Inventory control with manufacturing lead time flexibility

Inventory control with manufacturing lead time flexibility

Kok, de, A. G. (2011). Inventory control with manufacturing lead time flexibility. (BETA publicatie : working papers; Vol. 345). Technische Universiteit Eindhoven.

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Inventory control with manufacturing lead time flexibility

Ton G. de Kok

Beta Working Paper series 345

BETA publicatie WP 345 (working

paper)

ISBN 978-90-386-2490-7

ISSN

NUR 804

Inventory control with manufacturing lead time flexibility

Ton G. de Kok Unpublished Research Report TUE/TM/LOG November, 1998

Department of Technology Management P.O. Box 513, Pav. E4

5600 MB Eindhoven The Netherlands

Phone: +31-40-2473849 Fax: +31-40-2464596

E-mail: A.G.d.Kok@tm.tue.nl

This paper should not be quoted or referred to without the prior written consent of the author

Inventory control with manufacturing lead time flexibility

Ton G. de Kok

Department of Technology Management

Eindhoven University of Technology

The Netherlands

Abstract

In this paper we discuss an inventory control problem where manufacturing lead times can be shortened in case short-term stockouts are foreseen. This problem occurs in many practical situations where multiple items are produced in a production department using common capacity and customer lead time for the items is positive. In that case the common capacity provides the flexibility of scheduling in production orders of items for which a stockout occurs within the customer lead time, while scheduling out production orders for items that have sufficient stock available to satisfy short-term demand. We derive expressions for the probability of a stockout and the fill rate in case such replanning is allowed. We validate the heuristic analysis of the model by comparing model results with results from a case study. We discuss the importance of measuring the frequency of replanning for the application of the model.

1. Introduction

In this note we discuss the problem of modelling manufacturing lead time flexibility and its impact on the control of the inventory of finished products. To illustrate this problem we consider the following case situation: A glass bulb factory produces 110 different glass bulbs for ordinary lamps. There are 4 production lines, of which one is only producing the most common glass bulb, while the other lines produce a mix of different glass bulbs. Each production line produces approximately 6 glass bulbs per second. Due to the characteristics of glass itself and the characteristics of the production process changing over from a production run of one product to a production run of another product is time consuming. Change-over times range from several hours to several days. Over a range of years the following manufacturing, planning and control characteristics have been empirically established. The manufacturing throughput time ranges from 2 to 6 weeks, dependent on the item. Customer lead time equals 2 or 3 weeks, dependent on the item. This results in a mixed make-to-stock/make-to-order situation. Manufacturing lot sizes are more or less fixed and in principle a cyclical schedule is used: glass bulb types are produced in a fixed sequence in order to minimise set-up time. A finished goods stock of about one month production is held to buffer against uncertainty in demand during the manufacturing throughput time. This buffer is mainly used to enable the cyclical “minimum change-over time” schedule. Now and then a customer order is received that cannot be filled from stock nor from future planned orders. In that case a replanning process starts, where the available stock is analysed in detail and the existing production plan is adjusted in order to fill all future customer orders. Implicitly

information about future unknown customer orders, derived from the long history of collaboration with a relatively small number of customers, is taken into account in this replanning process. The overall result of planning, control and execution of the plans over the last 10 years has been a 100% customer service level. We consider the above case typical for factories producing items for other factories. In many cases customer delivery time is shorter than the manufacturing throughput time. This implies

uncertainty in demand during the throughput time and therefore finished goods stock is required.

The problem we are faced with in this particular case is the fact that 100% customer service is realised. To our knowledge application of almost any classical inventory

control model to this situation would result in an infinite finished goods stock. E.g. assuming normally distributed lead time demand and a P1-service measure (cf. Silver and Peterson [1985]) yields a safety factor of ∞, because the safety factor k should satisfy φ(k)=1. Apparently this is not the case in this real-world situation.

Key to the realisation of 100% service level are two aspects. Firstly, customer order lead time is positive, which gives the possibility to anticipate possible disservice. Secondly, in a multi-product situation one usually has some manufacturing flexibility by which possible stockouts for some items can be prevented by replanning without the expense of causing stockouts of other items. Hence a mathematical model should capture these two aspects. In traditional single echelon inventory literature (cf. Hadley and Whitin [1963], Lee and Nahmias [1993]) we assume that item inventory is controlled by comparing the inventory position, which is defined as the sum of physical stock and outstanding orders minus customer backorders, with some reorder level. If the inventory position is below the reorder level a replenishment order is initiated and communicated to the supplier of the item. After a positive, possibly random, lead time the replenishment order is added to the inventory. It can be proven that such rules are cost-optimal, yet only under the

assumption that the inventory state variable is the inventory position and the assumption that lead times are exogenous to the model. In the real-world the inventory state variable is a vector consisting of future planned inventories as presented by MRP-systems (cf. Volmann, Berry and Whybark[1992]). Furthermore, lead times are usually endogenous to the inventory control process: As in the case presented above lead times can be reduced when needed. In that way one may find a positive correlation between

manufacturing lead times and future stock on hand. This is to our knowledge hardly ever incorporated in inventory models. In this paper we propose a heuristic approach to cope with this type of manufacturing flexibility. The underlying idea is that planners are able to prevent stockouts by replanning. We do not explicitly model the replanning process. We restrict ourselves to measuring the frequency of replanning under the assumption that replanning prevents stockouts, indeed. Potential stockouts are identified by future negative planned inventory on hand in the same way as calculated by MRP-logic. In the mathematical literature little can be found on this type of manufacturing flexibility. In the context of repairable items pipeline flexibility is modelled in

Dada[1992] and Verrijdt et al.[1995]. An extensive discussion of manufacturing flexibility can be found in Gupta and Goyal[1989].

The paper is organised as follows. In section two we describe the mathematical model considered. In section 3 we present a heuristic analysis of the model. The case discussed above is described in further detail in section 4, thus providing empirical evidence of the validity of the approach. In section 5 we present conclusions and ideas for further

research.

2. Model description

We consider a stock item that is produced on a production line, where other items are produced as well. The item inventory is controlled by an (R, s, Q)-policy. I.e. at the beginning of each review period of R time units the inventory position is monitored. If the inventory position is below s, then a multiple of Q items is ordered at the production department, such that the inventory position is raised to a value between s and s+Q. If the inventory position is above s no order is initiated. The normal production lead time equals L, where L is a multiple of R. Without loss of generality we assume R=1. This implies that each production order initiated at the beginning of review period t is assumed to be produced in period t+L. Here period t is the interval (t-1, t]. We assume that customer orders arrive at the beginning of period t. These orders must be delivered in period t+ Lc with 0 ≤ Lc ≤ L. We assume that customer demand in subsequent periods is i.i.d. Let D denote the demand for the item in an arbitrary period. Furthermore we define D(t,t+s] as the cumulative item demand during the time interval (t,t+s]. At the beginning of period t the following quantities are determined.

I(t,t+s) := expected on hand stock at the end of period t+s, s ≥ 0. p(t,t+s) := expected amount produced during period t+s, s ≥ 0.

µ(t,t+s) := expected demand during period t+s, s ≥ 0 It follows from the definition of µ(t,t+s), D, D(t,t+s] and Lc that

µ( , ) ( , ]; [ ] t t s D t s t s s L E D s L c c + = + − + ≤ ≤ >    1 0 Furthermore we have I(t,t+s) = I(t,t+s-1) + p(t,t+s) - µ(t,t+s), s ≥ 0

Here I(t,t-1) is the actual stock at the beginning of period t. The planner observes I(t,t+s), s ≥ 0. If I(t,t+s) < 0 for some s, the planner may be able to replan p(t,t+s), s ≥ 0, such that I(t,t+s) ≥ 0 after replanning.

Define

π := probability that a replanning action is decided upon.

A successful replanning action eliminates negative I(t,t+s). given the model parameters R, s, Q, L, Lc and π we want to determine the service levels P1 and P2 achieved,

P1 := the probability of a stockout at the end of a replenishment cycle P2 := fraction of demand satisfied in time.

Here a replenishment cycle is defined as the time between arrival of two consecutive replenishments. Once we have an expression for P1 and P2 as a function of the model parameters we can conversely determine e.g. s* as a function of the other model

parameters, such that P1 (P2) is equal to a target value P1* (P2*). We claim that through the introduction of π a target value of 1 can be realised with a finite average on hand stock. This contrast with results for the classical single echelon one-product inventory models like (s,S), (s,nQ), (R,s,S) and (R,s,nQ) for e.g. normally distributed or gamma distributed lead time demand.

In this section we derive expressions for P1 and P2 as a function of the model parameters. The analysis consists of two steps. First of all we give an expression for P1 and P2 in case no replanning is possible. This is equivalent to the classical inventory model assumption. Secondly we use a number of assumptions through which we can use the results for the model without replanning to derive an expression for P1 and P2 in the model with replanning. To make a distinction between characteristics for the model with and without replanning we use upper indices (r) and (nr), respectively.

Let us now assume that replanning is not possible. In that case we have a classical (R,s,Q)-policy with the only difference that we have to account for the fact that at the beginning of period t the future demand D(t, t+Lc] is exactly known. Hence we have only uncertainty in lead time demand during (t+Lc,t+L]. In case Lc=0 the appropriate control variable is Y(t),

Y(t) := inventory position at the beginning of period t. In case Lc>0 we propose to use W(t) as a control variable, W(t) := Y(t) - D(t, t+Lc]

and to adapt the (R, s, Q)-policy such that an order of Q is initiated when W(t) < s. Here we assume that the undershoot of s is small compared to Q. In that case it can be shown that P₁ = P D L L{ ( , ]_c +U >s} P E D L L U s E D L L U s Q Q c c 2 1 = − [( ( , ]+ − ) ]+ − [( ( , ]+ − +( ) ]+ Here

Fitting mixtures of Erlang distributions to the first two moments of D(0, L-Lc] + U we can numerically obtain an excellent approximation for P1 and P2.

Now let us assume replanning is possible and π is given. Then we observe the following: The number of stockouts in the model without replanning equals the number of potential replanning actions in the model with replanning.

Here we assume that Q is large compared to E[D], so that replanning a quantity Q at the beginning of period t eliminates all stockouts in period t+s, 0 ≤ s ≤ L.

The number of stockouts per time unit in the model without replanning equals the number of replenishment cycles per time unit multiplied by the probability of a stockout at the end of a replenishment cycle. We assume that a replanning action eliminates a stockout at the end of a replenishment cycle. To give an expression for the number of stockouts in the model with replanning we define the following quantities.

N( )r _{:= the number of stockouts per time unit in the model with replanning} N(nr)_{:= the number of stockouts per time unit in the model without replanning}

Then we find the following relations

P nr P D L L U c 1 ( ) _{= { ( , ]+ >s}} π Pr Pnr 1 1 1 ( ) ₌ ( )_{( − )} N r Pr E D Q ( ) ₌ ( ) [ ] 1 N nr P nr E D Q ( ) ₌ ( ) [ ] 1

Note that Q/E[D] equals the average replenishment cycle length. The above relations imply that

Pr P D L L U

c

1 1

N r E D P D L L U

Q c

( ) ₌ (1−π) _{[ ] { ( , ]+ >s}}

From N( )r _{we can derive an expression for the key performance indicator , which is} defined as

f_r

f_r:= the number of replanning actions per time unit.

It is easy to see that

f_{r =}N( )nr ₋N( )r _, and thus f E D Q P D L L U s r = c + > π [ ] { ( , ] }

In order to find an expression for we observe that the average shortage at the end of a replenishment cycle in the model without replanning equals ( . The average shortage at the end of a replenishment, given that a shortage exists then equals

Pr 2 ( ) )Q ( ) 1−P₂nr ( ( )) ( ) 1 ₂ 1 −P Q P nr

nr . We assume that in the model with replanning this conditional expectation of a shortage at the end of a replenishment cycle is the same as in the model without replanning. Then the unconditional expectation of the shortage at the end of a replenishment cycle in the model with replanning equals ( )

( ) ( ) ( 2 1 ⋅ P Q P nr nr r ) 1 1 − P . Thus we find ( ( )) ( ( )) ( ) ( ) 1 ₂ 1 ₂ 1 1 −P Q= −P Q⋅ P P r nr r nr , which yields

P P P P r r nr nr 2 1 1 2 1 1 ( ) ( ) ( ) ( ) ( ) = − − .

Using the above derived expressions for P r , and , we obtain

1 ( ) _Pnr 1 ( ) _P nr 2 ( ) P Q E D L L U s E D L L U s Q r c c 2 1 1 ( ) _{= − −}( π)_{( [( ( , ]+ − ) ]}+ _{− [( ( , ]+ − +( )) ])}+

In practice it is often easier to measure than to determine . Therefore we write as f_r π Pr 2 ( ) P f Q E D P D L L U s E D L L U s E D L L U s Q r r c c c 2 1 1 ( ) ₍ . [ ] { ( , ] })( [( ( , ] ) ] [( ( , ] ( )) ]) = − − + > + − − + − + + +

Finally we assume that the replanning actions do not impact the average stock on hand. This yields lim [ ( , )] [ ] [ ( , ]] t c Q E I t t s E U E D L L →∞ ≈ − − + 2 .

This completes the heuristic analysis of the model. In the next section we discuss the application of the model to the glass bulb case.

4. The glass bulb case

In the introduction we described the planning and control process at the glass bulb factory. The combination of cyclic schedules and replanning actions in case of future stockouts yields a complex multi-item inventory problem. A related model has been studied by Fransoo et al[ ]. We decided to decompose the problem into single-item models. The multi-item aspect is dealt with through the measurement of . The number of replanning actions per time unit is a measure for flexibility. We measured by analysing together with the planner the production plans of 9 consecutive weeks. A replanning action was identified by the difference in the production planned for an item

f_r

for the same week in two consecutive production plans. We carefully identified with the planner the cause of the replanning action. this was very important, since a stockout for one item typically causes changes in future production orders for several items. Hence there is a great danger of double-counts. Through this process we identified 30

replanning actions. Assuming that all 110 products have equal stockout probability we derived that

f_r = _x30 week≈ week

9 110 0 03

( )/ . /

Since a 100% fill rate Pr is realised, we conclude that . This yields

2 ( ) _Pr 1 0 ( ) ₌ 0=P D L L{ ( , ]_c + > −U s} _{E D}f Qr [ ].

Thus the reorder level s can be determined from

P D L L{ ( , ]_c U s} _{E D}f Qr

[ ]

+ > = .

For each item we used historical data and information from the planner to determine Q, E[D], σ(D), Lc and L. Then we implemented our heuristic into a discrete event

simulation program. Reason for using discrete event simulation was the dynamics of the model due to the fact that customer demand during D(t, t + Lc] is known at time t. Furthermore the discrete event simulation program was used for the more complex situation, where so-called demand orientations for periods beyond t + Lc where

available. Finally for the last year we collected data about the average on hand stocks for each production group, i.e. the sum of on hand stocks for all items produced at the same production line. In the table below we compare the results of the simulation with the actual data.

production average on hand stock (million glass bulbs)

group simulation actual

1 14.0 33.7

2 25.6 26.7

3 32.4 35.7

4 6.8 8.1 total 78.8 104.2 Comparison of the model results for stock on hand with actual data

For production groups 2, 3 and 4 we find acceptable results. For production group 1 we see an unacceptably large difference between the model result and the actual result. Product group 1 in fact consists of only one product: the most fast moving glass bulb (used in households most often). Due to its importance for the total turnover of the glass bulb plant management had decided to maintain a so-called strategic stock of about 20 Million glass bulbs in order to guard against extraordinary accidents like a glass oven breaking down. Based on this and the acceptable results for production groups 2, 3 and 4 we concluded that our approach was valid.

5. Conclusions and further research

In this paper we discussed an inventory model that incorporates manufacturing lead time flexibility. It attempts to model the fact that production orders are replanned when needed. Instead of making a detailed model of the replanning process in relation to the inventory state of end-products we decided to measure the number of times that

production orders are replanned. By assuming that production orders are only replanned if a future stockout is foreseen we linked the model with replanning to the classical inventory model without replanning. Based on this assumption we derived expressions for the P1 and P2 service measures for the model with replanning. We validated our heuristic approach using case data.

The applicability of this approach depends on the appropriate measurement of the replanning frequency . This measurement is easily corrupted by the fact that one replanning action to prevent a stockout for a particular item causes a number of

replanning actions for other items to make a new feasible schedule. In the case situation we were aware of this problem and eliminated it by detailed analysis of a number of consecutive production plans. To make this approach applicable planning systems should register so-called replanning sessions, that start with the identification of the item for which a stockout is about to occur and end with the last replanning action needed to create a feasible schedule again. The planner should be aware of the importance of this measurement and if in the replanning process incidentally another stockout is prevented, then the planner should register that more than one stockout is prevented in this

replanning session. f_r

In this paper we validated the heuristic approach for the special case of 100% service level, only, by comparison of model results with actual data. Although this is an

important special case, validation of the approach for situations with service levels less than 100% is needed as well. Further research is needed to validate in a more rigorous way this black box approach with respect to the replanning process. It is likely that a combination of mathematical models and discrete event simulation fits this purpose.

6. References

Dada, M., 1992, A Two-Echelon Inventory System with Priority Shipments, Management Science 38, 1140-1153.

Fransoo, J.C., Sridharan, V. and Bertrand, J.W.M., 1995, A hierarchical approach for capacity coordination in multiple products single-machine production systems with stationary stochastic demands, EJOR 86, 57-72.

Gupta, Y.P. and Goyal, S.K., 1989, Flexibility of manufacturing systems: concepts and measurements, EJOR 43, 119-135..

Hadley, G. and Whitin, T.M., 1963, Analysis of Inventory Systems, Prentice-Hall, Englewoods Cliffs.

Lee, H.L. and Nahmias, S., 1993, Single-Product, Single-Location Models, in: Logistics of Production and Inventory, edited by Graves, S.C., Rinnooy Kan, A.H.G. and Zipkin, P.H., North-Holland, Amsterdam.

Silver, E.A. and Peterson, R., 1985, Decision Systems for Inventory Management and Production Planning, Wiley, New York.

Verrijdt, J.H.C.M., Adan, I. and De Kok, A.G., 1995, A Trade-off between Emergency Repair and Inventory Investment, Research Report TUE/TM/LBS/95-05, Eindhoven University of Technology (accepted for publication in IIE Transactions on Scheduling and Logistics).

Vollmann, T.E., Berry, W.L. and Whybark, D.C., 1992, Manufacturing planning and control systems, 3rd ed., Homewood : Business One, Irwin.

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281 2009 The link between product data model and

process model J.J.C.L. Vogelaar; H.A. Reijers

280 2009 Inventory planning for spare parts networks with

delivery time requirements I.C. Reijnen; T. Tan; G.J. van Houtum

279 2009 Co-Evolution of Demand and Supply under

Competition B. Vermeulen; A.G. de Kok

278

277 2010

2009

Toward Meso-level Product-Market Network Indices for Strategic Product Selection and (Re)Design Guidelines over the Product Life-Cycle

An Efficient Method to Construct Minimal Protocol Adaptors

B. Vermeulen, A.G. de Kok

R. Seguel, R. Eshuis, P. Grefen

276 2009 Coordinating Supply Chains: a Bilevel

Programming Approach Ton G. de Kok, Gabriella Muratore

under demand parameter update

274 2009

Comparing Markov chains: Combining

aggregation and precedence relations applied to sets of states

A. Busic, I.M.H. Vliegen, A. Scheller-Wolf

273 2009 Separate tools or tool kits: an exploratory study

of engineers' preferences

I.M.H. Vliegen, P.A.M. Kleingeld, G.J. van Houtum

272 2009

An Exact Solution Procedure for Multi-Item Two-Echelon Spare Parts Inventory Control Problem with Batch Ordering

Engin Topan, Z. Pelin Bayindir, Tarkan Tan

271 2009 Distributed Decision Making in Combined

Vehicle Routing and Break Scheduling

C.M. Meyer, H. Kopfer, A.L. Kok, M. Schutten

270 2009

Dynamic Programming Algorithm for the Vehicle Routing Problem with Time Windows and EC Social Legislation

A.L. Kok, C.M. Meyer, H. Kopfer, J.M.J. Schutten

269 2009 Similarity of Business Process Models: Metics

and Evaluation

Remco Dijkman, Marlon Dumas, Boudewijn van Dongen, Reina Kaarik, Jan Mendling

267 2009 Vehicle routing under time-dependent travel

times: the impact of congestion avoidance A.L. Kok, E.W. Hans, J.M.J. Schutten

266 2009 Restricted dynamic programming: a flexible

framework for solving realistic VRPs

J. Gromicho; J.J. van Hoorn; A.L. Kok; J.M.J. Schutten;