Engineering Costs and Production Economics, 9 (198.5) 165-168 Elsevier Science Publishers B-V,, Amsterdam - Printed in The Netherlands

165

HEURISTIC MODELLING OF A MULTI-ECHELON PRODU~lON=lNVENTORY SYSTEM

Attila Chiktin

Karl Marx University of Economics, Budapest (Hungary)

The model is based on a manufacturing company’s production-inventory system. The company consists of several plants which are inter-connected in a multi-echelon struc- ture. The company has only one end product, and it has a strong market position, con- sequently it can dictate its own delivery con- ditions.

The problem of programming the produc-

tion and distribution of similar systems (Fig. 1) is extensively studied in the liter- ature. There is an enormous number of papers exploring possible solutions to the problem under various conditions (For a recent good review see Williams [ 1 I). As a result of the difficulties arising from han- dling such systems, the use of heuristic algorithms is widely accepted and practised.

Our model also belongs to this family of ~ont~butions. Since it is based on a real life system*, we could use realistic assump- tions to simplify the problem. The heuris- tics of the model (or, more precisely, the model-system) is built on the fact that the plants schedule their respective production and delivery fairly ~dependent~y, and com- pany management, for various reasons, sup-

*The model has been worked out on the basis of a contract with the Technological Development Institute of the Machine Industry. The development of the model was only a part of

the whole project. The team working on the project was headed by the author; its members were J. Berries, P. KelJe,

M. Nagy and I. Vass. The computations were made by

I. Gygri.

ports the continuation of this independence.

On the other hand, inventory costs are judged from the point of view of the com- pany as a whole. Thus we had to find a model which interlinks the plants and their own production schedules in such a way as to ensure low inventory costs for the whole company.

The operation of the system is as follows: There is a scheduling period (usually 6

months) for which a Master Production Schedule is determined. (The MPS is based on market forecasts of the demand for the end item.) An explosure algorithm is used to determine the requirements (the planned production quantity) for each plant.

____--_ -_-_- 1 .k ’ 0

Pig. 1. The general multi-echelon system.

0167-188X/85/$03.30 o 1985 Elsevier Science Publishers B.V.

166

Each plant j on echelon i has a practically continuous production pattern, and deter- mines its delivery schedule independently.

Given the planned production quantity (rij), the following questions have to be answered :

(a) What should be the delivery lot sizes, or (which is obviously the same question) how many deliveries (rzii) should be made during the scheduling period (t)?

(b) How can it be achieved that the in- terest of the whole system is considered when preparing the plants own independent delivery plan?

(c) How can the various plants protect themselves against the uncertainty arising from the fact that although the lot size of their supplying plant is fixed, the delivery time is random in consequence of the ur.- certain character of the raw material input?

Various answers can be provided to these questions, with which the system can operate similarly well. We used the following ap- proach to the questions:

(a) The lot sizes can be determined by some variant of the economic order quan- tity.

(b) The connection between the subse- quent plants is made through the inventory between the two echelons. We assume that once an item is produced at a workshop on echelon i it is kept in stock at this workshop until the production of the whole delivery lot is finished and then it is delivered to the input

stocking point of a plant on the (i-l) eche- lon. This procedure leads to the fact that the lot size of plant i influences the input stock of plant (i-l) and needs to be con- sidered when determining lot sizes. This leads to an at least partly common interest between plants i and (i-l).

(c) Since the delivery schedule of plant i is to be considered random from the point of view of plant (i-l), a safety stock must be held at the input stocking point of plant (i-l) to ensure that its production schedule

can be met (at least with a high probabil- ity). Let us take the following notations:

C(nij) 1

Pij :

hij :

Iij 1

Yij 1

Wij 1

l--E :

the total cost associated with the number of deliveries nij at plant ij

the unit price (value) of the product of plant ij the unit inventory holding cost of the product of plant ij (hii is inde- pendent of the location of the stock) the average inventory level at plant ij

the planned production quantity for the product of plant ij (we assume that only one product is produced in each plant) the constant cost of one delivery from plant ij

the safety factor, the probability of meeting all demand in the sched- uling period.

Thus the cost function for the product of plant ij can be expressed in general form

as follows:

C(tiij)= hijIfj(Tij;?Zij) + Wij(?lij) +hqI(i-l)(nij; V(i-l)j;E)

(1)

This general cost function can be specified in many ways. In a practical case it is quite possible that the specific form of the func- tion is different for the various plants. If the system is fully worked out, an inventory of inventory models can be used for the specification, when after identifying the actual conditions for each plant the selec- tion of the appropriate models can be made from a model library (for such a library see Barancsi et al. [ 21).

In our case it has turned out that, as is quite general in practical applications, a simple model system gives rather good results.

We have used the classical EOQ formula for determining the number of deliveries, and a simple reliability type model [3] to determine the safety stock.

Since this latter model might not be generally known, its basic principles are as follows:

167

The model is built on the assumption that the demand for an item is a given num- ber for some given period. (In our case for the 6-month scheduling period.) The output flow from this inventory has a uni- form pattern. The input arrives in random but equal lots, which amount to the cumu- lated demand for the end of the scheduling period. The question is: how much initial inventory should be held to meet the demand with given probability? Using the notations given above the following formula is derived:

I$!l)j = ‘(i_l)j \I 11 1 --ln- (e#O) llij2 E

For details see Prekopa [ 3 1.

We use the following cost function to determine nii:

rij C(~Q) = hii - + wqnij + hvr(i-l)j (3)

nii

The optimal value of nii can be determined from the equation C’(n,) = 0.

The total inventory between plants i and i-l is maximal in the case when plant i de- livers immediately at the beginning of the

TABLE 1

period examined and then delivers regularly thereafter (since in this case the safety stock is unnecessary). In this “worst case” the total inventory of a product is rii/2nij at the out- put side of plant (ij) and a current stock rij/2nij plus the safety stock ~~‘l,j at the in- put side of plant (i-1)j. This sums to (rii/

nij) + @_rjj; we made our calculation assum- ing this case.

I _-_-__ _-

-7 -_---

I

I

I

I

I

- -I- - -r -

Fig. 2. The system analyzed the unit requirements.

Numerical results of the heuristic model of the system, in Fig. 2 hii = 0.3&i; (1-c) = 0.99

The structure of the plants and

Plant Demand Constant Unit value of Number of Current Safety Total Number of Stock as

(ii) (rij) delivery end product deliveries stock cost

(Wjj) (Pij) (nij) (I$

(100 X I&$

0 200 800 4740 13 8 - 8 25.0 4

1.1 1000 400 290 19 53 69 122 8.3 12 1.2 600 450 780 27 22 58 90 6.7 15 2.1 4000 100 18 20 200 339 539 7.1 14

2.2 6200 300 96 36 172 405 577 11.1 9

2.3 1200 150 50 18 67 214 281 4.3 23 3.1 12000 100 5 20 600 1357 1957 6.1 16 3.2 6200 120 20 43 144 1435 1579 3.8 26 3.3 12400 200 30 44 282 1418 1700 7.3 36

Total cost of the system, 131546; total inventory, 277069; total production value, 948000. Number of turnovers for the whole system = 3.5. Stock as a percentage of production = 29%.

168

There are three types of plants in the system which must be mentioned separately. In echelon 0 (at the plant where the end product is produced) the output stock is determined by using the EOQ formula, since the company has a strong market position and can make its own decision in its distribution system about the delivery dates.

In echelon 3 (raw material level) the company must suffer from the effects of the shortage economy, namely it has to deal with an uncertain supply. The stochastic input can be treated by several methods. To ensure the structural uniformity of our system we use the same approach as for the in-process inventory.

There is a slight problem with plant 2.2, since that is the point where the otherwise pure tree-structure is disturbed by the fact that this plant is connected to both plants in echelon 1. If it is assumed that plant 2.2 delivers to both users at the same time and proportionates the quantity delivered to the overall demand of the two users, then the plant can be handled the same way as any other.

To illustrate the operation of the system we show an example, which is a slightly simplified model of a real company. The system can be seen in Fig. 2 and the data and the results of the computations in Table 1. Results of a sensitivity analysis can be seen in Table 2.

It should be mentioned that we have carried out an analysis of two similar sys-

TABLE 2

Sensitivity analysis

1-e = 0.99 service level hii = @ii holding cost (a: % of unit value)

wij = constant delivery cost

Total costs (compared to the original)

q

0.5 0.59 0.68 0.76 0.82 0.99 1.00

1 .o 0.79 0.90 1.00 1.10 0.95 0.93

2.0 1.05 1.20 1.33 1.46 0.90 0.89

Total inventory

:

0.5 0.90 0.83 0.77 0.73 0.99 1.00

1.0 1.18 1.08 1.00 0.95 0.95 0.91

2.0 1.52 1.39 1.30 1.23 0.90 0.86

terns. The heuristic approach has also given satisfactory results for these systems. For example, the number of turnovers per year have been about 25-30% over the average of the Hungarian manufacturing industry.

REFERENCES

Williams, J.F., 1983. A hybrid algorithm for simultaneous

scheduling of production and distribution in multi-

echelon structures. Manage. Sci., 29: 77-92.

Barancsi, I?. et al., 1983. A Report of Research on In-

ventory Models. Engineering Costs and Production

Economics, Vol. 7. pp. 127-136. F’rekopa, A., 1981. Reliability type inventory models.

A survey. In: A. Chikan (Ed.), The Economics and Management of Inventories, Vol. 2. Elsevier, pp. 4777490.