routing flexibility and production scheduling in a flexible manufacturing system

344 European Journal of Operational Research 60 (1992) 344-364

North-Holland

Theory and Methodology

Routing flexibility and production scheduling in a flexible manufacturing system

Soumen Ghosh

Graduate School of Business Administration, Michigan State University, Department of Management, East Lansing, MI 48824-1121, USA

Cheryl Gaimon

School of Management, Georgia Institute of Technology, Atlanta, GA 30332-0520, USA

Received December 1989; revised July 1990

Abstract: A multiproduct, multiperiod, multistage network model is presented for the planning of order release and production scheduling in a flexible manufacturing system environment under the existence of alternate routings. It is assumed that premanufacturing decisions such as machine grouping and tool loading have been made, so that setup costs and setup times are negligible and can be included in the processing times. The decision process addressed by the model is the disaggregation of weekly production requirements to daily production requirements, the determination of production batch sizes for each operation of each part type, and the daily assignment of each batch to machine groups given the flexibility of alternate routings. The model also provides the interface and linkage between an MRP component planning system and the shop scheduling system. The model is solved using a price-directive decomposition technique with column generation. Experimentation is performed with the model for varying problem sizes to determine the impact of shop flexibility on total cost, inventory levels, existence of bottlenecks, shop utilization, and the number of setups and split lot production. The results indicate important cost-benefit trade-off implications for system design and acquisition. For example, if in fact setup costs and times are nonnegligible, then it is shown that increasing the routing flexibility of a system without a parallel decrease in setup costs and times is unlikely to reap significant benefits.

Keywords: Production planning; scheduling; flexible manufacturing systems

1. Introduction

A network model is presented for the planning of order release and production scheduling in a flexible manufacturing system (FMS) environment. Specifically, given input from a component production planning process such as MRP, the model optimally determines the quantity and timing of order releases for multiple part types to the shop floor. Further, the processing stage of order for each part type is optimally assigned to a machine while simultaneously considering all possible alternate routings and machine capacities associated with the given FMS shop structure. The problem is formulated as a

0377-2217/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved

s. Ghosh, C. Gaimon / Routing flexibility and production scheduling 345

multiproduct, multiperiod, multistage network model with bundle capacity constraints. Due to an FMS's ability to quickly (automatically) accomplish product changeovers, almost negligible downtime is experi- enced for setups (Groover, 1987; Klahorst, 1981; Voss, 1986). Taking advantage of this feature, it is assumed that setup costs and times are negligible, and can be included in the processing times.

Given the particular FMS environment addressed, the problem is too large to be solved straightforward as an LP, and therefore a price-directive decomposition procedure using column generation is applied to obtain solutions. The model is easily decomposed into pure network subproblems which are solved very efficiently. Experimentation with varying problem sizes is performed to show the computational efficiency of the decomposition technique compared to a straightforward LP solution. Experimen- tation is also performed to investigate the effect of shop flexibility on total cost, inventory levels, existence of bottlenecks, shop utilization, and the number of setups and split lot production.

In a recent comprehensive survey of 107 FMSs, Edgehill and Davis (1985) found that 50 percent are comprised of machining centers consisting of very versatile numerical control (NC) machines. Versatility of these machines has a significant influence over the overall flexibility of an FMS; particularly routing flexibility where each workpiece can have several alternate routings through the machines, i.e. each operation of a workpiece can be performed on any one of several machines. Therefore, a routing as well as a sequencing decision is required for all workpieces. While this added degree of freedom greatly reduces the possibility of bottlenecks in the system (and thereby increases the throughput rate), it also considerably increases the complexity of the scheduling function. The model presented here addresses these concerns.

There are two models in the literature that are structurally similar to the one here (allocation of production requirements to machines), but address different production environments. In Gaimon (1986), a serial (fixed routing), multistage, multiperiod, single product, capacitated environment is addressed. Capacities and operating costs of machines performing the same tasks at any stage in the production process are permitted to vary in order to consider the flexibility of being able to choose machines at each stage according to their individual characteristics. However, the model has a pure network structure (no bundle constraints), and is solved optimally using a specialized network algorithm, NETFLO (Kennington and Helgason, 1978). Second, Zahorik, Thomas and Trigeiro (1984) introduce a multiproduct, multiperiod, multistage model in which they allow bundle constraints on inventory and production (i.e. capacity constraints on total inventory and total production at each stage). Row operations are performed on the constraint matrix to transform it into a pure network problem for a three-period special case. Then this method is applied to solve larger problems heuristically. Like the Gaimon (1986) model, Zahorik et al. (1984) also require fixed and identical routings for each product, and do not consider setup costs. In contrast, the model introduced here not only allows bundle inventory and production constraints at each stage, but also explicitly incorporates the possibility of nonidentical routings for each product, as well as the possibility of alternate workcenters for each operation of each product.

2. Problem description

The FMS environment depicted here consists of logical clusters of identically tooled machines (numerically controlled machines with tool magazines capable of holding multiple tools) as machine groups, and a work-in-process (WlP) storage buffer or automated storage/retrieval system (AS/RS) for each group. It is assumed that the set of part types (set of unique workpieces) that are to be simultaneously processed by the system is known in advance. Given these part types and their required operations, it is assumed that the tool loading decisions (assignment of tools to the tool magazine of each machine) have been made. Therefore, the set of machine groups capable of processing a particular operation of a part type is known (see Stecke, 1983; Stecke and Talbot, 1983; Kusiak, 1985; Berrada and Stecke, 1986; Shanker and Tzen, 1985, for descriptions and techniques to solve these problems). If these major system setups (specifically tool loading) are performed before the actual start of production, and if each operation of each part type is assumed to be assignable only to those machines which have already

346 s. Ghosh, C. Gaimon / Routing flexibility and production scheduling

been loaded with the necessary tools required for that operation, then setup times can be assumed to be negligible. Minor setup operations such as workpart positioning on the machine bed, gauging and sensing, or retrieving the part program and required tool from the tool magazine are not just performed at the start of each production lot, but occur before each operation of each part, and therefore can be included in the processing times (see Groover, 1987; Jaikumar, 1984).

Unlike existing scheduling models, explicit linkage to the plant's MRP system is provided by directly taking the planned order release quantities in the MRP action buckets as input to the models, and disaggregating them into daily production requirements. Since the planning horizon of the model is one week, an MRP lead time of one week or less is assumed for each part type. This assumption can be easily relaxed for part types whose MRP lead times are more than one week, by allocating the total production requirement of those parts equally (or in some ratio) during each week of their lead time. Since the mature planned order releases from MRP act as input to this model, it is implicitly assumed that the components for each part type are available as a result of the lead time offsetting logic in MRP. Given the inputs, the decision process solved by the model includes (i) the division of weekly production requirements to daily production requirements and daily backorder levels, (ii) the determination of the production quotas for each operation of each part type, (iii) the assignment of these daily production quotas to machine groups given the flexibility offered by alternate routings, and (iv) the level of WIP inventory of each part type in the storage buffers of each machine group.

Before presenting the model, the following notation is introduced. First, the variable indices are defined, followed by the input parameters, and then the decision variables. Indices:

t = 1 . . . . . T = time periods (days), n = 1 . . . . . N = machine groups, k = 1 . . . . . K = part types, s = 1 , . . . , S = sequence of operations for part type k (S is the last operation, or finished part type).

Input parameters: w(k) = weekly production requirement for part type k, A(s, k) = set of alternate machine groups that can perform operation s of part type k, d(s, k, n) = unit processing time of operation s of part type k, on any machine in machine group n (in

hours), X(n , t) = capacity of machine group n, at time t (in hours per day), Y(n, t) = capacity of WlP storage buffer or A S / R S system of machine group n, at time t, b(k, t) = unit cost of backording part type k during period t, h(s, k, n, t) = unit cost of carrying inventory of part type k following the completion of operation s, in

buffer associated with machine group n, at time t, c(s, k, n, t) = unit cost of production associated with operation s of part type k, on any machine in

machine group n, at time t, ~,(k, t) = capacity (upper bound) on backordering arcs, y~(s, k, n, t) = capacity (upper bound) on inventory arcs, ~(s, k, n, t) = capacity (upper bound) on production arcs.

Decision variables: r(k, t) = production requirement for part type k, during time t, z(k, t) = level of backorder of finished part type k, during time t, y(s, k, n, t) = level of inventory of part type k following the completion of operation s, in buffer of

machine group n, at time t, x(s, k, n, t) = level of production of operation s of part type k, on any machine of machine group n, at

time t.

Formulation

Before presenting the formulation, several points must be made concerning (i) production batches, (ii) capacity feasibility, and (iii) time buckets. First, the exclusion of setup times in a production planning and

S. Ghosh, C. Gaimon / Routing flexibility and production scheduling 347

scheduling model facilitates an LP approach. Recall from our earlier discussion that setups in an FMS environment may or may not be insignificant depending on how the tools loading function is performed. If setups were nonnegligible, then 0-1 binary variables would have been necessary to correctly model the environment (this is the premise in Ghosh and Gaimon, 1988). Here, however, it is assumed that all required tools have been loaded prior to the actual start of operations with no dynamic tool exchange capabilities present, and consequently setup times/costs are considered negligible. It is important to understand that this model does not perform the actual sequencing of parts on machines (determination of start and finish times for each operation), but rather the immediate higher-level planning decisions regarding the determination of daily production quotas for each operation of each part type, and their allocation to machines. An individual part can proceed to a machine for an operation once its preceding operation is completed.

Second, precedence relations among operations in conjunction with capacity constraints prohibit the production of the same operation of the same part on multiple machines, or the simultaneous production of two operations for the same part. Third, it is conceivable that when the system starts from an idle state (no jobs in the system), the machines assigned to perform the second and succeeding operations of a part type may have to be idle until the preceding operation is done. This may lead to slight capacity distortion during startup, but can be avoided by incorporating some capacity slack in the model, as is usually done in higher-level decision models of a production planning hierarchy. The actual sequencing decisions are usually accomplished at the next lower level, where the daily production quota for each machine are scheduled into smaller time buckets such as hours and minutes.

The model is formulated as a linear multiproduct, multistage, multiperiod, capacitated network model. Complete freedom of movement is allowed (flow of parts) not only among all machines but among all WIP buffers as well. The network model is formulated by defining four kinds of arcs (flows): daily production arcs for each part type r(k, t); daily production arcs assigning each operation of each part type to a machine group x(s, k, n, t); daily inventory arcs for each operation of each part type in the WIP storage buffers y(s, k, n, t); and daily backordering arcs for each part type z (k , t).

Since a single-source network is formulated, the equilibrium between weekly and daily production requirements can be represented by the partial network shown in Figure 1. The corresponding flow equilibrium conditions can be written as in (2) in the formulation.

In Figure 2, the partial network representation of flows in and out of a particular node of the network corresponding to any operation s < S of part type k, at time t, is shown. Given the ending inventory from the previous period, the total production requirement for operation s of part type k at time t is generated from the sum of the ending inventory of the same operation and the production requirements for the next operation of part type k during time t. Note that the choice of alternate machines for any operation, and inter-inventory movement are included in the flow equilibrium condition written as in (3).

w(k)

r(k,1) r(k,t) r(k,T)

Figure 1. Flow equilibrium for weekly and daily production requirements

348 S. Ghosh, C. Gaimon /RoutingflexibUity and production scheduling

x(s+l,k,n,t)

y(s,k,n,t- 1) ~ y(s,k,n,t) n e A(s,k) n e Ms,k)

x(s,k,n,t) n e Ms,k)

Figure 2. Flow equilibrium for any operation s < S

In the final stage (finished parts), backordering arcs and daily production requirements are included. The production batch size for each operation of each part type for this last operation is generated from ending inventory, backorder, and daily production requirements. The partial network representation for the last operation S of part type k at time t is shown in Figure 3. The flow conservation is shown in (4).

Since a machine group can process multiple operations of different part types, the bundle production capacity constraints can be expressed as in (5). Similarly, the bundle inventory constraints for each buffer appear in (6). Note that since all part types do not necessarily share the same size and shape, a space index could easily be introduced in this constraint. This is omitted for the sake of simplicity. Further- more, the number of constraints of this type could be reduced from) n ( ( T - 1) to ( T - 1) by assuming a central storage system rather than local buffers. It is assumed that all machine and inventory buffer capacities (X(n , t) and Y(n, t)) are finite and known. Finally, for the sake of generality, individual capacity constraints on the backorder, inventory, and production arcs can be specified (7-9).

The objective (1) is to minimize the sum of linear costs incurred due to production, inventory and backorders. Only out-of-pocket costs, and not sunk costs, should be used in the objective function.

r(k,t)

y(S,k,n,t-1) n e A(S,k)

z(k,t-1)

y(S,k,n,t) n E A(S,k)

z(k,t)

~2~ x(S,k,n,t) n e A(S,k)

Figure 3. Flow equilibrium for the last operation S


Examples of sunk costs may be regular time labor and unit WIP inventory holding costs. The production and WlP inventory holding cost terms included here could easily be set to zero if they are sunk costs.

The complete mathematical formulation is stated below. (Problem LP1)

K T 1" S N

Minimize Y'~ Y'~ [b (k , t ) z ( k , t) + E Y'~ {h(s, k, n , / ) y ( s , k, n, t) k = l t = l s = l n = l

+c(s, k, n, t ) x ( s , k, n, t)}] (1)

T

subjectto ~ r ( k , t ) = w ( k ) , k = l . . . . . K, (2) t = l

E x ( s , k , n , t ) + E y ( s , k , n , t - 1 ) n~A(s, k) n~A(s, k)

= Y'~ x ( s + 1, k, n, t) + ~ y ( s , k, n, t ) , n~A(s+l , k) n~A(s, k)

k = l . . . . . K, s = l . . . . . S - I , t = l . . . . . T, (3)

Y'~ x ( S , k , n , t ) + E y ( S , k , n , t - 1 ) + z ( k , t ) n~A(S, k) n~A(S, k)

= E y ( S , k , n , t ) + r ( k , t ) + z ( k , t - 1 ) , n~A(S, k)

k = l . . . . . K, t = l . . . . . T, (4) K S

~_, E d(s , k, n ) x ( s , k, n, t) <~X(n, t) , n = l . . . . . N, t = l . . . . . T, (5) k = l s = l

K S

E E Y ( s , k , n , t ) < ~ Y ( n , t ) , n = l . . . . ,N, t = l . . . . . T - l , (6) k = l s = l

O < z ( k , t ) < < . ~ ( k , t ) , k = l , . . . , K , / = 1 . . . . . T, (7)

O<~y(s, k, n, t) ~ ( s , k, n),

s = 1 . . . . . S, k = 1 , . . . , K , t = 1 . . . . . T - 1, N ~ A ( s , k) , (8)

0 <~x(s, k, n, t) <<.~(s, k, n, t) ,

s = l . . . . . S, k = l . . . . . K; t = l . . . . ,T , n ~ A ( s , k ) , (9)

y(s , k, n, O) =y(s , k, n, T) = O,

s = 1 . . . . . S, k = 1 . . . . . K; n ~ A ( s , k) , (10)

r(k , t )>~O, k = l . . . . . K, t = l . . . . . T. (11)

The individual subnetworks for each part type lie on a plane (planar network) and have a pure network structure. For example, a subnetwork of a part type having three operations (S = 3) with two alternate machines for each operation, and three time periods (T = 3) is shown in Figure 4. The vertical arcs depict production flows (the set of arcs corresponding to alternate machines capable of performing the same operation for that part type), while the horizontal arcs depict flows due to inventory and backorder (in the final stage only). The complete network is made up of the entire set of these individual planar subnetworks corresponding to each part type, lying parallel to each other and coupled together by the production (eq. 5) and inventory (eq. 6) bundle constraints. Note that the individual subnetworks for each part type will differ according to the number of operations and alternate routes for each operation of that part type.

350 S. Ghosh, C. Gaimon / Routing flexibility and production scheduling

3. Solution approach

Although the above formulation is LP, in most practical applications the size of the problem makes the straightforward use of the simplex algorithm far too inefficient. For example, note that the problem has

2T ~ A ( s , k) + K ( 2 T + T S + 1) variables and ( K + KTS + 2NT) constraints, l s= l

excluding the upper bound constraints (7)-(9). Suppose there are 20 part types (K = 20), 10 machine groups (N = 10), 5 time periods (T = 5), 10 operations per part type (S = 10), and 5 alternate machine

Weekly Demand, w(k)

O

0

s=3

s=2

r(k,t) Inventory Arcs, "~ y(s,k,n, t)

Backordering Ares, z(k,t)

Production Arcs, x(s,k,n,t)

s= 1

t=l t=2 t=3 Time Periods

Figure 4. Subnetwork for a part type k

S. Ghosh, C. Gaimon / Routing flexibility and production scheduling 35 !

l k

subject to i = 1

n lk

E i = 1

E i = 1

groups for each operation (A(s, k) = 5, s = 1 . . . . . & k = 1 . . . . . K). Then a total of 11220 variables are required in the constraint matrix and 1120 constraints exist excluding upper bound constraints.

Without the bundle constraints (5) and (6), the problem defined by (1)-(11) has a pure network structure, for which very efficient solution algorithms exist. The presence of the bundle constraints prohibits the straightforward use of a network technique. However, price-directive decomposition techniques have been found to be quite efficient in solving multicommodity network flow problems (Kennington and Helgason, 1980). A column generation procedure using the price directive decomposition technique is used to solve the model defined by (1)-(11). While the theory of price-directive decomposition of linear programs is well known and can be found in several textbooks (e.g. Lasdon, 1970; Kennington and Helgason, 1980), the application of the technique may be unique for different problem structures. This decomposition technique is specifically used here to obtain subproblems whose pure network structure can be exploited so that they can be easily and efficiently solved using a minimum-cost flow network algorithm. The details of the decomposition approach are quite standard and are omitted in this section, but provided in the Appendix for the interested reader. However, the derivation of the subproblems and the master problem are not immediately transparent, and are shown here to demonstrate the application of the decomposition technique to this particular problem.

Using (2i(s, k, n, t), ~i(s, k, n, t), 2(k, t), P(k, t)} as a solution vector, lki as the convex multipliers, and denoting sl(n, t) and s2(n, t) as slack variables, the master problem can be given as (Problem LP2)

lk K T S N

Minimize E ~] E ~] E [{h(s, k, n, t )~i(s , k, n, t))}lki i=l k = l t=l s=l n=l

+ {c(s, k, n, t) . f i(s, k, n, t)}lki ] (12)

K S

Y'~ ~_, {d(s, k, n).~i(s , k, n, t )} lk i+Sl(n, t) = X ( n , t) k = l s = l

= 1 . . . . . N, t = 1 . . . . . T, (13)

K S

~_, ~ _ , ~ i ( s , k , n , t ) l k i + s 2 ( n , t ) = Y ( n , t ) n = l . . . . ,N, t = l . . . . . T, (14) k = l s = l

lki = 1 , k = 1 , . . . , K , (15)

lki>/O, k = 1 . . . . . K, i= 1 , . . . , I k , (16)

sl(n, t), s2(n, t) >>.O, n = l , . . . , N , t = l . . . . . T. (17)

Note that the foregoing problem LP2 defined by (12)-(17) only contains the bundle constraints (13)-(14) and the K convexity constraints (15), apart from the nonnegativity restrictions.

Since feasibility of the bundle capacity constraints are taken care of in the master problem, the subproblems are left with the flow equilibrium conditions. Denoting the dual variables for constraints (13)-(15) by ul(n , t), u2(n, t), and v(k) respectively, the column to enter the basis of the master problem can be obtained by solving a subproblem for each part type k as follows. (Problem LP3)

T N S

Minimize Y'~ • • [{c(s, k, n, t) - u l ( n , t )d ( s , k, n)}x(s , k, n , / ) ] t = l n = l s = l

T - 1 N S T

+ E ~-, ] ~ _ , [ { h ( s , k , n , t ) - u 2 ( n , t ) } y ( s , k , n , t ) ] + ~ , b ( k , t ) z ( k , t ) (18) t = l n = l s = l t = l


subject to T

E b(k, t ) z (k , t) = w(k) , t = l

(19)

Y'~ x(s , k, n, t) + Y'~ y ( s , k, n, t - 1) n~A(s, k) nEA(s, k)

= ~_, x ( s + l , k , n , t ) + Y'. y ( s , k , n , t ) , n~A(s + l, k) nEA(s, k)

s = l . . . . . S - l , t = l . . . . ,T,

E x ( S , k , n , t ) + Y'. y ( S , k , n , t - 1 ) + z ( k , t ) n ~ A ( S , k ) n ~ A ( S , k )

(20)

= ~ y ( S , k , n , t ) + r ( k , t ) + z ( k , t - 1 ) , t = l . . . . ,T, n ~A(S, k)

(21)

O<<.z(k,t)~<5(k,t), t = l . . . . . T,

O < ~ y ( s , k , n , t ) < y ( s , k , n , t ) , s = l , . . . , S , t = l . . . . , T - l , n E A ( s , k ) ,

O < ~ x ( s , k , n , t ) ~ < 2 ( s , k , n , t ) , s = l . . . . . S, t = l . . . . ,T, n ~ A ( s , k ) , y ( s , k , n , O ) = y ( s , k , n , T ) = O , s = l . . . . . S, n ~ A ( s , k ) ,

r (k , t )>/O, t = l . . . . . T.

(22)

(23)

(24)

(25)

(26)

Problem LP3 defined by (18)-(26) contains the flow equilibrium conditions and upper bounds. Since it does not contain any bundle constraints, it has a minimum-cost flow pure network structure and can be exploited to efficiently generate columns that are candidates to enter the master basis. The details of the solution approach are provided in the Appendix.

In the next section, the experimental results are discussed.

4. Experimental results

The purpose of the experimentation performed with this model is twofold. First, a measure of the efficiency of the column generation technique used is desired as compared with the straightforward application of the simplex method. Second, while the objective of the model is to obtain the minimum-cost production plan, it would be interesting to examine the solution performance relative to other system performance measures such as system utilization, the number of bottleneck machines, the number of split lots, and the number of setups. Furthermore, since the model introduced here is intended for a flexible manufacturing system environment, it is desirable to analyze the impact on the above-mentioned system performance measures as the degree of routing flexibility changes.

Exogenous variables

A series of test problems have been generated by varying the following four model parameters: (a) number of part types, K; (b) number of machine groups, N; (c) magnitude of weekly demand for each part type, w(k), k -- 1 . . . . , K; and (d) degree of routing flexibility.

The values chosen for K and N determines the total number of rows and columns of the problem (problem size). In the experimentation conducted, the number of part types have been assigned the values K = 5,10,15,20 and the number of machine groups have been assigned the values N = 5,10,15,20.


Little data is available defining the mean values of K and N for actual FMS installations. However, in a recent survey, Jaikumar (1986) reports that the average number of part types produced by US FMS systems is 10, while for Japan the value is 93. Due to problem size limitations imposed by the software and computer system (Prime 9950) used here (5000 constraints and 10000 variables), the largest value chosen for K is 20. Jaikumar's (1986) survey also reports the average number of machine groups per system as 7 in the US and 6 in Japan. These values lie within the experimental range specified here for N.

The magnitude of weekly demand for each part type was generated for 50% and 85% system load based on the average processing time for each part type. The actual system utilization will, of course, vary according to the manner in which the model actually assigns the operations of each part to the machine groups under the existence of alternate routings. However, demand generation with 50% and 85% system load based on average processing times is intended to act as a surrogate for planned capacity slack in the system. These two figures are consistent with Jaikumar's (1986) report which states that the average system utilization rate for FMSs in the US is 52% and for Japan 84%.

The degree of flexibility is an important system design parameter for flexible manufacturing systems. While there are various forms of flexibility (see Brown et al., 1984; Jaikumar, 1984), the routing flexibility is an indication of the different machine groups that can process an operation of a given part type. Chang et al. (1987) give a formula for defining the flexibility index of a job shop. Consistent with Chang's definition, the degree of flexibility is defined here as the number of machine groups that can process each operation of each part type, divided by the total number of machine groups. The values are set at 0% (only one machine group can process each operation of each part type), 50% (approximately half the total number of machine groups can process each operation of each part type), and 80%.

Performance measures

Results depicting the effect on various measures of system performance (such as total cost and utilization) corresponding to different levels of flexibility are desired. Such results are critical since it is the flexibility attribute of a flexible manufacturing system that results in its competitive edge and efficiency for batch manufacturing environments.

The performance measures investigated here are (a) total cost of backordering, inventory and production; (b) total inventory in the system; (c) total number of setups in the system; (d) number of split lots in the system; (e) number of bottleneck machines; (f) average system utilization; and (g) total CPU time required to obtain a solution. Note that there are two categories of performance being tested here. The first six are related to system performance, while the last one is a test of algorithm speed.

Five days of FMS operation (480 minutes per day) are assumed per week, and all performance measures are weekly values. While setup times are assumed negligible in this model, it is desirable to compute the actual number of setups required if in fact setups are significant for a given system configuration. The total number of setups is defined as the number of discrete lots produced by the model. Introducing 8x(s, k, n, t) as an setup variable, then

6x(s , k , n, t) = ( 1 if x ( s , k, n, t) >0 , 0 otherwise.

Summing 8x(s, k, n, t) over s, k, n, t gives us the total number of setups. Since sequence dependency is not addressed, and since the model does not perform actual sequencing of the lots, the possibility of eliminating a setup by assigning the same operation of the same part type to the same machine group for two consecutive periods is not captured. The number of split lots is defined as the number of different machine groups that are actually assigned for the production of an operation of a part type (within the set A(s, k)) during the same time period. As the degree of flexibility increases, this value is expected to increase. A bottleneck machine is defined as one which runs over 90 percent of its capacity. The average

354

Table 1 Data generation

S. Ghosh, C. Gaimon / Routing flexibility and production scheduling

Factors Data generation

Experimental factors: Number of part types, K Number of machine groups, N Shop flexibility levels Capacity utilization levels

Parameters: Weekly demand for each part type, w(k) Set of alternate machine groups A(s, k) Number of operations for each part type, S Unit processing time, d(s, k, n) Unit production cost, c(s, k, n, t) Unit holding cost, h(s, k, n, t) Unit backordering cost, b(k, t) Capacity of machine groups X(n, t) Capacity of WlP buffers, Y(n, t) Upper bounds:

.~(s, k, n, t) ~(s, k, n, t) ~,(~, t)

5, 10, 15, 20 5, 10, 15, 20 0%, 50%, 80% 50%, 85%

capacity utilization levels shop flexibility levels Uniform (1, N) Uniform (1, 20) hours d(s, k, n) for all time periods 30% of corresponding c(s, k, n, t) 150% of average production cost 40 hours per week 30 units

X(n, t) for all s and k Y(n, t) for all s and k 500 units

system utilization is computed as the ratio of the actual weekly system production and the total weekly capacity of the system.

Note that setup time is not reflected in the calculation of system utilization. Also, due to the cost minimization approach used, sensitivity analysis with respect to unit costs is not performed; rather model results with respect to non cost performance measures are investigated. The values of model input parameters are either set or generated using uniform distributions. The settings of all model parameters are shown in Table 1.

Results

The values of the above system measures appear in Tables 2-9 corresponding to different representa- tions of problem size (K and N), degree of flexibility, and shop load conditions. It is quite evident that the time taken by the column generation technique is a small proportion of the time required by the LP approach. Furthermore, in most cases the algorithm terminates with a solution well within five percent of the LP optimum. It should be noted that the number of part types K, the number of machine groups N, and the degree of flexibility increase the problem size, and hence effect the computation times. Figure 5 compares the CPU times between Simplex and the column generation technique for K = 10 at 85 percent shop load. It is evident that as the number of variables increases, CPU times for the column generation approach is of a lesser degree polynomial than Simplex. The time advantage of the column generation technique is even more pronounced for larger problems. Recall that the total number of part types is related to the number of subproblems, but does not significantly effect the size of the master basis matrix. Therefore, since subproblems are very efficiently solved as single-commodity network flow problems, as the number of part types increases, the efficiency of the column generation algorithm relative to the LP approach becomes more pronounced. It should be noted that while the worst case performance of Simplex and column generation techniques to reach the optimal solution is the same, the average case performance of column generation techniques is usually superior.

Performance of some of the system measures reaffirms prior expectations. For example, as the degree of flexibility is increased, the number of bottleneck machines is reduced. This result is illustrated

Tab

le 2

K

= 5

at

50 p

erce

nt u

tili

zati

on

No.

F

lex-

of

ib

ilit

y m

ach.

in

dex

grps

.

Cos

t ($

)

Bac

k-

Inve

n-

Pro

duc-

or

deri

ng

tory

ti

on

Tot

al

Tot

al

inv.

un

its

No.

N

o.

No.

of

Avg

. N

o.

No.

N

o.

CP

U

LP

(S

impl

ex)

of

of

bott

le-

syst

em

of

of

of

tim

e C

PU

T

otal

se

tups

sp

lit

neck

ut

ili-

ro

ws

colu

mns

it

era-

(s

ecs)

ti

me

cost

lo

ts

mac

hine

s za

tion

ti

ons

(sec

s)

(% l

ess)

5 1

6078

54

25

750

3188

2 3

315

0 19

666

1998

1 4

456

0 19

498

1995

4

10

1 86

69

33

3949

8 48

200

5 0

0 25

107

2510

7 8

0 0

2320

0 23

200

15

1 75

56

0 44

502

5205

8 8

0 0

3414

4 34

144

12

0 0

3255

9 32

559

20

1 52

41

0 54

003

5924

4 10

0

0 43

151

4315

1 16

0

0 42

108

4210

8

47 0 0 35 0 0 0 0 0 0 0 0

31

58

62

40

47

61

42

51

71

60

63

71

0 3

62.3

15

0 18

5 36

1.

1 2.

4 0.

05

3 2

50.2

15

0 45

5 42

3.

4 7.

1 0.

01

5 2

43.6

15

0 59

0 6

5.4

10.9

0.

05

0 6

59.6

19

5 18

5 31

1.

3 2.

8 0.

01

1 2

40.3

19

5 72

5 29

2.

9 6.

1 0.

02

4 2

38.1

19

5 11

30

26

3.5

7.0

0.03

0 4

69.2

24

0 18

5 15

1.

1 2.

3 0.

02

4 1

50.7

24

0 11

30

33

3.7

7.7

0.01

7

1 43

.2

240

1670

25

4.

5 9.

1 0.

01

0 5

71.6

28

5 18

5 24

1.

9 3.

9 0.

04

6 2

62.8

28

5 14

00

23

3.5

7.2

0.01

13

1

50.0

28

5 20

75

33

5.5

11.3

0.

01

Tab

le 3

K

= 5

at

85 p

erce

nt u

tili

zati

on

No.

F

lex-

of

ib

ilit

y m

ach.

in

dex

grps

.

Cos

t ($

)

Bac

k-

Inve

n-

Pro

duc-

or

deri

ng

tory

ti

on

Tot

al

Tot

al

inv.

un

its

No.

N

o.

No.

of

Avg

. N

o.

No.

N

o.

CP

U

LP

(S

impl

ex)

of

of

bott

le-

syst

em

of

of

of

tim

e C

PU

T

otal

se

tups

sp

lit

neck

ut

ili-

ro

ws

colu

mns

it

era-

(s

ecs)

ti

me

cost

lo

ts

mac

hine

s za

tion

ti

ons

(sec

s)

(% l

ess)

5 1

5930

11

28

3475

0 41

808

3 26

80

643

1695

0 30

273

4 99

1 51

2 26

486

2798

9

10

1 88

99

279

4065

0 49

827

5 0

0 30

650

3060

5 8

0 0

2765

0 27

650

15

1 79

25

197

5000

8 58

130

8 0

0 37

882

3788

2 12

0

0 35

246

3524

6

20

1 54

62

189

5760

0 63

251

10

0 0

4032

0 0

16

0 0

3768

1 37

681

78

21

19

109 0 0 64

0 0 47

72 0

27

50

52

32

66

72

46

61

72

60

12

81

0 4

90.1

15

0 18

5 26

1.

0 2.

1 0.

01

3 2

85.2

15

0 45

5 29

5.

4 10

.9

0.04

4

2 81

.3

150

590

26

5.4

10.9

0.

03

0 6

87.6

19

5 18

5 18

1.

3 2.

7 0.

02

6 2

75.0

19

5 72

5 26

7.

7 15

.4

0.01

12

1

72.6

19

5 11

30

35

6.9

14.1

0.

01

0 5

83.3

24

0 18

5 39

1.

3 2.

8 0.

03

9 2

72.6

24

0 11

30

46

6.3

12.9

0.

04

18

1 68

.5

240

1670

29

7.

1 14

.4

0.01

0 5

85.9

28

5 18

5 40

1.

3 2.

9 0.

01

2 2

78.6

28

5 14

00

40

4.2

8.6

0.02

22

2

71.2

28

5 20

75

32

8.1

18.2

0.

01

Tab

le 4

K

= 1

0 at

50

perc

ent

util

izat

ion

No.

F

lex-

C

ost

($)

of

ibil

ity

Bac

k-

Inve

n-

Pro

duc-

T

otal

m

ach.

in

dex

orde

ring

to

ry

ton

grps

.

Tot

al

No.

N

o.

No.

of

Avg

. N

o.

No.

N

o.

CP

U

LP

(S

impl

ex)

inv.

of

of

bo

ttle

- sy

stem

of

of

of

ti

me

CP

U

Tot

al

unit

s se

tups

sp

lit

neck

ut

ili-

ro

ws

colu

mns

it

era-

(s

ecs)

ti

me

cost

lo

ts

mac

hine

s za

tion

ti

ons

(sec

s)

(% l

ess)

5 1

8097

57

6 44

400

5307

3 17

5 3

1320

28

1 29

966

3156

7 57

4

780

0 29

450

3023

0 0

10

1 85

86

489

4820

4 57

279

136

5 0

0 42

015

4201

5 0

8 0

0 40

069

4006

9 0

15

1 83

24

319

4941

9 58

062

123

8 0

0 37

788

3778

8 0

12

0 0

3359

1 33

591

0

20

1 85

99

0 54

545

6314

4 0

10

0 0

5684

5 56

845

0 16

0

0 56

830

5683

0 0

60

0 3

60.7

39

5 62

2 74

3.

9 19

.9

0.02

74

3

2 51

.2

395

1666

86

12

.9

65.1

0.

02

89

3 2

50.3

39

5 21

88

23

18.6

93

.4

0.01

63

0 6

72.9

44

0 62

2 64

3.

2 16

.3

0.01

18

6 17

2

60.3

44

0 27

10

59

19.2

29

6.3

0.02

21

8 19

1

58.0

44

0 42

76

24

22.3

31

1.6

0.04

107

0 5

66.7

48

5 62

2 11

4.

8 24

.3

0.01

20

8 21

3

55.2

48

5 42

76

68

27.8

33

9.4

0.02

28

4 42

1

52.1

48

5 63

64

11

22.2

31

1.0

0.02

156

0 4

68.3

53

0 62

2 10

5.

2 26

.3

0.01

18

7 51

2

52.1

53

0 53

20

47

16.6

13

3.5

0.02

21

8 51

3 2

50.2

53

0 79

30

67

23.5

26

7.7

0.03

Tab

le 5

K

= 1

0 at

85

perc

ent

util

izat

ion

No.

F

lex-

C

ost

($)

of

ibil

ity

Bac

k-

Inve

n-

Pro

duc-

T

otal

m

ach.

in

dex

orde

ring

to

ry

tion

gr

ps.

Tot

al

No.

N

o.

No.

of

Avg

. N

o.

No.

N

o.

CP

U

LP

(S

impl

ex)

inv.

of

of

bo

ttle

- sy

stem

of

of

of

ti

me

CP

U

Tot

al

unit

s se

tups

sp

lit

neck

ut

ili-

ro

ws

colu

mns

it

era-

(s

ecs)

ti

me

cost

lo

ts

mac

hine

s za

tion

ti

ons

(sec

s)

(% l

ess)

5 1

1731

1 58

9 45

803

3 13

80

561

4027

0 4

929

130

3393

7

10

1 17

031

732

5012

0 8

0 12

3 41

989

15

1 98

31

628

6785

3 8

7805

28

3 48

132

12

6704

0

4432

6

20

1 13

542

543

6957

5 10

14

704

0 51

667

16

5801

0

5123

5

6370

3 17

0 42

211

127

3503

2 78

6788

3 16

5 42

112

56

7831

2 15

2 56

220

52

5103

0 0

8366

0 12

6 66

371

0 57

036

0

51

0 5

93.9

39

5 62

2 53

2.

6 13

.2

0.01

60

3

3 80

.2

395

1666

60

7.

1 35

.8

0.01

62

3

1 78

.3

395

2188

53

12

.1

60.8

0.

01

109

0 7

88.3

44

0 62

2 38

3.

9 19

.9

0.04

14

0 15

2

73.5

44

0 42

76

72

24.2

32

1.5

0.01

121

0 7

85.7

48

5 62

2 22

5.

6 27

.9

0.01

24

2 20

3

73.2

48

5 42

76

50

22.3

28

7.6

0.01

25

6 39

1

70.6

48

5 63

64

43

31.3

30

2.1

0.01

150

0 6

86.6

53

0 62

2 19

6.

3 22

.4

0.02

26

0 62

2

70.2

53

0 53

20

25

15.7

23

0.7

0.05

28

2 71

2

68.3

53

0 79

30

68

32.1

37

9.4

0.01

Tab

le 6

K

= 1

5 at

50

perc

ent

util

izat

ion

No.

F

lex-

C

ost

($)

of

ibil

ity

Bac

k-

Inve

n-

Pro

duc-

m

ach.

in

dex

orde

ring

to

ry

tion

gr

ps.

Tot

al

Tot

al

inv.

un

its

NO

. N

O.

NO

. of

Avg

. N

O.

NO

. N

O.

CP

U

LP

(S

impl

ex)

of

of

bott

le-

syst

em

of

of

of

tim

e C

PU

T

otal

se

tups

sp

lit

neck

ut

ili-

ro

ws

colu

mns

it

era-

(s

ecs)

ti

me

cost

lo

ts

mac

hine

s za

tion

ti

ons

(sec

s)

(% l

ess)

5 1

1529

0 98

9 45

712

6199

1 20

2 3

1131

2 52

0 39

033

4494

1 13

7 4

969

312

3310

9 34

390

59

10

1 15

916

763

5132

2 68

001

183

5 10

300

482

4360

1 54

383

140

8 81

6 21

2 40

344

4137

2 57

15

1 16

013

703

5513

2 71

848

161

8 ~

12 ~

20

1 12

612

896

6095

1 10

a

16 a

a P

rob

lem

7446

0 21

2

61

78

87

67

125

139

112

145

0 4

61.3

80

5 13

56

60

8.3

57.8

0.

05

3 2

52.1

80

5 37

68

38

4.1

32.6

0.

05

5 1

50.6

80

5 49

74

40

15.3

31

0.7

0.03

0 6

75.6

85

0 13

56

41

12.1

16

2.1

0.03

19

3

60.2

85

0 61

80

30

22.5

49

6.5

0.02

25

2

57.5

85

0 97

98

27

39.7

57

8.1

0.03

0 5

66.3

89

5 13

56

53

15.7

24

0.5

0.02

0 5

65.2

94

0 13

56

61

17.6

32

5.3

0.03

size

exc

eeds

sto

rage

lim

itat

ion

for

XM

P.

Tab

le 7

K

= 1

5 at

85

perc

ent

util

izat

ion

No.

F

lex-

C

ost

($)

of

ibil

ity

Bac

k-

Inve

n-

Pro

duc-

m

ach.

in

dex

orde

ring

to

ry

tion

gr

ps.

Tot

al

Tot

al

inv.

un

its

No.

N

o.

No.

of

Avg

. N

o.

No.

N

o.

CP

U

LP

(S

impl

ex)

of

of

bott

le-

syst

em

of

of

of

tim

e C

PU

T

otal

se

tups

sp

lit

neck

ut

ili-

ro

ws

colu

mns

it

era-

(s

ecs)

ti

me

cost

lo

ts

mac

hine

s za

tion

ti

ons

(sec

s)

(% l

ess)

5 1

2160

3 12

31

5768

0 80

514

242

65

3 15

722

670

5030

0 66

692

175

91

4 90

15

535

4725

1 56

801

157

105

10

1 18

671

1567

63

323

8356

1 27

6 77

5

1232

0 92

0 55

202

6844

2 18

1 14

4 8

9903

73

6 50

375

6101

4 16

1 16

2

15

1 19

433

1501

69

270

9020

4 28

8 12

7 8

a 12

a

20

1 15

161

2113

75

310

9258

4 33

9 17

3 10

a

16 a

0 5

89.7

80

5 13

56

38

7.1

57.8

0.

03

6 3

83.1

80

5 37

68

72

15.3

24

0.1

0.04

8

2 81

.0

805

4974

51

20

.1

412.

1 0.

05

0 7

91.7

85

0 13

56

101

30.1

48

8.6

0.01

25

4

83.2

85

0 61

80

81

33.6

66

6.5

0.05

31

3

80.6

85

0 97

98

62

42.1

79

8.3

0.03

0 8

88.6

89

5 13

56

95

30.7

38

8.6

0.04

0 8

86.1

94

0 13

56

57

42.6

87

6.2

0.04

a P

robl

em s

ize

exce

eds

stor

age

lim

itat

ion

for

XM

P.

"~

Tab

le 8

K

= 2

0 at

50

perc

ent

util

izat

ion

No.

F

lex-

C

ost

($)

of

ibil

ity

Bac

k-

Inve

n-

Pro

duc-

T

otal

m

ach.

in

dex

orde

ring

to

ry

tion

gr

ps.

Tot

al

No.

N

o.

No.

of

Avg

. N

o.

No.

N

o.

CP

U

LP

(S

impl

ex)

inv.

of

of

bo

ttle

- sy

stem

of

of

of

ti

me

CP

U

Tot

al

unit

s se

tups

sp

lit

neck

ut

ili-

ro

ws

colu

mns

it

era-

(s

ecs)

ti

me

cost

lo

ts

mac

hine

s za

tion

ti

ons

(sec

s)

(% l

ess)

5 1

1727

3 10

32

5670

7 3

9671

57

1 49

222

4 53

20

303

4560

5

10

1 19

861

1233

63

201

5 a

8 a

15

1 20

253

1567

72

771

8 a

12 a

20

1 18

706

1633

85

622

10 ~

16

~

a P

robl

em s

ize

exce

eds

stor

age

lim

itat

ions

for

XM

P.

7501

2 18

7 59

464

122

5122

8 65

8429

5 20

7

9459

1 26

6

1059

91

281

71

0 4

68.7

14

50

2513

66

46

.2

766.

1 0.

05

89

17

2 57

.2

1450

71

39

82

55.6

98

0.5

0.02

10

9 25

2

55.3

14

50

9452

31

63

.3

1321

.1

0.03

88

0 6

65.3

14

95

2513

13

2 82

.1

2027

.3

0.02

141

0 7

66.2

15

40

2513

10

3 66

.2

1821

.7

0.04

188

0 10

71

.5

1585

25

13

97

73.2

22

17.1

0.

04

Tab

le 9

K

= 2

0 at

85

perc

ent

util

izat

ions

No.

F

lex-

C

ost

($)

of

ibil

ity

Bac

k-

Inve

n-

Pro

duc-

T

otal

m

ach.

in

dex

orde

ring

to

ry

tion

gr

ps.

Tot

al

No.

N

o.

No.

of

Avg

. N

o.

No.

N

o.

CP

U

LP

(S

impl

ex)

inv.

of

of

bo

ttle

- sy

stem

of

of

of

ti

me

CP

U

Tot

al

unit

s se

tups

sp

lit

neck

ut

ili-

ro

ws

colu

mns

it

era-

(s

ecs)

ti

me

cost

lo

ts

mac

hine

s za

tion

ti

ons

(sec

s)

(% l

ess)

¢5

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S. Ghosh, C. Gairnon / Routing flexibility and production scheduling 359

Z 0 ,?,

L)

5 0 0 -

450-

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350 -

300-

250"

2 0 0

150.

100 -

5 0 -

0 622

=

5320 7930

NUMBER OF COLUMNS o SIMPLEX + COLUMN GENERATION

Figure 5. Comparison of CPU times

7

i 0 % 50% 8 0 %

LEVEL OF FLEXIBILITY o N = 5 + N = 1 0 o N = 1 5 t, N = 2 0

Figure 6. Number of bottleneck machines vs flexibility

graphically in Figure 6 for K = 10 at 85 percent shop load, and is quite intuitive since one of the major advantages associated with the capability of alternate routes is to relieve bottlenecks. Similarly, total inventory in the system also decreases as flexibility increases. This relationship is important since it suggests that for flexible systems the space allocation required for WlP buffers can be significantly reduced relative to conventional, less flexible, systems. Less WlP inventory also implies less shop floor congestion and queues. As a result of reduction in workcenter queues, a reduction in flow time and increase in throughput rate of jobs in the system may be realized. The decrease in the measure of total cost that occurs as a result of increasing the routing flexibility is mainly obtained due to the decrease in inventory holding costs. In Figures 7 and 8, for K = 10 and at 85 percent shop load, the total cost and inventory levels are shown, respectively. It is particularly interesting to note the relationship between the value assigned for routing flexibility and the values obtained for total cost and inventory. It can be observed that per unit improvements in the values obtained for each performance measure are more pronounced when flexibility increases from 0 percent (conventional job shop) to 50 percent, than from 50 percent to 80 percent. Therefore, the marginal benefit of additional flexibility decreases as flexibility is increased (i.e. diminishing returns). There also seems to be evidence that the change in total cost and inventory is nonlinear with respect to changes in the value of N. In particular, rate of change (increase) in total cost is larger as N increases, while the rate of change (decrease) in inventory is also larger for higher values of N. Therefore, a larger system size (higher value of N) helps to decrease inventory

90 K = I O at 8 5 % S h o p L o a d

4 002

~ 70'

~ 60 • o~

0% 50 '% 8 0 %

LEVEL OF FLEXIBILITY o N = 5 + N = 1 0 o N = 1 5 • N = 2 0

Figure 7. Total cost vs flexibility

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110 .

100-

90 " 80 -

70 . 60 .

5 0 -

10- 0

0 % 50% 8 0 %

LEVEL OF FLEXIBILITY o N = 5 + N = 1 0 ¢ N = 1 5 A N = 2 0

Figure 8. Total inventory vs flexibility


94

92 ' =

90-

88.

86-

84-

82-

80-

76

74

72

0% 50% 80%

LEVEL OF FLEXIBILITY o N = 5 + N = 1 0 o N = 1 5 a N = 2 0

Figure 9. System uti l ization vs flexibility

faster, but at the same time increases total cost at an increasing rate. These results have important implications for the cost-benefit trade-off analysis that must be addressed in system design and acquisition.

In Figure 9, the effect of increasing flexibility on system utilization is illustrated for K = 10, at 85 percent shop load. The decrease in system utilization (while still meeting demand) that results from an increase in flexibility demonstrates that the system can tolerate higher demand rates or process more part types if flexibility is high. In general, as shop load is increased (higher demand), total cost increases due to an increase in backordering and inventory. The number of setups, split lots, and bottleneck machines also increases as shop load is increased. However, with a higher level of routing flexibility, the shop tolerates higher demand rates more efficiently and shows a lower system utilization level. Here, the system utilization level is acting as a surrogate measure for the production rate or throughput rate. A higher throughput rate indicates a higher system capacity. The importance of this result is discussed in Jaikumar's (1986) survey of systems in the US and Japan. Whether the increased capacity achieved by introducing flexibility in a manufacturing system is used to increase throughput rates or to increase the number of part types to be processed by the system should probably be dictated by market conditions.

In addition to the above results, the impact of the degree of flexibility on the number of setups and split lots are presented (see Figures 10 and 11 respectively) for K = 10~ at 85% shop load. It is shown that as flexibility increases, both the number of setups and split lots increase. Recall that the number of setups is equal to the number of discrete production lots required by the solution. As a result of increasing the flexibility, the availability of alternate machine groups for each operation of each part type

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a

6o'% LEVEL OF FLEXIBlUTY

D N=5 + N=10 o N=15 A N=20

Figure 10. N u m b e r of setups vs flexibility

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6O

5 0 -

4 0 -

3 0

20.

1 0

0 O% 50%

LEVEL OF FLEXIBIUTY a N = 5 + N=10 o N = 1 5 a N = 2 0

Figure 11. N u m b e r of split lots vs flexibility

K = I O at 85% S h o p L o a d

80 %

s. Ghosh, C. Gaimon / Routing flexibility and production scheduling 361

increases. Therefore, the potential number of different machine groups that may be assigned production for each operation of each part type, and the total number of production lots, may increase. This relationship may be interpreted as the cost associated with increasing routing flexibility. However, it should be noted that since setup costs and times are usually very small for FMS systems, the benefits from flexibility in terms of decrease in total cost, inventory, bottleneck machines, and higher throughput rates, are likely to far outweight the disadvantages associated with the increased setups. By the same token, increasing the flexibility of a system without drastically reducing setup times and costs (if they are significant) is likely to result in poor system efficiency.

5. Conclusion

A multiproduct, multiperiod, multistage network model has been presented for the planning of order release and production scheduling in a flexible manufacturing system environment under the existence of alternate routings with negligible setup times. The model disaggregates the weekly production requirements to daily production requirements, determines the daily production quota for each operation of each part type, and allocates the production quotas to machine groups given the flexibility of alternate routings. The model also provides the interface and linkage between an MRP component planning system and the shop scheduling system. The model is solved using a price-directive decomposition technique with column generation.

Experimentation is performed with the model for varying problem sizes to determine the impact of shop flexibility on total cost, inventory levels, existence of bottlenecks, shop utilization, and the number of setups and split lot production. The results indicate important cost-benefit trade-off implications for system design and acquisition. While the total cost, inventory levels, number of bottleneck machines, and throughput times decrease as flexibility increases, the existence of diminishing returns, as well as the increase in the number of setups (if significant) and split lot production, suggests that the benefits attained are not without associated costs. Therefore, increasing the routing flexibility of a system without a parallel decrease in setup costs (if infact setup costs are nonnegligible) is unlikely to reap significant benefits.

Finally, several extensions can be made to the model to incorporate different implementation scenarios. The model presented here requires that all demand be met over the finite planning horizon. If the production requirements are set too large, then the model may not be able to obtain a feasible solution. To eliminate this restriction, an arc connecting the source node to the sink node can be added for each part type, constraints (2) can be suitably modified, and a penalty term can be inserted in the objective function corresponding to the flow on these arcs. As a result, the relative settings of the unit costs and the value of the unit penalty assigned to unmet demand would capture management's policy on the strategic trade-off between minimizing operational costs and maximizing customer service (meeting demand on time). Also, while the model does not perform the actual sequencing of parts within the day, the time buckets can be reduced from days to hours. This would increase the problem size, but would address the actual hour-to-hour operation of the system in more detail. The model could also be implemented in a rolling horizon basis, where it is solved at the start of each day and only the first day's results are actually implemented. Any unfinished work or additional work done that day could be adjusted when the model is rerun the following day.

Appendix

The column generation procedure using the price-directive decomposition technique is described to solve the model defined by (1)-(11). Let a solution vector corresponding to flows in problem (1)-(11) be denoted by f k , for part type k, k = 1,2 . . . . . K, i.e.,

f k = { x ( s , k , n , t ) , y ( s , k , n , t ) , z ( k , t ) , r ( k , t ) ,

s = 1 . . . . . S, t = 1 . . . . . T , n ~ A ( s , k)}. (A1)

362 S. Ghosh, C. Gaimon /Routingflexibility and production scheduling

Assuming finite upper bounds in (7)-(9) (i.e. ~,, y and $ are not infinite), the feasible region given by the constraint sets (2)-(4) and (7)-(11) and for each part type k (k = 1 . . . . , K) is a bounded polyhedral set. Let this set be denoted by Pk, and let an extreme point i, i = 1 . . . . . I k, of Pk be denoted by the vector Pki" Therefore, Pkl, Pk2 . . . . . Pklk represents the set of extreme points of Pk, k = 1 . . . . . K, where Pki is given by

Pki = {Xi( S' k, n, t), ~i(s, k, n, t), 2(k, t), f ( k , t ) ;

s= 1, . . . ,S , t= 1 . . . . . T, n ~A(s , k)}. (A2)

Clearly, a particular fk can be expressed as a convex combination of the extreme points of Pk for lki >>. O, i = 1,..., I k, as follows:

lk

where

fk = ~, lkiPki (A3) i = 1

/k

E lki = 1. (A4) i = 1

Substituting Iki and Pki in (1), (5) and (6), and denoting sl(n, t) and sE(n, t) as slack variables, we have the master problem (problem LP2) defined by (12)-(17).

While the master problem is of more manageable size than problem LP1, the number of extreme points, Ik, for a particular part type k = 1 . . . . . K may be quite large. Explicitly solving this problem is rather impractical. However, explicit enumeration of all the extreme points of problem LP2 can be avoided by generating candidate entering columns for the master basis through the solutions of a sequence of subproblems.

Denoting the dual variables for constraints (13)-(15) by ul(n, t), u2(n, t), and v(k), respectively, the dual feasibility of the master problem LP2 is guaranteed by the following conditions:

(i) ul(n , t) ~< 0 corresponding to each sl(n, t),

(ii) u2( n, t) <~ 0 corresponding to each s2( n, t ), T N S

(iii) ~] ~ ~] [{c(s, k, n, t) - u , ( n , t )d(s , k, n)}2i(s, k, n, t)] t = l n = l s = l

T - 1 N S

+ • E Y'~ [{h(s, k, n, t) -uE(n , t)}~i(s, k, n, t)] - v ( k ) >10, t = l n = l s = l

corresponding to each Iki. Note that (iii) above represents the reduced cost corresponding to columns lki in problem LP2. If any

of conditions (i)-(iii) is violated, the column corresponding to sl(n, t), Sz(n, t) or Iki is a candidate to enter the basis of problem LP2. Following the conventional Simplex rule to identify the entering variable, we want to find the l~i column that violates condition (iii) the most. This information can be obtained by solving a subproblem (problem LP3), given by (18)-(26), for each part type k.

Problem LP3 contains the flow equilibrium conditions and upper bounds. Since it does not contain any bundle constraints, it has a minimum-cost flow pure network structure. Such special structures can be exploited and solved very efficiently using one of several network algorithms (e.g., Kennington and Helgason, 1980).

Let the optimal value for the k-th subproblem LP3 be denoted by ZLP3(k). The values of ZLP3(k), k = 1 . . . . . K, can be used to identify the entering column for the master problem LP2 as follows. Define

[Mi( }] V ( I ¢ , i ) = n Z L P 3 ( k ) - v ( k ) - Y ' . b ( k , t ) z ( k , t ) < 0 (A5) t = l


for iteration i. If V(/~, i) exists, then that column corresponding to lkl enters the basis of problem LP2 for a new iteration. Therefore, the master problem offers an improved feasible solution to the overall problem LP1, while the subproblems LP3 identify the entering column for the master problem, if any. To avoid the use of artificial variables (that is necessary if a bundle constraint (13)-(14) is violated), a good initial solution is obtained with a heuristic that is feasible to both, the master problem LP2 and the subproblems LP3, and hence the overall problem LP1. The heuristic works by first dividing the weekly production requirement for all part types equally for each day of the week, and then assigning them to the minimum-cost arcs for each time period subject to the availability of capacity. If capacity is unavailable, production is shifted to other arcs or other time periods until a feasible solution is obtained.

Note that the optimal solution to problem LP2 (and hence problem LP1) is obtained when V(k, i) >i O, k = 1 . . . . . K, holds at any iteration i. Computational evidence suggests (see Lasdon, 1970) that the rate of convergence for column generation schemes is typically slow near the optimal point. Therefore, here, the algorithm given below terminates (convergence achieved) when the difference between the values of problem LP1 obtained for two successive iterations is within 0.1 percent. The algorithm used to solve problem LP1 is presented below.

Step O. solution

(Initialization). Set iteration i = 1. Use the above procedure to obtain an initial feasible

Pki = {.~i(S, k , n, t) , ~i(s, k , n, t ) , 2 ( k , t ) , P(k, t);

s = l . . . . . S, t = l . . . . . T, n ~ A ( s , k ) } , k = l . . . . . K, i = 1 .

With this solution, the basis of the master problem LP2 is defined. Obtain the dual vectors (u~, u 2, v). Step 1. (Subproblems). Set i = i + 1 and save the current value of problem LP1 as SAVE. Solve the

subproblems using the latest dual vectors (ul, u2, v). Compute V(/c, i). If no such V(/~, i) exists, then the basic feasible solution corresponding to the last master step lkl(k = 1 . . . . . K), Pk~ (k = 1, . . . , K), and eq. (A3) is an optimal solution of the overall problem LP1 defined by (1)-(11). Otherwise, let the subproblem solution corresponding to V(/~, i) be denoted by Pki, and note that lki is a candidate to enter the master basis.

Step 2. (Master Step). Perform the necessary revised simplex forward transformation to update the column corresponding to lki , and enter it into the master basis. Reoptimize the master problem LP2 to obtain the new master basis, dual vectors (u 1, u 2, v), and multipliers lki (k = 1 . . . . . K). Obtain the current solution of problem LP1 using eq. (A3), and compute the value of the objective function of LP1 as ZLP1.

Compute GAP = I SAVE - ZLP11/SAVE Step 3. (Termination). If GAP is less than 0.001, then stop. The current solution meets the

convergence criteria defined. Otherwise go to Step 1. The solution for the master problem LP2 and the straightforward LP solution (for comparison

purposes) of problem LP1 have been achieved using the revised simplex method with the aid of Roy Marsten's (1979) XMP code for linear programming problems. Solutions to the subproblems have been obtained using the NETFLO algorithm (Kennington and Helgason, 1980) for single-commodity, minimal-cost network flow problems with upper bounds.

References

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Brown, J., Dubois, D., Rathmill, K., Sethi, S.P., and Stecke, K.E. (1984), "Classification of flexible manufacturing systems", The FMS Magazine April, 114-117.

Chang, Y.L., Matsuo, H., and Sullivan, R.S. (1986), "A beam search heuristic method for job scheduling in a flexible manufacturing system", Working Paper 86/87-4-3, School of Business, The University of Texas at Austin, TX.

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Edghill, J.S., and Davies, A. (1985), "Flexible manufacturing systems - The myth and reality", International Journal of Advanced Manufacturing Technology 1/1, 37-54.

Gaimon, C. (1986), "Optimal inventory, backlogging and machine loading in a serial, multi-stage multi-period production environment", International Journal of Production Research 24/3, 647-662.

Ghosh, S., and Gaimon, C. (1988), "Production scheduling in a flexible manufacturing system with setups", Working Paper, College of Business, Michigan State University, East Lansing, MI, forthcoming in l ie Transactions.

Groover, M.P. (1987), Automation, Production Systems and Computer-Integrated Manufacturing, Prentice-Hall, Englewood Cliffs, NJ.

Jaikumar, R. (1984), "Flexible manufacturing systems: A managerial perspective", Working Paper 1-784-078, Harvard Business School, Cambridge, MA.

Jaikumar, R. (1986), "Postindustrial manufacturing", Harvard Business Review 64/6, Nov-Dec, 69-76. Kennington, J.L., and Helgason, R.V. (1980), Algorithms for Network Programming, John Wiley, New York. Klahorst, H.T. (1981), "Flexible manufacturing systems: Combining elements to lower costs, add flexibility", Industrial Engineering

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Flexible Manufacturing Systems and Allied Areas, North-Holland, Amsterdam. Lasdon, L.S. (1970), Optimization Theory for Large Systems, MacMillan, New York. Marsten, R.E. (1979), "Experimental mathematical programming (XMP) code", University of Arizona. Shanker, K., and Tzen, Y.J. (1985), "A loading and dispatching problem in a random flexible manufacturing system", International

Journal of Production Research 23/3, 579-595. Stecke, K.E. (1983), "Formulation and solution of nonlinear integer production planning problems for flexible manufacturing

systems", Management Science 29/3, 273-288. Stecke, K.E., and Talbot, F.B. (1983), "Heuristic loading algorithms for flexible manufacturing systems", Proceedings of the 7th

International Conference on Production Research, Windsor, Canada. Voss, C.A. (1986), Managing New Manufacturing Technologies, Monograph No. 1, Operations Management Association, Waco, TX. Zahorik, A., Thomas, L.J. and Trigeiro, W.W. (1984), "Network programming models for production scheduling in multi-stage,

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routing flexibility and production scheduling in a flexible manufacturing system

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