Production and Inventory Control: Theory and Practice

Any system of production and inventory control must be based upon a preliminary study of the relationship between the factors: DEMAND-INVENTORY-PRODUCTION. Each of these factors has certain characteristic features.

The pattern of co-operation in modern industry has become more complex as a result of an increase in the range of products and the increasing technical complexity of the production processes. The whole scale of events has widened, raising problems which were not even perceived, let alone solved, up till now.

We have shown in ‘An Introduction to Production and Inventory Control’ how important it is to acquire a proper understanding of probability theory and a knowledge of statistical laws as applied to production and inventory control. Natural laws were discussed in Chapter 3 of that book, the cumulation of deviations was covered whilst the terms ‘standard deviation’ and ‘fluctuations’ were defined in connection with the re-order level.

We shall confine ourselves as far as possible to the notation advocated in the Netherlands by, amongst others the ‘Mathematisch Centrum’ (Mathematics Centre) (M.C.) in Amsterdam [Kr 1].

Let us now consider the distribution density of the interval t between two consecutive events in circumstances identical with those in which the number of events per time unit follows a Poisson distribution: 15<math display='block'> <mrow> <mi>P</mi><mrow><mo>(</mo> <mn>0</mn> <mo>)</mo></mrow><mo>=</mo><msup> <mi>e</mi> <mrow> <mo>&#x2212;</mo><mi>&#x03BC;</mi> </mrow> </msup> </mrow> </math>$$P\left( 0 \right) = {e^{ - \mu }}$$

In industrial practice, demand is invariably defined as a quantity per period. The succession of figures representing the demand for a given product constitutes a time series. Let d _t denote the demand in period t.

Examination of the annual turnover figures of an assortment of products (individually in the case of comparable products, in terms of money in the case of others) often reveals that these figures follow a curve which is a reasonable approximation of a lognormal distribution. The cumulative distribution, plotted on a logarithmic scale, will then constitute a straight line. For example, Fig. 10 illustrates the conversion of the lognormal, skew, distribution of the annual turnover figures D, through the different transformations and transitional stages, into a straight line for the cumulative distribution.

The idea of dynamic programming was conceived by Bellman and developed by Howard [Ho 2], amongst others, into a method of analysis for discrete problems which is particularly useful in dealing with numerical data. Its scope is not confined to the applications discussed in this book. A brief, general introduction to the subject is provided by Van der Veen [Ve 1] and by Kaufmann/Faure [Ka 2]. Bellman’s principle of optimality plays a fundamental part in this: a problem which is insoluble in itself and involves an object function of n variables, can be overcome by solving n problems, each with one variable, one after another.

Humanity has been making predictions for hundreds of years, and employing some very strange devices for the purpose. It is common knowledge that the solar system, for example, has always appealed tremendously to the imagination of human beings, some of whom have ventured to make pronouncements concerning future events, based upon certain astronomical phenomena. At the same time, we would be very surprised to read an advertisement in which astrologers were asked to apply for the position of ‘forecaster’ in an industrial company.

So far we have discussed one or two methods of forecasting the expected demand. We wish to emphasize that in a given concrete case, all the customers, orders or articles need not necessarily be treated in exactly the same way.

In every system of stock replenishment it is necessary to calculate the re-order quantity and the re-order date. There are numerous factors which may add to the difficulty of this calculation. Some of them will now be defined. In the short term there may be limitations in the form of money shortage or lack of space, or other resources. The re-order quantity (chosen by the customer) affects the buffer stock which the supplier must maintain. Nor would it be in the interests of the customer himself to calculate the re-order date and the re-order quantity independently, since both affect the costs of placing an order, stock keeping and stockout, or the avoidance of same. Again, much depends upon the re-order system employed, whilst the stock variation and associated costs also depend upon a variety of factors, so that it is necessary to formulate rules covering various circumstances. Last but by no means least, there is the question of the values to be substituted for the different factors in the formula.

Camp’s formula, discussed in ‘An Introduction to Production and Inventory Control’, will now be briefly re-iterated. We shall do so on the same assumptions as in Section 1.1 and add that N = +∞ thereby eliminating the risk of stockout (b* = 0).

The factors occurring in Camp’s formula, discussed in ‘An Introduction to Production and Inventory Control’, will now be explained more fully.

The stock variation associated with production for stock and with abrupt replenishment of the stock with a batch size Q* has been described earlier. It was then assumed that the stock replenishment takes place in a very short time, or in other words that the ratio of the rate of usage (D) to the rate of production (P) is zero (D/P = 0), D representing the usage per time unit and P the production per time unit (e.g. per year). In reality the production of a replenishment order may take some time, so that this assumption is no longer valid.

So far, Q* has been calculated on the assumption that the inventory is controlled according to a (B, Q) method and that B* and Q* are established independently of each other and separately for each article. We shall now define a suitable approach to the calculation of batch sizes in a periodic re-ordering system, for example a (s, S) method.

The batch size can be determined by means of certain techniques which, although they do not produce very exact results, are nevertheless very useful for practical purposes.

The function of buffer stock and re-order level is explained fully in ‘An Introduction to Production and Inventory Control’. To sum up.

The preceding example of the newspaper vendor involves two cost item, namely TK and TV. These amounts should be regarded as ‘opportunity costs’; considerable discretion is called for in fixing these amounts, as explained in some detail by Starr and Miller [St 1]. The ‘opportunity costs’ TK and TV are established by solving an arithmetical problem, as follows (applicable to the situation of the newsvendor).

A number of variants of the graphical-empirical method are described in the literature. It is the simplest and most understandable, but in most cases also the most laborious method. Despite this drawback, however, the graphical-empirical method constitutes an excellent introductory study not only for those engaged in mastering the principles of inventory control, but also for companies planning to introduce these principles for the first time, since it provides a step-by-step analysis of events. At the same time, every effort should be made to ensure an eventual change to one of the other methods, described later, since they are less time-consuming.

A situation which is readily calculated occurs when theoretical considerations suggest, and practical tests confirm, that the demand follows a Poisson distribution, the average value of which can be forecast with reasonable accuracy.

The possibility of variation in the lead time t itself has been ignored in the calculations so far. It was argued in paragraph 18.2 that this method of calculation is usually justified. At the same time, there are two possible situations requiring a calculation procedure which takes account of a standard deviation in the delivery time.

If the interval (S—s) in an (s, S) system is large in relation to the demand per re-order interval and if the lead time t is constant, then the following will hold good to a reasonable approximation (see Fig. 56).

The levels s and S in a (s, S) system can be established relatively simply for two extreme cases, namely: 1s = S, the system of fixed-interval replenishment;2(S — s) ≫ d, or in other words, where the need to place an order only arises infrequently (less than once per 5 periods). In other cases dynamic programming (DP) may provide a solution.

The ‘classical’ inventory studies are planned for 1 article, held in stock at one point. This applies not only to Camp’s publication (1922), but equally also to that of Galliher, Morse and Simon (1959).

This situation is also described in ‘An Introduction to Production and Inventory Control’, paragraph 17.2.

The situation ‘several articles; stock points parallel’ has a number of features in common with the situation ‘one article; stock points parallel’ discussed earlier, but is more prone to complications.

Inventory control in the true sense of the term involves making a regular comparison between 1, the actual stock on hand and 2, the norm to which this stock must conform. It is not easy to determine the actual stock on hand at any given moment and, if anything, still more difficult to establish norms. In order to provide norms which are in any way realistic, the first essential is to make a careful study of what the practical application of inventory control really involves.

Given an assembly plant comprising a number of small, independent assembly lines, each assembly group operating such a line can produce one complete article. A plant might consist of, say, a hundred or more of such groups.

To calculate a capacity it is necessary to know the total amount of work involved per product or per unit product. Work measurement has already been under review in the work-study sector for some decades. Originally this was merely for the purpose of establishing norms and calculating wages; in many cases work study and wage calculation were virtually synonymous.

Controlling a stock of spare parts very often presents a problem, various solutions to which have already been evolved and introduced. One of these will be discussed in some detail in the present chapter.

A plant produces domestic consumer goods. The production process involves the assembly of parts and subassemblies. The parts are either obtained from sources within the company or purchased from outside suppliers. Parts in the latter category include fixing parts such as screws, nuts and washers. The fixing parts store contains about 500 different articles, varying in price from 0.005p to 0.05p each. The total monthly turnover is about £20 000. Despite the fact that the fixing materials are obtained from many different suppliers, there is no appreciable difference in the lead times of the various articles.

The present chapter deals with the situation in a woodworking factory manufacturing mainly cabinets. A brief account of the organization and successive phase of the production process is followed by a description of the system of scheduling and accepting incoming orders and planning the components required.

The department concerned produces small, invidual products.

The right approach is vital to the success of all activities connected with production and inventory control. Alas, there are all too many examples of investigations which bring no real improvement but merely result in a more expensive system. This is either abandoned as soon as the investigators have gone or simply falls apart in course of time. All such cases constitute a sad waste of time, money, effort and misplaced enthusiasm.

The term simulation as employed in this book means: the step-by-step imitation of an inventory and/or production pattern such as might be encountered in reality.