Designing an optimal production system with inspection

European Journal of Operational Research 52 (1991) 4 5 - 5 4 North-Holland

45

Theory and Methodology

Designing an optimal production system with inspection C h r i s t o p h e r S. T a n g

Anderson Graduate School of Management, University of California, Los Angeles, Los Angeles, CA 90024, USA

Abstract: This paper presents a model for designing a production system with inspection. This production system has N stages, where each stage performs manufacturing operations that are followed by a potential inspection location. The proposed model integrates the issues of inspection location, inspection capacity, and production capacity. To design an economical system, a system design problem is defined that consists of finding the inspection location, the number of testers at each inspection location and the number of machines at each production stage. By exploring the special structure of a subproblem, the system design problem is formulated as an 'integrated' dynamic program that can be solved in pseudo-polynomial time. An illustrative example is presented.

Keywords: System design, quality control, resource allocation, dynamic programming, queueing system, combinatorial optimization

Introduction Competitive pressures have caused many manufacturers to place a greater emphasis on production planning and quality control. For instance, Toyota has focused on designing an efficient production system for reducing the work-in-process inventory while producing high quality automobiles (cf. Monden, 1983; Stout, 1985). This success story has triggered US manufacturers to advocate the concept of 'zero-defect' for increasing product quality and the concept of 'zero-inventory' for reducing work-in-process inventory (cf. Starr, 1989). This paper presents a model for designing a production system with inspection. Specifically, we consider a serial production system that consists of N stages. Each stage consists of a 'bank' of identical machines that are followed by a potential inspection location (or check-point), where each inspection location may consist of identical testers. We shall restrict our attention to a '100% inspection or none at all' policy that has been shown to be optimal under fairly general conditions (cf. Lindsay and Bishop, 1964). This policy has been adopted by many researchers for constructing various models in quality control (cf. Menipaz, 1978). To monitor the quality of the product, one needs to inspect the work-in-process inventory at various production stages (cf. Sinclair, 1978). The location and the capacity of the check-points are important Received August 1989; revised November 1989 0377-2217/91/$03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)

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C.S. Tang//Designing an optimal production system with inspection

factors for two reasons. First, high investment and high inspection costs are incurred at the inspection locations when they are 'too close' to each other, say, 'inspect every stage'. When the inspection locations are 'too far apart' from each other, processing effort is wasted due to 'late detection' (i.e., the defective items are detected many stages after the stage at which the defect occurs). Second, excessive inventory may accumulate at an inspection location which is the bottleneck of the system. The production capacity at each stage is another critical decision for designing a production system with low work-in-process inventory. For instance, machine investment is wasted when there is excessive production capacity. In addition, excessive inventory may accumulate at those bottleneck stages when the production capacity is not 'balanced'. To design a cost effective production system with inspection, many researchers have analyzed the interactions among inspection location, inspection capacity, and production capacity (see, for example, Ballou and Pazer, 1982; Lindsay and Bishop, 1964; Menipaz, 1978; White, 1969). We highlight two observations that are pertinent to our model. First, suppose that the defective items are 'identified' and 'disposed' only at an inspection location. Then the job arrival rate (or the workload) at a production stage may drop after each inspection location. Second, the management would like to identify the defective items as early as possible so that they can make necessary adjustments or corrections at the earliest time. The first observation indicates the relationship between the inspection location and the workload at each subsequent production stage. The second observation reveals the tradeoff between 'early detection' and the inspection cost. These two observations motivate us to construct a model that integrates the decisions of inspection location, inspection capacity, and production capacity when designing a production system with inspection. To our knowledge, the model presented in this paper is studied here for the first time. Given a budget for designing a production system with inspection, we consider a system design problem that determines the optimal inspection capacity at various locations and the optimal production capacity at each stage so that the total cost (in terms of production, inventory, inspection, and disposal costs) is minimized. We formulate this system design problem as an 'integrated' dynamic program that can be solved in pseudo-polynomial time. This paper is organized as follows. Section 1 reviews related literature. In Section 2 we present a queueing model for designing a production system with inspection. Section 3 analyzes a subpr0blem that can be formulated as a simple resource allocation problem that can be solved quickly by using marginal analysis. Then this subproblem is 'integrated' into a dynamic program for determining an optimal solution to the system design problem. An example is illustrated in Section 4. Section 5 concludes this paper.

1. Literature review

While our model integrates the decisions of inspection location, inspection capacity and production capacity, most models in quality control and production planning have treated these issues separately. For instance, given the production capacity of each stage, Ballou and Pazer (1982), Britney (1972), Lindsay and Bishop (1964), Eppen and Hurst (1974), Pruzan and Jackson (1967), and White (1966, 1969) develop various models for determining the optimal inspection location for different production systems. (However, these models do not address the issue of inspection capacity.) More recently, Menipaz (1978) provides an excellent survey on this subject. Given the system configuration (in terms of the production capacity and the inspection location), Lee and Yano (1988), and Tang (1988) develop different models for determining optimal production level; Lee and Rosenblatt (1986), and Tayi and Ballou (1988) construct models that analyze the interaction between the production lot size and reprocessing lot size. More recently, Porteus (1986) and Fine and Porteus (1987) investigate the tradeoff between lot size, setup reduction, and quality improvement. This paper presents a system design model that integrates three types of decisions: inspection location, inspection capacity, and production capacity. We hope that this model would provide some meaningful insights for designing production systems with inspection.

C.S. Tang / Designing an optimal production system with inspection

47

2. A queueing model for system design 2.1. A production system with inspection

Consider a multiple stage production line that produces a single product. As shown in Figure 1, the stages are numbered so that the first stage is denoted as stage 1 and the final stage is stage N. Each stage j has x/ (a decision variable) identical machines, where each machine has exponentially distributed service times with mean 1 / # j . Each stage performs manufacturing operations and each stage is followed by a potential inspection location. Let I _c {1 . . . . . N } (the location decision) represents the set of inspection locations. (For example, we inspect every stage if I = (1 . . . . . N }.) To guarantee 'zero defect', inspection must take place at the final stage N. Thus, N e I. For any given inspection location i ~ I, let Yi (a decision variable) denote the number of identical testers at location i, where each tester has exponential distributed service times with mean l / r , . For each item, let pj be the probability that 'no defect' is generated at stage j, where p / s are assumed to be known. Rework is not allowed; a defective item is disposed once detected at an inspection location 1. (We shall discuss the case in which each defective item can be repaired in the conclusion.) Finally, we assume that the inspection is perfect. Let d be the demand rate of the (non-defective) finished product. To meet the demand rate d, stage 1 must process at a rate R, where R=d/(p * P2 * "'" * PN)" We assume that items are arriving at stage 1 according to a Poisson process 2 with rate )~1 = R. In this case, the entire production-inspection system can be modeled as a tandem queue 3 (cf. Kleinrock, 1975). For any tandem queue with Poisson arrivals at the first stage, it is well known that each production stage j and each inspection location i ~ I behave like an M / M / x j queue and an M / M / y , queue, respectively. In sum, the production system is assumed to have the following characteristics: (a) Inspection is perfect. (b) No rework is allowed. (c) Each item must be inspected at the final stage. (d) The arrival process at the first stage is a Poisson process. (e) The processing times are exponentially distributed. (f) The demand is being satisfied in one production run, which implies no finished good inventory. However, in practice demand occurs over time may require more than one production run. In that case, the model needs to accommodate finished goods inventory (safety stock) and the time between production runs. potential inspection locations pro uction stages --->~--->

0 R= d/(p1*...*p~)

---> ®

®

]

[

i i

I t

* disposal of defective items

demand rate = d

&

disp I of defective items Figure 1.

i For example, rework is not allowed during the fabrication process of silicon wafers (cf. Chen et al., 1988). 2 The exponential service times and Poisson arrivals provide meaningful insights about system design. In the context of manufacturing system design, Solberg (1977) and Suri (1981) have shown that Poisson arrivals lead to fairly robust conclusions even when the arrivals are not Poisson. In any event, we can apply the approximation analysis developed by Whitt (1985) for the case when the service times are not exponential and the arrivals are not Poisson. 3 Our model can be extended to the case of multiple products by using the steady state analysis developed by Kelly (11979).

48

C.S. Tang / Designing an optimal production system with inspection

The above assumptions are quite reasonable for modelling manufacturing facility that produces electronic components such as silicon wafers (cf. Gise and Blanchard, 1986; Chen et al., 1988). First, in wafer fabrication, the testing procedure is elaborate so that nearly all defects can be detected at an inspection station. Second, due to technical difficulty, repairing a defective wafer is not practical. Third, to guarantee customer satisfaction, it is rather common to inspect all items before shipment. Finally, the processing time is uncertain because of the complexity of the operation performed at each stage. 2.2. Work in process inventory We now turn our attention to establish the relationship between the decisions (i.e., inspection location, inspection location, and production capacity) and the work in process inventory (i.e., the number of items waiting to be processed). The arrival rate at each production stage and each inspection location depend on the location at which the last inspection is taken place. To see that, let ~¢(i) be the 'last' inspection location before inspection location i, where &a(i)---max{k: k e I and k < i}. (For notational convenience, we let 0 e I and Po = 1.) Since the inspection is perfect, all items that pass the test at location .La(i) are non-defective up to stage ZP(i). Thus, the probability that an item is non-defective up to stage .5~'(i) is equal to P0 * P~ * "'" * P~to" In this case, it is easy to check from figure 2 that the arrival rate Aj for each production stage j between inspection locations Z~a(i) and i can be expressed as Aj=R*

P 0 * P~ * "'" * P-~(o

forj=Z~'(i)+l

. . . . . i.

(1)

The arrival rate at inspection station i, a~, can be expressed as a i = R * Po * Pl * " ' " * P~(i)"

(2)

Let L ( x j , #j, A j) and L ( y , , r~, a,) be the number of items waiting (in queue and in process) at production stage j and at inspection station i, respectively. Since each production stage j behaves like an M/M/xj queue and each inspection location i behaves like an M / M / y ~ queue, L ( x j , #j, A j) and L(Yi, ri, ai) can be expressed as xj

/x~_~

} xj 1

k + -I

t(y,,

~,, ~ ) . . . .

~ yi!(1 _ ai/(YiT,) )

k-----T---. +

~'i

arrival rate

arrival rate

AZ~I)+I

Ax = R*P0*Px*'''*Pz(i) \ \ \

=

R*P0*Pl*'''*Pz(~)

N\ _-->

+ --

y,!(1---~',))

~"

-->

z(i)+l

--->

z(i)+2

"'--> ---> .....

inspection

l[

i ~-->~ /

production stages

I

arrival rate

111 ~iI- R*p0*pI*...*pz(i)

I

- -->...

inspection

I[ [

I

, disposal of defective items

disposal of defective items

Figure2.

C.S. Tang / Designing an optimal production system with inspection

49

2.3. A system design problem We now define a system design problem that addresses the issues of inspection location, inspection capacity and production capacity. Let uj be the investment cost of a machine at stage j and let v, be the investment cost of a tester at location i ~ I. (Since 0 ~ I is a fictitious inspection location, v0 = 0.) Let BP and BI be the 'production' and 'inspection' budgets for the total investment in machines and testers, respectively. To guarantee that these budgets will not be exceeded, the number of machines (x j) and the number of testers (yi) must satisfy the following constraints:

E u j * xj

BP,

(3)

J

(4)

E Vi * Yi <~ BI. i~l

To ensure that the utilization factor is strictly less than 1 at each production stage and each inspection station, i.e., h J ( x j # j ) < 1 an a~/(yir,) < 1, it is easy to check that xj and y/ must satisfy the following constraints: IX j/g1] + l ~ < x j

forj=l

. . . . . N,

[a,/~ i ] + l <~yi f o r i ~ I ,

(5) (6)

where ~j and a i are given in equations (1) and (2). Let gj (hi) denote the inventory cost of holding one item (in queue or in process) at a production stage j (at an inspection location i). In this case the total inventory holding cost is equal to HC, where HC = ~"~gj * L ( x j , Xj, gj) + ~_~ h i * L ( y i, a i, ri). j

(7)

i~1

Let rj (s,) denote the operating cost of processing (inspecting) one unit at a production stage j (at an inspection station i). In this case the total operating cost (processing and inspecting costs) is equal to OC, where OC=E~* j

Xj + E s , * ~ i -

(8)

iEl

Let w~i), i be the expected cost of disposing of a defective item that is found to be defective at inspection location i given that the item was non-defective when inspected last at location .~(i). (White (1969) has provided a thorough discussion on how the disposal cost is computed.) Notice that all items that pass the test at inspection location &a(i) are non-defective up to stage Aa(i). This observation implies that the probability of an item to be non-defective at inspection location i given that it is non-defective at location Aa(i) is equal to p.~,)+~ * - - - * Pi. In this case the disposal rate of defective items at inspection location i is given by a i * (1 - P - ~ o + ~ * "'" * Pi), where a, is the arrival rate of jobs to be inspected at inspection location i. Since the defective items are detected and are disposed only at the inspection location, the disposal cost is incurred only at each inspection station i. Thus, the total disposal cost is equal to DC, where DC = E w.~,i).i * ai * (1 -P-~u)+' * "'" * Pi)"

(9)

i~l

For a given production budget and a given inspection budget, it is desirable to determine a production-inspection system so that the inventory holding (HC), operating (OC) and disposal (DC) costs are minimized. For this reason, we formulate the system design problem that finds the optimal inspection location (I), the optimal number of testers at each inspection location (yi), and the optimal number of

C.S. Tang / Designing an optimal production system with inspection

50

machine at each production stage (xi) as the following mathematical program. (P)

m i n m i n , min. 1

y,

L(xs, hj, #j)+ ~_,h i* L(y,, a,, ri)

HC+OC+DC=~_,gj*

x~

j

i~l

+~

* Xj+

j

+ E

~s,*,~, i~l

*

*

• ---

•

i~I

subject to

constraints (1) through (6).

Program (P) is a fairly complex non-linear integer program that has three types of discrete variables: the inspection location I, the number of testers Yi at each inspection location i ~ I, and the number of machines xj at each production stage j. Thus, program (P) is not easy to solve. For this reason, we shall develop an approach for determining an optimal solution to program (P) in the next section.

3. Dynamic programs Our approach consists of two major steps. The first step deals with a subproblem of program (P) that finds the optimal number of machines for those production stages that are located between any two consecutive inspection locations. This subproblem is called the machine planning problem, which is formulated as a simple resource allocation problem. The second step 'integrates' this simple subproblem into a dynamic program for determining an optimal solution to the system design problem.

3.1. A machine planning problem Suppose that the inspection locations I and the number of testers y, at each inspection location i ~ I are fixed. Then it remains to determine the number of machines x / a t each production stage j. In this case program (P) is reduced to the following program (Q)" (Q)

rain. x)

J

J

subject to As= R * P 0 * Pl * "'" * P~
E Uj

*

forj=&a(i)+l

xj ~ BP,

. . . . . i,

and f o r i ~ I ,

(10) (11)

J

[Tlj/'/*j] + 1 ~< xj

for j = 1 . . . . . N.

(12)

Notice that 0 ~ I and that N ~ I, the production stages 1 through N can be divided into 'intervals', where each interval consists of production stages Aa(i) + 1 through i that are located between two consecutive inspection locations Aa(i) and i. Program (Q) has a special structure that can be described in terms of the production stages within each 'interval' [Z~'(i)+ 1, i]. First, constraint (10) suggests that the arrival rate Xj's are constant for those production stages within the 'interval' [.$a(i) + 1, i] and that Xj depends only on the locations Aa(i) and i. This observation implies that the objective function of program (Q) can be expressed as i

E

E

i ~ l j =.~°(i) + 1

Next, suppose we 'divide' the total production budget (BP) into 'interval budgets', where each 'interval budget' (BP,.) is the production budget for machines at those stages within the 'interval' [Aa(i) + 1, i]. (Of course, Ei ~ +BP, = BP.) Then it is easy to check that program (Q) can be decomposed into subproblems,

C.S. Tang / Designing an optimal production system with inspection

51

where each subproblem determines the optimal number of machines xj for those production stages within the interval [£P(i) + 1, i]. This subproblem is called the machine planning problem, which can be expressed as i

(S,)

{ gj • / - (x j, L , "J) + rj * Xj }

O(.~q'(i) + 1, i, BP~) = nun. x:

subject to

j~So(i)+ 1

uj * x 9 ~ BPi,

Y'.

(11')

j =~(i ) + 1

[XJ/~9] + 1 ~ xj

for j - - ~ ( i ) +

1 . . . . . i,

(12')

where Xj= R * P0 * "'" * P-~(o" Observe that 0(~8(i) + 1, i, BPi) is the minimum total cost for allocating the 'interval' budget BP~ to production stages within the interval [Z#(i) + 1, i]. Notice that the Xj's, j = ~ ( i ) + 1. . . . . i, are constant for any two consecutive inspection locations ~ ( i ) and i. The only decision variables of program (Si) are the xj's, the number of machines at stage j, j =ZP(i) + 1 . . . . . i. Thus, the relevant term in the objective function is Y.jgj • L ( x j ) , where L ( x j ) is the number of items waiting (in queue or in service) at an M/M/xj queue. It is well known that L ( x j ) is convex and decreasing in x j (cf. Dyer and Proll, 1977). Hence, subproblem (S~) is a simple resource allocation problem with separable convex objective function (cf. Denardo, 1982). Consequently, subproblem (S,) can be solved quickly by using marginal analysis (cf. Fox, 1966). In brief, the marginal analysis method starts with a feasible solution, and it assigns an additional machine to a stage at which the 'marginal improvement' is maximized. This method terminates when no additional machine can be assigned without violating the budget constraint (11'). It is well known that marginal analysis is efficient and it takes no more than O(N * BP) number of operations to determine an optimal solution to subproblem (Si) (cf. Denardo, 1982). Oiven an interval budget (BP,) for the production stages within the interval [ ~ ( i ) + 1, i], the marginal analysis provides a simple way to determine the optimal number of machines xj and the minimum total cost 8 ( Z # ( i ) + 1, i, BP~) for those production stages ~ ( i ) + 1 through i. Utilizing this special structure of the machine planning problem (S~), we 'integrate' the subproblem (Si) into a dynamic program that finds the optimal inspection location I, the optimal number of testers y~, and the optimal number of machines xj. 3.2. Program integration

Suppose that the inspection location and inspection capacity have been set for locations 1 through i and that the production capacity has been set for production stages 1 through i. In addition, suppose that b i is the remaining inspection budget for testers at those 'potential' locations i + 1 through N and that bp is the remaining production budget for machines at those production stages i + 1 through N. Then the triplet (i, b i, bp) constitutes a state. Let ~p(i, b i, bp) denote the minimum total cost (in terms of holding, operating, and disposal costs) for allocating the inspection budget b i and the production budget bp for those (potential) locations and stages i + 1 through N. Let k be the 'next' inspection location after location i. Let 3 be the cost of the testers assigned to the inspection location k. Let fl be the 'interval budget' for machines at those production stages within the interval [i + 1, k]. In this case, we can 'integrate' O(i + 1, k, fl), the minimum cost function of the machine planning problem, into the following functional equation: ~b(i, bi, b p ) =

subject to

min.

k, Yk,fl, 8

(O(i+l,k,

fl)+hl

, L ( y k , ak,.rl~)+Sk ,OQ~

+wi, k * a/, * (1 - P i + l * "'" * p k ) + ~ b ( k , N > ~ k > ~ i + l, a k = R * Po * Pl * "'" * Pi, °k * Yk = 3,

+ 1
bi-&

bp-fl)} (13) (14) (15) (16)

52

C.S. Tang / Designing an optimal production system with inspection

The first term of the functional equation corresponds to the minimum cost for allocating a given budget fl to those production stages within the interval [i + 1, k]. The second, third and fourth terms are the holding cost, inspection cost and the disposal cost incurred at the inspection location k, respectively. The fifth term is the minimum cost for allocating the inspection budget b i - ~ (the production budget bp - fl) for locations (stages) k + 1 through N. Since k is the 'next' location after location i, constraint (14) specifies the arrival rate of items to be inspected at location k. Constraint (15) specifies the cost of the testers at location k while constraint (16) ensures that the utilization of the testers at location k is strictly less than 1. Since there is no penalty cost for surplus (i.e., unspent budget), ~k(N, a, b ) = 0 for any non-negative values of a and b. To solve the system planning problem, one needs to determine ~(0, BI, BP) by solving the above functional equation recursively. For any given state (i, b i, bp), constraint (13) suggests that the total number of potential locations k is bounded by N. Notice that the inspection budget for the testers at location k is bounded by BI, and that the production budget for the machines at those stages i + 1 through k is bounded by BP. In addition, observe that it takes no more than O ( N * BP) operations to evaluate the function 8(i + 1, k, fl). Thus, it takes no more than O(N 2 * BP 2 * BI) operations to evaluate the functional equation for a given state. Since there are N * BP * BI states, we can determine the optimal inspection locations, the optimal inspection capacity and the optimal production capacity within O ( N 3 * BP 3 * BI 2) operations. Notice that the total number of operations depends on the parameters BP and BI. Hence, the system planning problem can be solved in pseudo-polynomial time.

4. Numerical example We illustrate our model with a simple example. Consider a production system that has 3 stages. The probability that no defect is generated at a production stage is given as Pl -- 0.9, P2 = 0.7, and P3 = 0.8. The demand rate d is equal to 50. The service rate of each machines at each stage is given as ~1 = 1 0 0 , #2 = 80, and /% = 60, and the service rates of the testers zi's are all equal to 120. Each machine costs u j = $50000 for j = 1, 2, and 3, and each tester costs o i = $ 1 0 0 0 0 0 for i = 1, 2 and 3. There is a BP = $300 000 production budget and a BI = $300 000 inspection budget. It costs $2 to hold an item at each state and at each inspection location; i.e., g j = 2 for all j and h i = 2 for all i. In addition, it costs $3 to process (inspect) an item at each stage (each inspection location); i.e., rj = 3 for all j and s i = 3 for all i. Finally, the cost of disposing a defective item at location i given that the item was non-defective when inspected last at location *oca(i) is equal to wzeti), i, where w ~ u ) , i = 5 * ( i •~a(i) + 1). (Notice that the disposal cost depends on the 'distance' between two consecutive inspection locations.) By solving the 'integrated' dynamic program presented in Section 3, we can conclude that it is optimal to have 2 testers at location 2 and 1 tester at location 3. In this case we have I = {2, 3}, Y2 = 2 and Y3 = 1. In addition, it is optimal to have 2 machines at each production stage; i.e., x 1 = 2, x 2 = 2, and x 3 = 2. The minimum total cost (inventory, processing, inspection, and disposal costs) is equal to $1710.50. Clearly, our model can be used for budget planning. To do so, simply perform the parametric analysis on the production budget BP and the inspection budget BI. Thus, we can analyze the tradeoff between the initial investment in the production-inspection system (i.e., BP and BI) and the total operating cost (inventory, processing, inspection, and disposal costs).

5. Concluding remarks In this paper we have considered a system design problem that integrates the issues of inspection location, inspection capacity and production capacity for designing a production system with inspection. By exploring the structure of the problem, we have formulated the system design problem as an 'integrated' dynamic program that can be solved in pseudo-polynomial time.

C.S. Tang / Designing an optimal production system with inspection

53

Our model provides meaningful insights for designing a production system with inspection. Our model enables us to analyze the tradeoff between the initial investment in the production-inspection system and the total operating cost, which could be useful for budget planning. In our model, we assume that rework is not allowed. However, our model can be extended to the case in which repair work is performed right after each inspection location and each repaired item resumes (continues) its 'path' by entering the following production stage after the repair operation. More formally, our model can be extended to the case when the repaired items are not 're-entering' the system at an earlier stage. When the repaired items re-enter the system at an earlier stage, the subproblem (Q) will not be separable or decomposable. Nevertheless, the production-inspection system with feedback can be modelled as a queueing network. In that case, one may consider to apply the existing results in queueing networks (cf. Kelly, 1979; Bitran and Tirupati, 1988) to develop approach for determining an optimal system design.

Acknowledgement The author would like to thank one anonymous referee for providing helpful comments on an earlier version of this paper.

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