Models for evaluating the performance of serial and ... - Science Direct

European Journal of Operational Research68 (1993)49-68 North-Holland

49

Theory and Methodology

Models for evaluating the performance of serial and assembly MRP systems A.G.

Lagodimos

University of Cambridge, Engineering Department, Management Studies Group, Cambridge CB2 IR)L, United Kingdom Received November1990; revised June 1991 Abstract: This paper studies the problem of evaluating the level of customer service that can be achieved in serial and assembly production networks operating under MRP. Using results on the relationship between MRP and traditional inventory policies, we develop exact and distribution-free expressions for the inventory fluctuations at all stock points of these networks, and use them to derive analytic models for three customer service measures. All models make use of non-dimensional ratios, which drastically reduce the number of variables involved. These ratios are a direct extension to the multi-level context of previous ratios derived for single-level systems. Our analysis is concluded by an equivalence theorem, which allows the evaluation of the service obtained in two-echelon assembly networks through the direct application of the results obtained for serial networks.

Keywords: MRP; Inventory; Multi-echelon; Safety Stock; Service level I. Introduction

For over twenty years material requirements planning (MRP) has been the predominant production planning and control system in manufacturing. As a result, many MRP-related issues have become focal points of interest for both practitioners and academics. One issue that has dominated the research agenda is the development of strategies to allow the essentially deterministic MRP logic to operate effectively in uncertain environments. Most popular among the strategies proposed is the use of buffer inventories, introduced via some buffering policy. Three major buffering policies exist: planned safety stocks, safety lead times, and hedges to the master production schedule (MPS). A detailed discussion of these policies is given by New (1975) and Wijngaard and Wortmann (1985). Having selected a buffering policy, practitioners often need to quantify the buffering policy parameters in order to achieve some predetermined performance objective. Despite its importance (see Chu and Hayya, 1988), this problem has hardly been rigorously addressed. Most existing results are empirical or heuristic. Since they usually rely on single-echelon inventory theory, they fail to capture the internal interactions taking place in multi-echelon production networks, and so provide little insight into the effects of the policy parameters on buffering decisions. The aim of this paper is to develop and test analytic multi-echelon models for evaluating the customer service impact of buffering decisions in the context of serial and assembly production networks in MRP

Correspondenceto: Dr. A.G. Lagodimos, 6 FokylidouStreet, 106 73 Athens, Greece. 0377-2217/93/$06.00 © 1993 - ElsevierSciencePublishers B.V. All rights reserved

50

A.G. Lagodimos / Performance o f serial and assembly M R P systems

environments. These models can be used either to quantify the parameters of a buffering policy or to study the customer service impact of alternative network designs. Since we are modelling three different measures of service, these models can form the basis for establishing relationships between these measures (Wagner, 1980). We start by presenting some typical examples of previous research. In an early article, New (1975) stressed that the important consideration in establishing safety stocks for MRP is not the variation of the requirements plans but the forecast errors over the lead time. He then proposed a heuristic and a rule-of-thumb for evaluating safety stocks for independent and dependent demand items respectively. In a study addressing multi-echelon networks, but assuming that each stock point has individual service requirements, Meal (1979) proposed safety stock and safety lead time norms to compensate for the effects of quantity and timing uncertainty. These are the usual demand standard deviation multiples associated with the fractiles of the normal distribution. Similar models of various degrees of sophistication were later proposed by Chang et al. (1983) and Wijngaard and Wortmann (1985). The above papers considered one measure of customer service (the probability of a stockout per replenishment cycle) and effectively ignored the effects of lot-sizing rules other than lot-for-lot (I_AL) on the resulting service. Considering FOQ lot-sizing (see Orlicky, 1975, for a review of the lot-sizing rules used in MRP), Luyten (1987) proposed a heuristic for evaluating the service impact of the resulting increased cycle inventories. Assuming that stockouts occur only at the last period of a replenishment cycle, he evaluated the probability of a stockout per period as the weighted average of one term representing the cycle-based service and another representing a perfect service. A similar heuristic, tailored for the fill rate, was proposed by Etienne (1987). One characteristic of these studies is that, by relying on single-echelon inventory theory, they have effectively excluded the production network configuration from their analysis. In contrast, Miller (1979) proposed a heuristic to determine the MPS hedges necessary to achieve a required service level (the probability of a stockout per period) for a general serial network with L4L ordering throughout. Using the echelon inventory concept of Clark and Scarf (1960), this heuristic plans sufficient safety stock at each echelon, so as to cover the demand uncertainty over the entire lead time of this echelon. Having established echelon-based buffers, the stock point buffer inventories follow directly. Using simulation, Guerrero et al. (1986) tested the service performance of this heuristic and found it to perform as expected. In a recent study, however, Graves (1988) argued that Miller's heuristic over-estimates the resulting service levels. Finally, Wijngaard and Wortmann (1985) proposed Miller's heuristic to determine safety stocks in two-echelon assembly networks.

2. Assumptions and definitions

We will use the following notation and abbreviations: = Constant MPS requirements per period. C d(t) = Demand at period t. D = Demand in [t - L 1 , t], D = E~=od(t - L 1 + j ) . Ii( t ) = Net inventory of stock point i after the ordering decision at period t. Ji(t) = Inventory position of stock point i after the ordering decision at period t. t i = Lead time of stock point i. L i = Echelon lead time of stock point i, £ i = ~ = 1 Z j • Si = Order-up-to level of stock point i. Si = Reorder level of stock point i. SSi = Planned safety stock at stock point i. L~ 1 = Demand in [t - L i, t - L i - 1 ) , Yii = E/i-o d ( t - L^i ^ + j ) , for serial network. L 2 Ie2, Demand in It - L^ 21, t - L1), Y2I = ]~J=~-1d ( t - L21 + j ) , for assembly network. ^ Demand in [t - L 2 , , t -L2i_~), Y2i = EL2'-L2i-~-ld(t --L21 + J), for assembly network. Y2 i ^ Vi, Zi, M = Non-dimensional ratios defined by expressions (10).

A.G. Lagodimos / Performance of serial and assembly MRP systems

5!

Figure 1. Schematic representation of the production networks studied

a, fl, V

= Measures of service. = Lead time of stock point i augmented by the planned safety lead time at that stock point. Ai /~,tr = M e a n and standard deviation of the period demand. A = H a t to indicate echelon variables and parameters. MPS = Master Production Schedule. MRP = Material Requirements Planning. L4L -- Lot-for-Lot. FOQ = Fixed O r d e r Quantity. FPR = Fixed Period Requirements. SIC = Statistical Inventory Control. = Stock Point i. WIP = Work-in-Progress. The serial and assembly networks we study are shown in Figure 1, where stock points are depicted by a triangle and processes by a circle. Both networks operate under MRP, implementing a policy of rolling schedules. So, although planning, based on the standard M R P logic, takes place at each period, only the production and purchase orders scheduled for release at the current period are implemented. Other than some additional assumptions determining the stochastic nature of the imposed demand, our analysis assumes that: 1. the MPS remains constant over time, 2. the entire demand not satisfied from stock is backordered, 3. there are no capacity constraints and the lead times are fixed, 4. production and purchase orders that cannot be immediately satisfied are split, 5. the rescheduling of open orders (that is, production orders that have already been released) is not allowed, 6. all uncertainty is due to the inability of the system to exactly predict the external demand, 7. I A L lot-sizing is used at all stock points except those at the supply side of each network, which can use any lot-sizing rule.

A.G. Lagodimos / Performance of serial and assembly MRP systems

52

We now define some of the terms we use. For any stock point, the net inventory is the inventory that it physically holds, reduced by the amount of backorders. The net inventory, increased by its associated work-in-progress (WIP) (being the orders released but not yet received) gives the inventory position. In addition, we use the echelon net inventory and echelon inventory position of a stock point (see Clark and Scarf, 1960). The former is defined as the sum of the net inventory and all inventory that has passed through this stock point but has not left the system. The echelon net inventory, increased by its associated WIP, gives the echelon inventory position. For the end stock point of a network the local and the echelon inventory variables and parameters are identical. We are interested in three service measures. While the measures are used in a variety of forms (see Silver and Peterson 1985, for a review of some existing variants together with the terminology used in practice), the form that we consider here is~.the non-dimensional one in Schneider (1981). Details concerning the representation of these measures as functions of the end stock point inventory position can be found in the literature (see, for example, Schneider, 1981, and Lagodimos, 1990a). All the expressions we will use are valid under the common assumption that backorders are given priority over the current demand. 1. The a measure: This is defined as the fraction of periods for which the current demand is completely satisfied and corresponds to the probability that an end item shortage occurs at an arbitrary period. Hence a = P r [ J l ( t - L1) > D ] .

(1)

2. The 3' measure: This is related to the average cumulative backorders in the system expressed as a fraction of the average current demand and is defined as 1 /z

3' = 1 - - - E [ D - J I ( t - L O I D

>Jl(t-L1)]Pr[O

>J,(t-L1)

].

(2)

3. The/3 measure or fill rate: This is defined as the fraction of the average period demand satisfied on request, and is the measure most often used in practice. It can be shown that

1 [3 = 1 - - - { E [ D - J I ( t - L 1 ) Ix

ID > J,(t-L1)]Pr[D

> J,(t-L,)]

- E [ Y 1 - J , ( t - L , ) I Y 1 > J , ( t - L , ) ] P r [ Y 1 > J , ( t - t , ) ] }.

(3)

As they were expressed, all three measures represent averages over time. Hence, provided we can determine the effects of some given lot-sizing rule on the stochastic behaviour of Jl(t), we can develop exact service models for this rule.

3. The MRP ordering policies One prerequisite to the development of multi-echelon service models for any inventory policy is to establish how the ordering decisions relate to the policy parameters and the inventory content at all stock points of a network. While for some inventory policies this is relatively straightforward, there are two major problems associated with MRP: (i) the algorithmic form of the MRP ordering logic, and (ii) the apparently local nature of the information used for the release of production orders at each stock point. Recent research, however, has provided us with some insight into the operation of MRP systems. The first attempt to study the MRP ordering policies is the simulation-based investigation by Lambrecht et al. (1984), dealing with a two-echelon serial network. In an environment effectively determined by the assumptions of Section 2, they observed that the MRP policies are nothing m o r e than (s, S) policies, provided that echelon inventory quantities are considered. They demonstrated this through a set of tables, representing the results of a few realisations of their simulations. Arguments essentially

A.G. Lagodimos / Performance of serial and assembly MRP systems

53

supporting these observations were later presented by Wijngaard (1984) and Wijngaard and Wortmann (1985). However, there are shortcomings associated with these results: • there is no rigorous justification of their validity, • there is no evidence that they apply in less restricted environments, as when the MPS is not constant, or when there is supply uncertainty, or when different lot-sizing rules are used, • they provide very little information on the relationship between the (s, S) policy parameters and the parameters of the MRP system such as the lot-sizing rules or the parameters of the buffering policy used. Aiming to overcome these shortcomings, Anderson and Lagodimos (1989) developed an algebraic framework to study the MRP ordering policies in single-level MRP environments. Under assumptions identical to those of Lambrecht et al. (1984), they showed that the MRP ordering logic implements stationary periodic review (s, S) ordering policies and demonstrated how the parameters of these policies relate to those of the MRP system. In a later study, Lagodimos (1990a) extended this work to multi-echelon production networks and showed that the results of Lambrecht et al. (1984) apply only under very particular conditions. Without giving the details of this work, we present without proof two necessary results. Under the assumptions we are considering here, these are extensions to the results of Lambrecht et al. (1984) and are virtually identical to those reported by Axs~iter and Rosling (1990).

Proposition 1. For n-echelon serial networks operating under the assumptions of Section 2, the MRP ordering logic implements periodic review two-critical-number ordering policies in terms of the echelon inventory position at each stock point. Specifically, at all except the lowermost stock point, it implements a periodic review (s, S)policy, with parameters Sk~-gk = Essiq

-

i=1

EAiq-1 i=1

)

c.

(4)

The ordering policy at the lowermost stock point depends on the lot-sizing rule. Particularly, under: 1. Fixed Period Requirements (FPR) lot-sizing, with batch quantities of p n periods, it is a periodic review (s, S) policy with parameters hi + 1 c

SSi +

S n -~-

i=1

i

and

S,

SSi +

=

i=1

Ai + p , c.

(5)

i

2. Fixed Order Quantity (FOQ) lot-sizing, with batch quantity of Q units, it is a periodic review (nQ, s) policy with parameter g,, --

SSi + i=1

Ai + 1 c.

(6)

i

Proposition 2. For two-echelon assembly networks operating under the assumptions of Section 2, each of the serial networks, formed by the final assembly and one component, operate as independent two-echelon serial networks facing identical demand processes. The satisfaction of the orders released by the final assembly, however, is contingent on the availability of all components. There is an important implication of these results. By establishing formal relationships between the ordering policies of MRP and those of SIC systems, they permit the interchangeability of the findings associated with the two systems that have for years being considered as distinct. Hence, the results of this paper also apply to production networks operating under the relevant SIC policies and form part of the established body of research on multi-echelon SIC systems. We should, in fact, note that similar results to those in Sections 4.1 and 6 were independently derived by Rosling (1989), when determining cost-optimal policies for a particular SIC system, the base stock control system.

54

A.G. ISagodimos / Performanceof serialand assemblyMRPsystems

4. Serial networks

In this section, we establish exact and distribution-free expressions for the dynamic inventory fluctuations at all stock points of n-echelon serial networks operating under MRP. Assuming normally distributed period demands, we subsequently use these results to develop exact models for a, /3 and 3'. 4.1. The system dynamics In order to develop service models, we need to characterise the dynamic behaviour of Jl(t - L 1 ) , the inventory position of the end stock point (SP1), immediately after ordering at some period t - L 1. Using the variables Y~, ~(t) and f ( t ) defined in Section 2, we study the behaviour of the inventory content of the system over time in terms of these variables and the echelon ordering parameters in Proposition 1. Lemma 1. For an n-echelon serial network, the echelon inventory position, Jk(t), of any stock point k with k -~ n, at any period t, can be expressed as: fk(t) = min[fk+x(t), Ski.

(7)

Proof. See Appendix. Theorem 1. For an n-echelon serial network, the echelon inventory position, Jk(t -- L k), of any stock point k

with k v~ n, at any period t - L k, is given by:

p=k+l ~ rp,

Jk(t--£k) = k2i-2k+wFi-wF k=l

k ["] v +

p=2

k~i

Vpxp>Zi+wF i

.

(15)

p=2

Using this and introducing the non-dimensional ratios from (10) in (14), we get

Vpxp-2,-wF,l~

~=;E v +

(1-y)M= i

Pr(~¢/).

(16)

p=2

As in the case of a, each term in (16) corresponds to an integral over the probability space discussed earlier. It might be useful to point out that ( 1 - y ) M has a physical interpretation: it represents the average backorders in the system, made non-dimensional by dividing them by the standard deviation of D. We can also derive a model for the fill rate ft. From (3) we see that fl consists of two otherwise identical terms, except that YI replaces D in the second term. The first term is identical to the expression derived for y. Hence, without giving the details, we can show that

,i=1 -E

Vpxv-2i-M-wFil~

,. P r ( ~ ' i )

,

(17)

1 where

~,.=

Vpxpk=l

k~i

Y] V p x p > 2 i - 2 k + w F ~ . - w F p=2

k [7

>Zi+M+wF

i

,

(18)

A.G. Lagodimos / Performance of serial and assembly MRP systems

i!:!i!i;i!iii!i

57

0

,ii!iii!ilZ!iii~ii!iliZiiii!iiiiiiii~ ~'~:

x z.+

Figure 2. Region of integration for evaluating a over the standarised bivariate normal probability plane

and where ~¢i is given by (15). Since x 1 is also a standardised normal variable, the probability space in the second term of (17) is identical to that of the first term.

5.

Two-echelon serial networks

One problem associated with the exact models presented earlier is that they cannot be evaluated analytically. Concentrating on two-echelon serial networks with L4L ordering at both stock points, we develop and test analytic approximations that could be used in practice.

5.1. Analytic seruice approximations We first consider a. From (13), with n = 2 and w = 0, we can write:

a = Pr[v21

andx222 andx2>A }, A----V7

(23)

59

A.G. Lagodimos / Performanceof serialand assemblyMRP systems

Several transformations are necessary to evaluate the integrals above. It can be shown (see Lagodimos, 1990a), however, that the following exact expression for 3' holds:

(]-T)M---[1-~R(z1)l[v2~(A)+qb(A)(22-Jl) ]+ ~(A)~(Z,) ~

[ 1 - q~(B)] - 2 2 ( 1 - ~),

where

22 21(v? + 1) , v2~+ 1 -

B=

(24)

and A is given by (23). Of all terms in the above expression, only a cannot be evaluated analytically. By introducing c~ for a, an analytic approximation, 9, is obtained. We finally consider/3. Using (17) and (18), with n = 2 and w = 0, we obtain: (1 - / 3 ) M = E ( v - 2 , [ ~¢l)Pr(~gl) + E ( v I d 2 ) P r ( ~ 2 ) + V 2 E ( x 2 [ d 2 ) P r ( ~ 2 ) - Z2Pr ( H 2 )

VII I~'1 Pr(~l)+E(xll~'2)Pr(,~'2)

-g 1 E x 1

V!

'

(25)

where ~-~l =

Observe (22 + M ) / V maintaining algebra, we

x 1 > ~--~-1

and x 2 < A

,

~2=

xl + - - x z > - - a n d

V1

V1

x2 > A

•

that T~ = ( 1 - y ) M , and that T 2 has the same form as T1, except that (21 + M ) / V 1 , 1 and V z / V l have replaced Z,1, 22 and V2 respectively. Hence, we can evaluate T 2 by simply the equivalences among the corresponding parameters of the two expressions. After some obtain the following exact expression for/3:

+¢V22+1¢5

- v,

~

a- •

[1-O(B)I-Zz(1-a)

~(A) [ZI+M'

+ e(A)

~22+V2(21Vl+M)[I_dp(C)]

22 + M ( ] _ ~:)) (26)

60

A.G. Lagodimos / Performance of serial and assembly MRP systems

where

~---(~

( 21 "~-M ) V1 - ½ { a ( 2 ( q ~ l + 01)' 6 1 ) - a ( 2 ~ P l , al) },

01 = COS-

~1

~

,

~1 = COS-1

I ( 2 1 + M)2V2 + (22-- Zl) ^ 2VI2 VIV2

~

. . . .

C~

(22-21)V?-(Z

,

1+M)V 2

V1V2~~ + V 2

'

and where A and B are given by (23) and (24). Using O ( . , • ) from (22) in order to approximate s¢ and introducing 5, we can obtain an analytic approximation for the fill rate/3. 5.2. Simulation results

The proposed service approximations were tested by simulation. Using a specially built multi-level, multi-product M R P simulator written in FORTRAN 77, we simulated the two-echelon serial network studied earlier. The following p a r a m e t e r s were held constant throughout: the MPS (c = 1000), the lead times at both stock points ( L 1 = 15 and L 2 = 36) and the period demand distribution N(1000, 2002). All

i i i i i i i i i i

100

....

I i i i l l l l l

I

2000 90

80

70

60

50

40

30

. . . .

0.0

I

0.2

. . . .

I

0.4

. . . .

I

. . . .

0.6

I

0.8

. . . .

1.0

Positioning R a t i o p Figure 4. Predicted simulated a values as function of the positioning ratio p with parameter SS~ + SS2

A.G. Lagodimos / Performance o f serial and assembly M R P systems

,

0.8

,

,

,

I

,

,

,

,

I

. . . .

I

. . . .

I

61

,

,

,

cD 0 0.6

t~ . F.4

0.4 r"4 . ~

I O

Z

Q)

0.2.

2OOO

0.0

.... 0.0

I .... 0.2

I ....

I ....

0.4

0.6

Positioning

Ratio

I .... 0.8

f.O

p

Figure 5. Predicted and simulated (1 - y ) M values as function of the positioning ratio p with parameter S S 1 -~- S S 2

buffer inventories resulted exclusively from the use of planned safety stocks. Two p a r a m e t e r s were varied, to form a set of 22 cases: the system-wide safety stocks SS 1 + SS 2 (1200 and 2000) and their allocation, expressed as the positioning ratio p = SS2/(SS 1 + SS2) , between the two stock points (from 0 to 1 in increments of 0.1). Our experimental procedure consisted of two phases. At first, each of the 22 cases was simulated once (using the same random d e m a n d stream with a run length of 4000 periods), in order to obtain some indication of the behaviour of the service models of interest. Subsequently, three cases were selected and thirty-five dependent replications were performed. Figures 4 and 5 show the simulation results (solid lines) and the analytic approximations (dotted lines) for a and the non-dimensional average backorders (1 - y ) M , as functions of the positioning ratio p, with the two system-wide buffer inventory quantities as parameters. The approximations were obtained by first forming the non-dimensional ratios discussed earlier (using the p a r a m e t e r s of the theoretical d e m a n d distribution) and then substituting in the appropriate expressions. For example, the value a = 0.6401, shown in Figure 4 for p = 0.6 and SS] + SS 2 = 1200, corresponds to 22 = 1.5, V2 = 1.5 and 21 = 0.9. Both figures reveal a good agreement between the approximations and the simulation results. However, a consistent difference between the simulated and the predicted values is present. In order to assess its importance, the simulations corresponding to three p values (0, 0.5 and 1.0), all with 1200 units of systemwide safety stock, were repeated thirty-five times. It was found (see Lagodimos, 1990a) that all the predictions were well within the 95% confidence interval for the mean of the experimental values. Thus, we concluded that the biases in the graphs were due to some deviation of the sampled demand

A.G. Lagodimos / Performance of serial and assembly MRP systems

62

from its theoretical distribution (which formed the basis for obtaining the predictive expression) caused by the relatively short simulation runs used. The consistency of the bias accross all positioning ratios p (recall that we used the same random demand stream in all simulations) further supports this conclusion.

5.3. Comparison with previous results As we have noted, there have been no other exact service models for serial networks under MRP. There are two studies, however, whose results are comparable to ours. Miller (1979) developed a heuristic for jointly determining safety stock norms at all stock points of n-echelon serial networks (with L4L ordering throughout) in order to achieve a required a level: 21 = q0-1(a)

and

2k l+EV

= q~-J(a)

for all k ~ 1.

(27)

2

p=2

The end stock point safety stock planned by (27) corresponds to the quantity needed by a single-level system to achieve the same level of service (this can be seen from Figure 2 with 22 tending to infinity). This implies that (27) effectively ignores the impact that all stock points upstream could have on the system service and, consequently, would tend to overestimate it. Moreover, when used to evaluate safety stocks in order to achieve some desired service, this heuristic will result in an understocked system. A similar comment, but using different arguments, was made by Graves (1988). Donselaar and Wijngaard (1986) considered a two-echelon network under a SIC ordering policy which, on the basis of Proposition 1, is identical to that implemented by MRP with L4L ordering throughout. They developed an exact expression for a that is, in fact, equivalent to (19). They also proposed an empirical approximation for a, valid only when 21 = 22//~/1 + V2 , which corresponds to the relationship between 21 and 22 proposed by Miller. We can express this approximation as: ( 1 - & ' ) = [1 + q b ( 2 1 ) 1 / 2 ]

[1 - qb(Z1) ] .

(28)

Table 1 gives results corresponding to our analytic approximation (21) and to expressions (27) (shown as &") and (28) for different values of V2 and 22. In all cases, 21 was selected to satisfy the constraint for which (27) and (28) are valid. For example, for 22 = 2.5 and V2 = 1.5, then 21 --- 1.3868. From these results we can see that &' is often as much as 13% different from our approximation. Notice, however, the behaviour of Miller's heuristic, whose performance markedly differs from that of the other two approximations and gives estimates which could become as much as 54% higher than &. 6. Two-echelon assembly networks

The assembly networks we study consist of m distinct components and one final assembly, as shown in Figure 1. Without loss of generality, we will use the conventions: 1. the components are indexed in order of increasing lead time; so k > j implies that L2~ > Le/ 2. exactly one unit from each component is required to produce one final assembly. As for serial networks, in order to study the dynamics of assembly networks, we will observe their inventory behaviour over time. Using the notation Y21 and Y2k shown in Section 2, and Lak = L2k + L 1 for the echelon lead time of the assembly branch containing SPE~, we can state: ^

Lemma 3.

period t -

For two-echelon assembly networks, the inventoryposition Jl(t is given by

tl)

of

the final assembly at any

Z1

Jl(t-Zl)=

l . . " > 5 2 > 8 1 .

(32)

Since f~/> f~k for all j > k, these inequalities imply that Sj > Sk for L i > Lk- This, however, is not always true for assembly networks. Since the buffer quantities that can be allocated at the components are unconstrained, we can always think of an allocation which gives S2j > S2k for some L2~