It is noted that procedure BD is clearly a winner when the AVM restrictions are not too severe. This is in fact the case for the hard-disk drive example, where customers specify the HSA-disk combinations that ,are not allowed in their orders. Most HSA-disk

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2005

: i

i :

~ i i

0 ~!: ,92 :~,~. ~ 4 ~ 5 ~,8. ~ ? g1~ G;O 1

Figure 2 CPU Times vs. A~qvI Restriction Level: Problem Set 1

• .4..-- ~ G~a"~r~ t

/ i "

0

Figure 4 CPU Times vs. AVM Restriction Level: Problem Set 3

Figure 3 CPU Times vs. A r M Restriction Leveh Problem Set 2

assignments are typically acceptable to the customers, and the disallowed assignments turn out to be quite sparse and few in the AVM but cannot be ig- nored during the course of planning. In such situa- tions, the number of feasible build-pack assignments and hence flow-paths in formulation 7 9 become ex- tremely large. Solving 7 9 directly is clearly inefficient if not impossible. Procedure C G , though relatively robust, is not able to use this information to its advantage. We hence offer procedure B D as an alternative approach for such scenarios. On the other hand, in scenarios where the AVM restrictions are so fight that possibly only vel 3, few build types are allowed for each order, it is conceivable that enumer- ating all the possible flow paths would possibly be the best solution approach. Finally, another possible advantage of using a solution method based on procedure B D over the column-generation approach is as follows. One possible extension of the model is

to consider additional customer requirements. For example, customers may require shipments to be in specified lot-sizes. This can be modeled by augment- ing discrete variables and additional constraints in the formulation ~L~, that is, the Benders master problem. ~ B is then solved as an mixed-integer programming (MIP) problem. The advantage of the multistage formulation is that the Benders cut-generation process in the subsequent stages remains the same. On the other hand, for the column-generation algorithm and the model by Lee, Chew, and Ng (2005) (or equivalently formulation 79 in section 3), a branch- and-bound (B&B) algorithm will need to be devised so that the LP relaxation at each B&B node is now solved via colmrm generation. If many nodes need to be searched in the B&B tree, the difference in computation times between the two methods demonstrated in this paper may then grow significantly because a major bottleneck issue in computation time for B&B algorithms is solving the LP relaxation. In addition, for the Benders version, the integer variables and constraints are confined to the master problem and one can thus exploit efficient and available MIP solvers to solve the Benders master problem. In the column-generation case, because each node needs to be solved via column generation, a commercial solver might not be directly ap- plicable and additional effort may be required to code the branch-and-bound algorithm.

6. Conclusion This work has investigated a production planning

problem of emerging importance. Mass manufac-

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turers in the modem day face complex complica- tions imposed by customers. This work focuses on one such complication known as the approved vendor matrix based on an actual manufacturer of hard- disk drives. The AVM complication is not well addressed as yet in existing literature to the best of our knowledge. The problem is termed the Build- Pack problem as there are two types of planning decisions, that is, what to build and who to pack for. It has been shown that the build-pack problem can be modeled as a MCNF problem, hence allowing available solution techniques to solve the problem efficiently. In the work, the colunm-generation technique was implemented to solve the equivalent MCNF problem. Lastly, a solution procedure was developed using a multistage Benders decomposition approach that solves the problem in a sequential manner. It has been shown using industrial-strength problems that the approach outperforms both the CPLEX general-purpose LP solver and the column-generation technique in the case when the AVM restrictions are not too severe. The multistage decomposition is thus offered as an alternative solution approach in such scenarios. Other possible future extensions of the problem can include some enhancements to the solution approach in order to accelerate convergence. For instance, the solution scheme in the work gener- ates a single cut at each stage of the problem. In anticipation of a possibly large number of infeasible assignments generated when the AVM restriction level is high, it may be profitable to attempt generat- ing multiple cuts at each stage during a single iteration. One possibility is to make use of the existing pool of transportation solutions generated from the Dantzig-Wolfe subproblem to form demand alloca- tions for the successor problem stage. New feasibility cuts can then be generated. Alternative decomposition schemes other than the sequential procedure proposed can be considered, too. For example, in the first stage, independent transportation feasibility problems can be solved for the individual components in parallel. In the second stage, the vendor assignments from the first stage are used to build the transportation problems for the component pairs. Again, feasibility cuts can be generated from both stages when the AVM restrictions are violated, with the cuts from the second stage linking all the first- stage problems together in general. Dantzig-Wolfe decomposition can again be used to solve the first- stage problem with the cut constraints.

Acknowledgments

This work is sponsored by NUS RP R-266-000- 013-112 and NUS RP R-266-000-026-490.

References Ahmed. S.; Parija, G.R.; and King, A.J. (2003). "A multi-stage sto-

chastic integer programming approach for capacity planning under uncertainty." Journal oJ" Global Opthnization (v26, nl), pp3-24.

Ahuja, R.K.; Magnanti, T.L.; and Odin, J.B. (1993). Network Flows: Theory, Algorithms, and Applications. Englewood Cliffs, NJ: Prentice-Hall.

Benders, J.F. (1962). "Partitioning procedures for solving mixed variables progranmaing problems." Numerische Mathematik (v4, n2), pp238-252.

Chu, S.C.K. (1995). "A mathematical programming approach to- wards optimized master production scheduling." Production Eco- nomics (v38, n2) pp269-279.

Cordeau, J.F.; Laporte, G.; and Mercier, A. (2000). "A Benders decomposition approach for the locomotive and car assignment problem." Tlz~nsportation Science (v34, n2), pp133-149.

Cordeau, J.F.; Stojkovic, G.; Soumis, F.; and Desrosiers, J. (2001). "Benders decomposition for simultaneous aircraft routing and crew scheduling." Transportation Science (v35, n4), pp375-388.

Dantzig, G. (1963). Linear Programming and Extensions. Princeton, NJ: Princeton Univ. Press, pp342-354.

Floudas, C.A.; Aggarwal, A.; and Ciric, A.R. (1989). "Global opti- mum serach for nonconvex NLP and MINLP problems." Com- puters and Chemical Engg. (v 13, n 10), pp I 117-1132.

Geoffrion, A.M. (1972). "Generalized Benders decomposition" Journal of Optimization Theory and Applications (vl0, n4), pp237-260.

Geoffrion, A.M. and Graves, G.W. (1974). "Multicommodity distribution system design by Benders decomposition." Mgmt. Science (v20, n5), pp822-844.

Gilmore, P.C. and Gomory, R.E. (1961). "A linear programming approach to the cutting stock problem." Operations Research (v9, n6), pp849-859.

Graves, S.C. (1982). "Using Lagrangean techniques to solve hierarchical production planning problems." Mgmt. Science (v28, n3), pp260-275.

Hax, A.C. and Meal, H.C. (1975). "Hierarchical integration of production planning and scheduling." Studies in the Management Sciences, M.A. Geisler, ed. (vl, Logistics). Amsterdam: North- Holland and American Elsevier, pp53-69.

Ho, K.H. and Sundarraj, R.P. (1989). DECOMP: An Implementation of Dantzig-Wolfe Decomposition for Linear Programming. New York: Springer-Verlag.

Infanger, G. (1994). Planning Under Uncertainty - Solving Large- scale Stochastic Linear Programs. Boyd and Fraser.

Jacobs, J.; Freeman, G.; Grygier, J.; Morton, D.; Schultz, G.; Staschus, K.; and Stedinger, J. c1995). "SOCRATES: a system for scheduling hydroelectric generation under uncertainty." Annals of Op- erations Research (v59, nl) , pp99-133.

Kennington, J.L. (1978). "A survey of linear cost multicommodity network flows." Operations Resealz'h (v26, n2), pp209-236.

Kouvelis, P. and Yu, G. (1997). Robust Discrete Optimization and Its Applications. Kluwer Academic Publishers.

Lee, L.H.; Chew, E.P.; and Ng, T.S. (2005). "Production planning with approved vendor matrices for a hard-disk drive manufacturer." European Jot4~Ttal of Operational Research (v162, n2), pp310-324.

Magnanti, T.L.; Mirchandani, R.; and Wong, R.T. (1986). "Tailoring Benders decomposition for uncapacitated network design." Math- ematical Programming Study (v26, nl) , pp112-154.

Montemanni, R. and Gambardella, L.M. (2004). "The robust shortest path problem with intelwal data via Benders decomposition." Sub- mitted for publication.

326


2005

Nielson, S.S. and Zenios, S.A. (1997). "Scalable parallel Benders decomposition for stochasitc linear programming." Parallel Com- puting (v23, n8), pp1069-1088.

Richardson, R. (1976). "An optimization approach to routing aircraft.'" Transportation Science iv 10. n 1 ), pp52-71.

Wang, D.W. (1995). "Earliness-tardiness production planning approaches for manufacturing systems." Computers and Industrial Engg. (v28, n3), pp425-436.

Authors' Biographies Loo Hay Lee is an associate professor in the Dept. of Industrial

and Systems Engineering, National University of Singapore, since 1997. He received his BS (electrical engineering) degree from the National Taiwan University in 1992 and his SM and PhD degrees in 1994 and 1997, respectively, from Harvard University. He is currently a member of IEEE, ORSS, and INFORMS. His research interests include simulation-based optimization, production scheduling and

sequencing, logistics and supply chain planning, and vehicle routing. His e-mail and Web addresses are [email protected] and wwm ise.nus.edu.sg/staff/leelh, respectively.

EK Peng Chew is an associate professor and deputy dead (academic) in the Dept. of Industrial and Systems Engineering, National University of Singapore. He received his PhD degree from Georgia Institute of Technology. His research interests include logistics and inventory management, system modeling and simulation, and system optimization. His e-mail and Web addresses are [email protected] and www.ise.nus.edu.sg/staff/chewep/, respectively.

Tsan Sheng Ng is currently a research fellow in the Dept. of Industrial and Systems Engineering, National University of Singapore. He received his BEng (electrical engineering) degree and PhD (industrial and systems engineering) from the National University of Singapore in 2000 and 2005, respectively. His research interests include production planning, advanced planning systems, and modeling of service supply chains. His e-mail address is [email protected].

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