Combining make to order and make to stock

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OR Spektrum (1998) 20:73-81 @ Springer-Verlag 1998 Combining make to order and make to stock Ivo J.B.F. Adan, Jan van der Wal Eindhoven University of Technology, Department of Mathematics and Computing Science, P.O.Box 513, NL-5600 MB Eindhoven, The Netherlands Received: 12 November 1996 / Accepted: l0 November 1997 Abstract. In inventory control and production planning one is tempted to use one of two strategies: produce all de- mand to stock or produce all demand to order. The disad- vantages are well-known. In the 'make everything to order' case (MTO) the response times may become quite long if the load is high, in the 'make everything to stock' case (MTS) one gets an enormous inventory if the number of different products is large. In this paper we study two simple models which combine MTO and MTS, and investigate the effect of combining MTO and MTS on the production lead times. Zusammenfassung. In Lagerhaltung und Produktionspla- nung werden tiberwiegend zwei alternative Strategien be- trachtet: Produktion auf Bestellung oder auf Vorrat. Die Nachteile sind bekannt. Die erste Strategic kann zu sehr lan- gen Durchlaufzeiten ftihren, die zweite zu hohen Best~inden, insbesondere bei groBer Produktanzahl. In dieser Arbeit wer- den zwei Modelle betrachtet, in denen die beiden Betrach- tungsweisen kombiniert werden. Key words: Inventory control, production planning, differ- ence equations, queueing model, Markov process Schliisselwiirter: Lagerhaltung, Produktionsplanung, Dif- ferenzengleichungen, Warteschlangen, Markoff-Prozeg 1 Introduction Lesser and lesser production systems are fully Make to Stock (MTS) organized. In MTS systems the clients receive their products from a warehouse near the client. A lot of re- search concerns the performance and control of these sys- tems (multi-echelon inventory control). Some other systems are completely Make to order (MTO). No product is made without a client. The analysis of these systems calls for queueing models. An important problem in MTO systems is the relation between the lead time and the utilization of Correspondence to: I.J.B.F. Adan the capacity. It is well-known that the lead time grows with the utilization as 1/(1 - p) . Utilizations of 80% are often considered to be far to low. On the other hand a utilization of 95% may give lead times which are not acceptable any more. If one wants high utilizations and short lead times then one has to be able to deal with the natural day to day fluctu- ations in the demand (we are not thinking of seasonal vari- ations). One possibility is flexible capacity. Another option studied in this paper is the combination of MTO with some MTS. We will consider two models in which this mixture is studied. In the first one we assume that there are two types of products, standard products and non-standard products. If there are no orders the production resource (e.g. machine or worker) will produce a limited number of the standard products to stock. The (slow moving) non-standard products are not made to stock but always made to order, because of, e.g. the obsolescence risk and the fact that 'slow-mover stock' has hardly any effect on the short term fluctuations in the demand. In the second model we think of products being produced in two phases by a common tight production resource (e.g. worker). The first phase is standard and common for all products. The second phase is customer specific. If there are no orders the production resource can continue to produce a limited number of 'first phases' as a capacity inventory. Assuming exponential production and interarrival times the models can be described by Markov processes on a two- dimensional grid. One dimension indicates the number of products on stock, and thus is finite. We show that the equi- librium distributions of the Markov processes can be deter- mined analytically. From this distribution it is straightfor- ward to compute the distribution and moments of the pro- duction lead time. Numerical results show that combining MTO and MTS effectively reduces production lead times. The technique used to analyze the two models has also been applied successfully in other applications, see e.g. [1] where a queueing model is considered for a traffic regulation mech- anism at the edge of an ATM network. Notice that realistic models will be far more complicated. The processing times will be nonexponential and different 74 per product type. The number of different product types may be large. But the simple models in this paper can be used to study the effect of combining MTO and MTS, and the insights obtained may help to understand the more complex realistic systems. Further, in the final section we will relax some of the assumptions concerning the arrival process and the processing times. The literature on these hybrid systems is growing fast. The papers [9, 3, 8, 5] study models where several items are produced by a common facility with limited capacity. Some of these items are produced to stock, others are produced to order. The MTO items have priority over MTS items. For these systems a variety of questions is addressed, such as: 'Which items should be stocked or made to order?' and 'How must production capacity be allocated among MTO and MTS items?' In Sect. 2 we first consider some simple queueing models for inventory systems related to the produce-to-stock systems in [2], and investigate the relation between inventory and the production lead time. In particular tails of the production lead time distribution tend to be rather thick, Section 3 deals with the two type system, standard products to stock, specific products to order. In Sect. 4 we consider the production in two phases, the first one being standard, the second one customer specific. Section 5 is devoted to comments and conclusions. 2 Inventories and production lead times Let us first consider the following model. Orders arrive ac- cording to a Poisson process with rate p(< 1), and require one item at a time. The production time of an item is ex- ponential with mean 1. In this system the production lead time is exponential with mean 1/(1 - p). We now assume that some of the time during which there are no orders is used to produce items on stock, say at most M items. As state space for this Markov process we take the set { M, -M+ 1, . . . , 0, 1, 2 , . . .} where the negative numbers refer to a number of items on stock and the positive numbers to orders waiting to be processed. The equilibrium probabil- ity Pn for the system to be in state n clearly equals Pn =(1 p)p~+M, n = -M, -M + I, . . . . So the production lead time S(M) is equal to 0 with prob- ability 1 - pM and exponential with mean 1/(1 - p) with probability pro. Hence, for p close to 1, the maximum stock M has to be fairly large to get a significant effect. Table 1 shows the mean production lead time and the probability of a production lead time greater than 10 times the mean production time as a function of p and M. In case the production times have an Erlang distribu- tion with r phases, each phase with mean 1/r (so the mean production time is 1), and the maximum stock is M items the analysis proceeds in an almost identical way. Now we take as state space the set {-Mr , -Mr+ 1 , . . . ,0 , 1 ,2 , . . .} where the negative numbers refer to a number of phases completed in advance (so in state - r there is one item on stock) and the positive numbers to the number of uncom- pleted phases in the system. The equilibrium probabilities p~ are for n -- -Mr , -Mr + 1, . . . given by (see e.g. [6]) I.J.B.F. Adan, J. van der Wal: Combining make to order and make to stock Table 1. Performance characteristics for exponential production times E[S(M)] M 0 8 16 24 32 40 0.8 5 0.839 0.141 0.024 0.004 0.001 p 0.9 10 4.30 1.85 0.798 0.343 0.148 0.95 20 13.3 8.80 5.84 3.87 2.57 P[S(M) > 101 M 0 8 16 24 32 40 0.8 0.135 0.023 0.004 0.001 0.000 0.000 p 0.9 0.368 0.158 0.068 0.029 0.013 0.005 0.95 0.607 0.402 0.267 0.177 0.118 0.078 Table 2. Performance characteristics for Erlang-10 production times E[S(M)] M 0 8 16 24 32 40 0.8 3.2 0.140 0.006 0.000 0.000 0.000 p 0.9 5.95 1.32 0.291 0.064 0.014 0.003 0.95 11.5 5.47 2.61 1.24 0.592 0.282 P[S(M) > 10] M 0 8 16 24 32 40 0.8 0.027 0 .001 0.000 0.000 0.000 0.000 p 0.9 0.172 0.038 0.008 0.002 0.000 0.000 0.95 0.420 0.200 0.096 0.046 0.022 0.010 ~ ~ n+Mr Pn = A iX i 1 i=l where the xi 's are the roots of the equation x ~" - p(x ~-1 +x "r-2 +. . . + 1)/r = 0, and the coefficients Ai satisfy mi = (1 - p) /H(1 - x j /xO. The production lead time S(M) is equal to 0 with probability ~-M~ Pn and it is the sum of n+r exponentials with mean 1/r with probabilitypn for n=- r+ l , r+2, . . . So for t > 0 we obtain that n+~-I ( l P[S(M) > t] = Pn ~ e -~'~ I. n=-- r + l l=O 7 ~ = ~-~ A i X=-r+l+Mr - r (1-x~)t i=l and thus also d i .-r+l+Mr E [S(M) ] = r(1 xi) 2xi " i=l Table 2 shows how the production lead times are reduced by the production to stock of the standard items. By comparing the results in Table 1 and Table 2 we see that in case of Eflang production times the reductions in the production lead time are much larger than in case of exponential production times. This is also illustrated in Fig. 1. We finally note that for general phase-type production time distributions the analysis is equally simple. I. Adan, J. van der Wal: Combining make to order and make to stock 75 \ %, \ '.% 0 i 5 exponential erlang-lO -+.- -+. "+"+--+. -~.+_ I I I I I I 10 15 20 25 30 35 40 Fig. 1. The mean production lead time as a function of the maximum stock M for p = 0.95 in case of exponential and Erlang-10 production times, resp. 3 Two types of orders, standard (MTS) and customer specific (MTO) We now turn to a model which combines make to order and make to stock. There are two types of products, standard and non-standard products. When there are no orders, the pro- duction resource (e.g. machine or worker) is used to produce standard products until the stock reaches a a certain maxi- mum level, M say. A customer asking for a standard product receives it directly (if possible) from stock. If the stock is empty the customer order joins the queue. Non-standard, customer specific products are never delivered from stock, but always produced to order. Orders for standard products arrive according to a Poisson process with rate AI, the ar- rival of customer specific orders is Poisson as well with rate A2. Further we set A = A1 +A2. Orders require one product at a time. The production times for both types of products are exponential with the same mean 1/#. Production to stock is preempted by production to order. For stability we as- sume that p = A/# < 1. Below we will first present an exact analysis of this system and then investigate the effect of pro- ducing to stock on the production lead times of standard and non-standard orders. The system can be described by a Markov process with states (n, m) where n is the number of orders in the produc- tion queue and m the number of items on stock. If m > 0 all n orders in the queue are type 2, but if m = 0 the n orders are a mixture of type 1 and type 2 orders. The flow diagram of this Markov process is shown in Fig. 2. By balancing in each state (n, m) the flow out of and into that state we get the following set of equations: p(n,M)(A+#) =p(n- 1,M)A2+p(n+ 1; M)#, (1) n>O, p(n, m)(A + #) = p(n, m + 1)/~1 + p(n - 1, To,),-~ 2 (2) +p(n+l ,m)#, n>O,O +p(1, M)>, (4) m bt ~2 g ~2 9 ,,:2 ~/T ~' 1 )g2 ~ s )vl + )v2 I.t )vL + X2 Fig. 2. Flow diagram for the model with standard and customer specific orders p(O, m)(A + p) = p(0, m + 1)A1 + p(0, m - 1)# (5) +p(1,m)#, 076 IJ.B.F. Adan, J. van der Wal: Combining make to order and make to stock where for k = 1 , . . . , M - 1 the constants Ak,j satisfy A1A~ 1,k-1 Ak,k - A + p - 2px' Ak,j = #xAk,j+l + A1Ak 1,j 1 A+#-2px , j= l , . . . , k 1. The constants Ao,o, A~,o,..., AM_ t,o are still free. Equation (3) is a difference equation in p(n, 0) with the p(n, 1)'s given by (9) as inhomogeneous terms. The solution of this equation is given by p(n, O)= B + ~ AM,y x ~. (10) j=o The first term with B free is the solution of the homogeneous equation. The sum is a particular solution of the inhomoge- neous equation. The constants AM,j in this sum satisfy AM,M--1 = --AM-I,M-IX~ and AM,j = AI 1 [{(A + #)x -- 2px2}AM,j+I --#2C2AM,j+2 -- /~1 xAM_ I,j] , for j = 0 , . . . , M-2 , where AM,M = 0 by definition. We can conclude that the solution of the equations (1)-(3) is given by (9) and (10) in which the constants Ao,o,... ,A~-l,o and B are free. Notice that A0,0 = p(0, M) = 1 - p. The other constants can finally be determined from the boundary equations (4)-(5). Once the probabilities p(n, m) are known, performance characteristics, such as e.g. the mean number of orders in the production queue and the mean production lead time for the two types of orders can be easily obtained. For in- stance, let L(M) be the number of orders in the production queue. Then, by inserting the expressions for the probabili- ties p(n, m) we find that = ~ np(n, E[L(M)] m) =t31-A /~/z ~=o j=0 ~=o where the infinite sum can be easily calculated exactly (cf. relation (24)). The mean overall production lead time E[S(M)] (i.e. for type 1 and 2 orders together) can be ob- tained from E[L(M)] by application of Little's law, i.e., E[S(M)] = E[L(M)]/A. Denote by S~(M) and S2(M) the production lead time for a standard and non-standard order, respectively. From the PASTA property (see [10]) we obtain E[S2(M)] = (E[L(M)] + 1)/#. Further, it holds that J~[S(M) ] - - ~-E[S l (m) ] .-.F ~2 E[S2(M)] , from which E[SI(M)] directly follows. i f=3~ f= l +- f=1/3 [] t~, D "D I 5 10 15 20 25 30 35 40 Fig. 3. The mean production lead time for standard orders as a function of M for # = 1 and p = 0.9 where f = AI/A2 is varied as 3, 1 and 1/3, resp. i f=3 ~- - f= l + f=1/3 El- I?1 i 0 5 I 10 ................. ++ . . . . . . i i i i i 15 20 25 30 35 40 Fig. 4. The mean production lead time for non-standard orders as a function of M for # = 1 and p = 0.9 where f = AI/A2 is varied as 3, 1 and 1/3, resp. Figures 3 and 4 illustrate for p = 0.9 and # = 1 how the mean production lead times for standard and non-standard orders decrease with M for the ratios Aj/A2 equal to 3, 1 and 1/3, respectively. Note that for M = 0 and M = oc we have E[S, (0)] -- E[S2(0)] - 1/# 1 - P l - P2 ' E[&(oc)] = 0, E[S2(oc)] - 1/# 1 P2 ' where pl = A1/# and p2 = A2/#. We see that significant reductions in the lead time are possible, and that, apparently, the reductions for the standard orders are quite insensitive to the ratio A1/A2. The reductions are larger when the ratio At/A2 is larger, i.e. when most of orders are standard. So producing fast movers to stock has far more effect than producing slow movers to stock. Most of the reduction, 80% say, is already achieved for moderate M. This is also shown in Table 3 where we list the minimal I. Adan, J. van der Wal: Combining make to order and make to stock 77 Table 3. Minimal M to achieve c~% of the possible reduction in the lead time 3// At/A2 = 3 ~ 20 50 80 90 .8 2 5 9 12 p .9 4 8 17 23 .95 6 15 33 46 A~/A2 = 1 .8 2 4 8 11 p .9 4 8 17 23 .95 6 15 33 46 ~1/~2 = t /3 .8 2 4 7 10 p .9 3 7 16 22 .95 6 15 32 46 M for which c~% of the possible reduction in the lead time is realized, i.e. c~ E E[Sf f0) ] E [S I (M) ] > ~ [S1(0)], E[&(0)] - E[&(M)] >_ 1@0(El&(0)] - E[& (oo)1). Again observe that the minimal M is quite insensitive to the ratio At/A2. The Figs. 3 and 4 and Table 3 illustrate the reductions in the lead time achieved for given maximum stock M. The maximum stock size is important for the allocation of space needed for storage, the average stock is relevant for the calculation of holding costs. The average stock can be determined as follows. There is a simple relation between the probability q(m) that there are rn items on stock and p(0, m - 1). Namely, balancing the flow between the sets {(0, ra), (1, rn ) , . . .} and {(0, m - 1), (1, rn -- 1 ) , . . .} yields that q (m. )A l=p(0 ,m-1)#, rn=l , . . . ,M . From this relation we can compute the probabil it ies q(m) and thus also the average stock. In Fig. 5 we show for p = 0.9 and # = 1 the average stock as a function of M for the ratios AI/A2 equal to 3, 1 and 1/3, respectively. Observe that from M = 20 say, the lines are nearly straight with slope 1. It is also possible to compute the distribution of the pro- duction lead t ime from the equil ibrium probabil it ies p(r~, m). The production lead time $I (M) for a standard order is the sum of n + t exponentials with mean 1/t~ with probabil ity p(n, 0) and it is equal to 0 otherwise. For a non-standard order the production lead time S2(M) is equal to the sum of n + 1 exponentials with probability p(n, ra). Hence, for t > 0 we obtain that co ~ ~rz --btt (/L~)t P[S~(M) > t] ZP(n ' = u~2__,e ~ ' n=O I=O oc M n n=O rn=0 /=0 Table 4 lists the percentages of the standard and non-standard orders that wil l get a production lead t ime more than 10 times the mean production time. 40 35 r f=3 -~-- 30 25 20 15 10 5 0 0 f= l f = 1/3 43- I 215 I I 5 10 15 20 30 35 40 Fig. 5. The average stock as a function of M for # = 1 and p = 0,9 where f = AI/A2 is varied as 3, l and 1/3, resp. Table 4. Percentages of the standard and non-standard orders with a lead time longer than 10 times the mean production time P[SI(M) > 10] x 100% AI/A2 = 3 M 0 8 16 24 32 0.8 14 2.3 0.38 0.06 0.0t p 0.9 37 16 6.8 2.9 1.3 0.95 61 40 27 18 t2 AI/A2 = 1 0.8 14 2.3 0.38 0.06 0.01 p 0.9 37 16 6.8 2.9 1,3 0.95 61 40 27 18 12 ~/x2 = 1/3 0.8 14 2.l 0.32 0.05 0.01 p 0.9 37 15 6.5 2.8 1.2 0.95 61 40 26 17 12 P[Sz(M) > 10] x 100% AI/A2 = 3 .$f 0 8 16 24 32 0.8 14 2.3 0.41 0.10 0.04 0.03 p 0.9 37 t6 6.8 3.0 1.3 0.04 0.95 61 40 27 18 12 0.05 A~/X2 = 1 0.8 14 2.4 0.61 0.31 0.26 0.25 p 0.9 36 16 7.1 3.3 1.6 0A1 0.95 6l 40 27 t8 12 0.52 ,X~/,X2 = 1/3 0.8 14 3.4 2.1 1.9 1.8 1.8 p 0.9 37 17 9.4 6.2 4.9 3.9 0.95 61 41 29 21 16 5.6 4 Half-finished products to stock In this section we consider products that are produced in two phases. The first phase is standard and identical for all prod- ucts. The second phase is customer specific. We will analyze how the production lead t ime decreases if we store some, at most M, half-f inished products, that is products with phase 1 completed. The production of the two phases is done by a common tight resource (e.g. worker). The first phase of a product takes an exponential t ime with mean 1/#1, the second phase is exponential with mean 1 /#> Orders arrive for one i tem at a t ime according to a Poisson process with rate A. For stability we assume that 78 I.J.B.F. Adan, J. van der Wal: Combining make to order and make to stock )v )v ~Aq X / ~ b t >" i i k Fig. 6. Flow diagram for the model with half-finished products = + < 1. (11) The system can be described by a Markov process with states (n, m) with n the number of orders in the system and m the number of half-finished products in stock or in use. So state (1,2) denotes the situation with 1 order in the system for which the production resource is processing phase 2, and 1 half-finished product on the shelf. If phase 2 is completed the state changes to (0, 1) and the resource continues with the production of phase 1 products until the limit M is reached, or a new customer order enters. So, a newly arriving order preempts a stock production. In state (n, 0) with n positive the resource is working on phase 1. If phase 1 is completed the state changes to (n, 1) and the resource continues with phase 2. The flow diagram of this Markov process is shown in Fig. 6. By equating in each state (n, m) the flow out of and into that state we get: p (n ,M) (A+p2)=p(n- l ,M)A , n>0, (12) p(n,m)(A+#2) =p(n 1 ,m)k+p(n+ 1,m+ 1)p2, n>0,1 0, p(O, M)A = p(O, M - 1)p~, (16) p(0, m)(A + pl ) = p(0, m 1)pl +p(1 ,m+ 1)/z2, (17) 21 + p(1, 3)#2, (18) p(0 , 1)(A + /~1) ---- ,D(0, 0 )~ 1 +p(1 ,2 )#2, (19) 1"0(0, 0)(/~ -I- /~1) --= ,D(1, 1)/~ 2 . (20) The approach to solve the equations (12)-(20) is similar to the one in the previous section. We start with the equations for m = M and then subsequently solve the equations for m = M- 1, m = M- 2 and so on. Equation (12) is a difference equation in n only. Its solution is given by p(n, M) = Ao,ox ~ , where Ao,o is free and A X -- - - A+#2 By substituting p(n, M) = Ao,ox n into (13) for m = M - 1 we get an inhomogeneous recurrence relation for the prob- abilities p(n, M - 1), with the p(n, M) as inhomogeneous terms. The solution of this recurrence relation can be written as (cf. (8)) 1 j=0 J Next we insert this expression into (13) for m = M - 2. This yields an inhomogeneous recurrence relation for the probabilities p(n, M - 2), which can readily be solved. By repeating this procedure for m = M - 3 down to m = 2 we obtain k j=0 k=O, . . . ,M 2, n=0,1 , . . . , where for k = 1 , . . . , M - 2 the constants Ak,j satisfy X~2 Ak- l , k - l , Ak,k - X_t_ [z 2 x~2 a Ak,j = Ak,y+l + ~ ~k-U- I , J = k - 1, . . . , 1. This completes the solution of the equations (12)-(13). The constants Ao,0, . . . , AM-z,o in this solution are still free. We now consider the equations (14)-(15). Note that these equa- tions are difference equations in p(n, 0) and p(n, 1) with the p(n, 2)'s given by (21) as inhomogeneous terms. The solu- tion is given by M--2 p(n, 1)=Bly'l~+Bzy~'~+ ~ AM l,j . xn, (22) j=0 A + #2 -- AYl 1 A + #z -- AY2 I p(n, O) = BI y + B2 y~ l~l #1 M-2 +~-~Am4(n+J )x ~. (23) j=0 " J The first two terms in (22)-(23) are the solution of the ho- mogeneous equations, where B1 and B2 are free and yj and Y2 are the two roots of /~I/~2Y 2 -- [ A2 + A(,/~I q- #2)]Y + A2 -- 0 . Condition (11) guarantees that the roots Yl and y2 are both positive and less than 1. The two sums in (22)-(23) are a par- ticular solution of the inhomogeneous equations. Note that the particular solution has the same form as the inhomoge- neous term p(n, 2) given by (21). For the constants Am,j and Am 1,j in the two sums the following relations can be derived. The constants Am,j in (23) satisfy AM,M/ -2 - r Am/ -z ,M-2 , and Ax 1 Am,y - [(pl - #2)Am/,j+l + .~x 1AM,j+2] ,U,1 ~2X 3//--2 3/ / -2 - Z Am, , - .2~ Z( l+ l - j )Am 2,z, l=j+l /Q l=j I. Adan, J. van der Wal: Combining make to order and make to stock 79 for j = M - 3 , . . . , 0, where AM,M-1 = 0 by definition. The constants Am-Lj in expression (22) follow from the relations AM-1 j - 1 I/*IAM/,J 1+/ .2 x 3//-2 , ~X_ I ~ Am 2,m k rn=j 1 for j = 1 , . . . ,M- 2 and Am 10- 1 [ , /*2X- 1 ~()~ +/*1 -- ~x-1)Am/,o + Ax 1AM, 1 3//-2 /*2x Z A3//-l,z l=l We can now conclude that the solution of the equations (12)- (17) is given by (21)-(23) in which the constants A0,0,.. . , Am-z,0 together with B 1 and B2 are free. Notice that A0,0 = p(0, M) = 1 - p. The other constants can finally be solved from the boundary equations (16)-(20). It is straightforward to determine performance charac- teristics, such as the mean number of orders in the system and the mean lead time from the equilibrium probabilities p(n, m). For instance, for the mean number of orders in the system we obtain E[L(M)] = 1 + -- - /*1 (1 7y l )2 ( "~'t-/.2 ~y21) J~eY2 + 1+ - - /*! (1 y2) 2 M-2 k X-" * (1 + j)x ~c=0 j=0 3/I 2 + ~(Am t,j +AM,j)( j=O where we used that ~-~(n ; J )nx~ (1 + j )x (24) ~=0 (1 - x) 2+j " The mean production lead time follows from E[L(M)] and Little's law, i.e., E[S(M)] = E[L(M)]/A. Note that for M = 0 and M = ec it holds that 1//*1 + 1// .2 P2//*l E[s(0)] = 1 - P l - - P2 E[S(c~) ] - 1// .2 1 - -P2 ' where pl = A//*I and pz = A//*2. Figure 7 illustrates how the mean production lead time decreases with M for the ratios /*1//.2 equal to 3, 1 and 1/3, respectively. In each case the mean production time (i.e., 1//.1 + 1//.2) is equal to 1. We see that significant reductions in the lead time are possible. The reduction is large if phase 1 constitutes a large part of the total production time, thus if/.1//*2 is small. In i f=3 -e~ f= l + '~ ".,% '~ -. DNS1 *++ N "+--- i i i i i i i 5 10 15 20 25 30 35 40 Fig. 7. The mean production lead time as a function of M for mean pro- duction time 1 and p = 0.9 where f = #1//'2 is varied as 3, 1 and 1/3, resp. Table 5. Minimal M to achieve c~% of the possible reduction in the lead time M ~1/#2 = 3 ~ 20 50 80 90 .8 3 8 20 29 p .9 6 20 48 69 .95 14 43 102 146 #1/~2 = 1 .8 2 5 11 16 p .9 4 10 23 33 .95 7 21 48 68 u~/m = 1/3 .8 2 4 8 12 p .9 3 8 17 24 .95 5 15 34 49 Table 5 we list the minimal M for which c~% of the possible reduction in the lead time is realized, i.e. E[S(0)] - E [S (M) ] > I@0(E[S(0)] - E [S(oc) ] . Figure 7 and Table 5 show the reductions in the lead time for given maximum stock level M. It is interesting to know the average stock in these cases. To compute the average stock we can use the following relation between the probability q(m) that there are m half-finished products in stock and p(0, m - 1). By balancing the flow between the sets {(0, m), (1, m) , . . .} and {(0, m - 1), (1, m - 1) , . . .} it follows that (q (m) -p(0 , m) )p2=p(0 , m-1)#l , m=2, . . . ,M . Also note that q(1) = p(0, 1). From this relation we can calculate the probabilities q(m), and thus also the average stock. Figure 8 illustrates the average stock for the examples in Fig. 7. It is also possible to compute the distribution of the pro- duction lead time from the equilibrium probabilities p(n, m). Let S(n, m) be the lead time if on arrival the system is in state (n, m). So, if n + 1 80 IJ.B.F. Adan, J. van der Wal: Combining make to order and make to stock 35 ~f=3 f= l f - 1/3 .D. 30 o': []" / ,E] f 25 ~' .+~ D / 7 20 o'" /+" ,U~/ [] + . [] , 15 .D j .[] o S" El S ~ D +-~ 10 ~ ;+;-+" i i i 5 10 15 20 25 30 35 40 Fig. 8. The average stock as a function of M for mean production time l and p = 0.9 where f =/*]//z2 is varied as 3, I and 1/3, resp. Table 6. Percentages of the orders with a lead time longer than I0 times the mean production time P[S(M) > 10] 100% /./,1///,2 = 3 M 0 8 16 24 32 oc .8 8.8 5.3 3.2 2.0 1.4 .48 p .9 30 23 18 14 11 1.3 .95 55 48 43 38 34 2.2 #1/#2 = l .8 7.2 2.2 .68 .21 .07 .00 p .9 27 16 8.9 5.1 2.9 .00 .95 52 40 30 23 18 .00 m/~2 = 1/3 .8 8.8 1.7 .32 .06 .01 .00 p .9 30 14 6.3 2.9 1.3 .00 .95 55 37 26 18 ! 2 .00 system with probability p(n, rn) in state (n, rn) we obtain for t _> 0 that P[S(M) > t] = ~ p(r~, m)P[S(r~, m) > t]. From this relation it is straightforward to calculate the prob- ability P [S (M) > /J] once the probabilities p(r~, m) are known. This is done for the examples in Table 6 where the percentages of the orders that will get a production lead time more than 10 times the mean production time are listed. 5 Conc lus ions and comments In this paper we studied two production-to-order systems and we investigated whether producing to stock while there are no orders in the system can effectively reduce the production lead times. Indeed, it appeared that this simple combination of make to stock and make to order can significantly reduce the production lead time. In particular, we can draw the following conclusions from the two systems studied: - In a system producing (fast moving) standard products and (slow moving) non-standard products one should stock the standard products, not the non-standard ones. The larger the stream of standard orders, the larger the effect of stock keeping will be on the overall mean pro- duction lead time; - In a system with a two phase production, a standard phase and a customer specific phase, it pays to produce the standard phase to stock. Deferring the customizing phase (so that a larger part of the total production time is standard) will lead to a larger reduction in the production lead time; - Most of the possible reduction in the production lead time is achieved for already moderate stock levels; - The higher the system utilization, the higher the stock level will be to achieve a given reduction in the produc- tion lead time. To keep the analysis simple we assumed that the orders have unit size and that the production times are exponential and independent of the product type. Below we will com- ment on these assumptions, as well as on alternative solution techniques. Unit order sizes. In case the number of items required by an order is not one but possibly larger and random, the analy- sis of the two models proceeds in the same way, provided this number is bounded. Then arrivals may lead to large (bounded) jumps in the flow diagrams shown in Figs. 2 and 6 where n then denotes the number of items (instead of or- ders) to be produced. The resulting solutions will of course be more complex (e.g., they will involve more powers); Exponential production times. When in the system with stan- dard and non-standard products the production times are not exponential, but phase-type, a more complex state descrip- tion is needed. Then the system can be described by states (r~, m, i) where r~ and m are as before and i is the produc- tion phase for the item in process. The solution technique for this problem is similar to the one presented in this paper, although the resulting solution will be more complex (e.g. it will involve an extra term vi, say, corresponding to the variable i). The same holds for the model with two produc- tion phases, where each production phase has a phase-type (instead of an exponential) distribution. Alternatively, this model (and possibly also the previous one) can be analyzed by using generating function techniques. Type-independent production times. The model in which the production times of the standard and non-standard products are exponential with different parameters #1 and #2, say, will be (much) harder to analyze than the model with #1 = #2. In that case, the description of the production queue is more complex. It may be thought of as consisting of two parts: at the front there are i type 2 (i.e. non-standard) orders that arrived when the stock was nonempty and the rest of the queue (i.e. n - i orders) arrived when the stock was empty, so that is an unknown mixture of type 1 and type 2 orders (with probability A1/A an order is of type 1 and, otherwise, it is of type 2). If i = 0 one also has to specify whether the order in process is of type 1 or 2. Hence, this system can be described by a Markov process with states (n, i, j, m) where the variables r~ and i are unbouned, and the other two are bounded (j specifies the order in process). This means that I. Adan, J. van der Wal: Combining make to order and make to stock 81 we have to deal with an essentially 2-dimensional (instead of 1-dimensional) Markov process. It is not clear whether an approach similar to the one used in this paper also works ~br this two-dimensional problem. Alternative solution techniques. The equilibrium equations in the Sects. 3 and 4 have been solved by" using standard results from the theory of linear recurrence relations (which is analogous to the theory of linear differential equations, cf. [4]). Another approach which is well suited to solve these equilibrium equations is the matrix-geometric approach de- veloped by Neuts [7]. In fact, the corresponding rate matrix R can be solved by a simple and exact recursion, instead of the usual iteration scheme (cf. Sect. 6.5 in [7]). So one may expect that the efficiency and stability of this approach will not differ much from the approach employed in this pa- per. Further, the model with the two-phase production (and probably also fl~e model with two t~q?es of products) may be analyzed by using generating functions. This method is more complex and less efficient than the one used in this paper (in particular for the calculation of the lead time distribu- tion). But the advantage is that it also works when the two production phases have a general distribution (e,g., uniform or deterministic). References 1. Adan IJBF, van Doorn EA, Resing JAC, Scheinhardt, WRW (1996) A queueing system with two different service speeds, Memorandum COSOR 96-12, Department of Mathematics and Computing Science, Eindhoven University of Technology 2. Buzacott JA, Shanthikumar JG (1993) Stochastic models of manufac- turing systems, Prentice Hall, New York 3. Cart AS, Gtillti AR, Jackson PL, Muckstadt JA (1993) An exact analy- sis of a production-inventory strategy for industrial suppliers, Working paper, School of Operations Research and Industrial Engineering, Cor- nell University, Ithaca, NY 4. Coddington EA, Levinson N (1955) Theory of ordinary differential equations. McGraw-Hill, London 5. Federgruen A, Katalan Z (1994) Make-to-stock or make-to-order: that is the question; novel answers to an ancient debate. Working paper, Graduate School of Business, Columbia University, New York 6. Kleinrock L (1975) Queueing systems, Vol. I. Theory. Wiley, New York 7. Neuts MF (t981) Matrix-geometaic solutions in stochastic models, Johns Hopkins University Press, Baltimore 8. Sox CR, Thomas LJ, McCIain JO (t994) Jellybeans: Sorting capacity to improve service. Working paper, Department of Industrial Engineering, Auburn University', Alabama 9. Williams TM (1984) Special products and uncertainty in produc- tion/inventory systems. EJOR 15:46-54 t0. Wolff RW (1982) Poisson a~Tivals see time averages. Opns. Res. 30: 223 -231

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