Every manufacturer is confronted with the problem


 Oswin Flowers
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1 HOW MANY PARTS TO MAKE AT ONCE FORD W. HARRIS Prodction Engineer Reprinted from Factory, The Magazine of Management, Volme 10, Nmber 2, Febrary 1913, pp , 152 Interest on capital tied p in wages, material and overhead sets a maximm limit to the qantity of parts which can be profitably manfactred at one time; "setp" costs on the job fix the minimm. Experience has shown one manager a way to determine the economical size of lots. Every manfactrer is confronted with the problem of finding the most economical qantity to manfactre in ptting throgh an order. This is a general problem and admits of a general soltion, and, however mch it may be advisable to exercise jdgment in a particlar case, sch exercise of jdgment will be assisted by a knowledge of the general soltion. The writer has seen the practical workings of a firstclass stock system and does not wish to be nderstood as claiming that any mere mathematical formla shold be depended pon entirely for determining the amont of stock that shold be carried or pt throgh on an order. This is a matter that calls, in each case, for a trained jdgment, for which there is no sbstitte. There are many other factors of even more importance than those given in this discssion. Bt in deciding on the best size of order, the man responsible shold consider all the factors that are mentioned. While it is perfectly possible to estimate closely enogh what effect these factors will have, the chances are many mistakes costing money will be made. Hence, sing the formla as a check, is at least warranted. Given the theoretically correct reslt, it is easy to apply sch correction factors as may be deemed necessary. In determining the economical size of lot the following factors are involved: Unit Cost (C). This is the cost in dollars per nit of otpt nder continos prodction, withot considering the setp or gettingready expense, or the cost of carrying the stock after it is made. (5"). This involves more than the cost of getting the materials and tools ready to start work on an order. It involves also, the cost of handling the order in the office and throghot the factory. This cost is often neglected in considering the qestion. Operations Research Vol. 38, No. 6, NovemberDecember 947 Most managers, indeed, have a rather hazy idea as to jst what this cost amonts to. If sch is the case an investigation will show that the cost of handling, checking, indexing and sperintending an order in the offices and shops is a considerable item and may, in a large factory, exceed one dollar per order. The setp cost proper is generally nderstood. Indeed, shop foremen in general appreciate only too well what the cost of setp means on small orders, and so, if left to themselves, will almost invariably pt their work throgh in large qantities to keep down this item. So doing, however, affects nfavorably the next factor. Interest and Depreciation on Stock (/). Large orders in the shop mean large deliveries to the storeroom, and large deliveries mean carrying a large stock. Carrying a large stock means a lot of money tied p and a heavy depreciation. It will here be assmed that a charge of ten per cent on stock is a fair one to cover both interest and depreciation. It is probable that doble this wold be fairer in many instances. Movement (M). It is evident that the greater the movement of the stock the larger can be the qantities manfactred on an order. This, then, is a ital factor. Manfactring Interval (7"). This is the time reqired to make p and deliver to the storeroom an order, and, while it seldom is a vital factor, it is of vale in the discssion. There is another factor, X, the nknown size of order which will be most economical. Ths smmarizing, there are the following factors in the problem: M eqals the nmber of nits sed per month (movement). C eqals the qantity cost of a nit in dollars or the nit cost X/90/ $ Operations Research Society of America
2 948 / HARRIS S eqals the setp cost of an order in dollars. r eqals he manfactring interval in months. / eqals the nit charge for interest and depreciation on stock. X eqals the nknown size of order, or lot size, which is most economical. The manfactring interval is sefl only in that it enables s to fmd the safe stock minimm, or smallest qantity the storekeeper may allow his stock to fall to before he mst enter an order for more. At first sight this minimm qantity wold seem to inflence the amont of stock and therefore the interest charges. It does nothing of the kind, however, and it will be fond that the stock consists of additions in lots of X and a gradal exhastion of the stock to nothing. The stock minimm simply serves to notify the storekeeper when to enter an order for new stock, so that he will se p his stock clean before deliveries on the new order are made and, at the same time, never be withot stock for any considerable interval. The average stock, if the movement is reglar, it will be evident, is onehalf of X. If the movement is irreglar, and it generally is, there is introdced an additional complication. This, however, can generally be neglected or applied as a correction factor to the final reslt. The average stock being X/2, the vale of this stock will evidently be C times this, or CX/2 (vale of average stock on hand). This is the qantity cost only. To it mst be added the setp cost for the average stock. Since the setp cost per order is S, and the average stock is half the size of an order, the setp cost of the average stock will be S/2. The total vale of the average stock will then be V2{CX + S). The annal interest and depreciation cost at ten per cent will be onetenth this or VioiCX+S). Now since M nits per month are sed, this will be 12M nits per year, and this interest charge mst be divided by the nmber of pieces sed in a year to get the interest charge in dollars per nit, which gives (/240M)(CX + S) eqals /. The total setp cost for X nits being S dollars, the setp cost per nit mst be S/X. This now gives, as the whole cost of a nit, the interest charge per piece pls the setp cost per piece pls the nit cost per piece, or Let this smmation eqal Y. The problem then is the old one of finding the vale for X that will give the minimm vale to Y. As the soltion of this problem involves higher mathematics, sffice it to say that the vale for X that will give the minimm vale to Y, redces to the sqare root of (240MS divided by C). Now 240S/C may be calclated at once and the sqare root taken. Call this reslt K, becase it will be a constant for any case. Then X eqals K times the sqare [root] of M Now let an actal example be taken and see what the reslts will be. Sppose that an article has a movement, M, of 1,000 nits per month with a setp cost of two dollars and a nit cost of ten cents. Applying the formla, it is fond that the theoretical economical size of lot is 2,190 nits. This shows the setp cost to be abot 0.1 cent and the interest charges abot the same amont. Referring to the Figre I a crve will be seen representing the cost per piece of setp for varios manfactring qantities and an interest and depreciation charge nder the same conditions. The sm of these two is marked the total cost, althogh it does not inclde the nit cost of ten cents, which is not added becase assmed constant. U 03 J2 0.2 < Q a 0.1 Manfactring 1 V T EcoDomical Lot Size, Giving Mtnlmm 1otal Colt?< fotal Cost (Not Incldinc Unit Coil ol Article) Size of Order Crves Interest and Dfprcciailon on Siork Figre I. An increase in the size of the order reslts in an increased interest charge and a decreased setp cost. The crves show this graphically and indicate a minimm total cost in this case at 2,200 nits. _ y
3 OR Form / 949 It shold be noted that this socalled total cost can vary between wide limits only when the manfactring qantity is selected with very poor jdgment. For example, in the case given, the least total cost possible will be abot cents at 2,190 nits on an order. This qantity can vary from 1,000 to 5,000, and the additional cost will be only abot 0.05 cent. This on an article costing ten cents, is a very small percentage. While this is tre for the vales given it is not niversally tre, and ths it is seen that the general law can be applied with some profit to the specific problems of manfactre. Some actal examples of costs may be illminating. Take, for example, the copper connector shown in Figre II. Here is an article that is being sed at the rate of 1,230 pieces a month, with a nit cost of $ per piece, and a setp cost of $2.15. This latter cost incldes the clerical work, sperintendence, and so on, as well as the actal cost of getting ready in the shop. It is fond by applying the formla that the correct manfactring qantity is 6,850 pieces. The different vales of the cost are shown in the crves on the Article A Small Connector Material =Copper.OOiO a ^.0006 Q C.0005 ^0004 U w Monthly Qantity  M Cost per Piece  C i.o135 S = 2.15 Mfg. Qantity =X=6850 Total Cost(Noi Incldins Unit Coil o( Article) '"., ' Inicrcsl and Depreciation n Sinrk diagram. It will be noted that the loss is mch greater if too small a qantity is pt throgh, than it is for too large a one. For example, if this article were made in 2,200 lots, $0,001 wold have to be added to the original cost of $0.0135, which wold involve a loss of $ On the other hand, if they were pt throgh in lots of 10,000, which is abot as far the other way, there wold be a loss of only abot oneeighth of this amont. Or, in percentages, in the first case there wold be incrred an nnecessary expense of abot two per cent on the original cost, while in the latter case the loss wold have been only abot one qarter of one per cent. It shold be noted that in this case this article wold be pt throgh in lots that wold last for nearly six months. Take, as a frther example, the std shown on Figre III. Here is a piece that is sed in lots of thirty. It is an expensive piece and costs in qantity $5.65. The setp cost is $1.85. The correct qantity is 48.5 or, say, 49. Here the least overhead for setp and interest charges wold be $0,076 per piece. Making the article in lots of twenty, there wold be abot the same nnecessary loss as when they were made in lots Article = A Std Material = Copper Qantity per Month '^ M = 30 Cost per Piece. C Dollars, = S = 1.85 Dollars Manfactring Qantity:.X = 48.5 =.10 o Q a O U Total Cost(Not iticld'mi: Unit Cost of Arliclf) ' Interest and Depreciation on Siork V / / ^ o 240 X 30 X U Thosands of Pieces ioo Pieces on Order Figres II and III. In each of these diagrams is shown the effect of the size of the lot on the setp and interest charges per nit. The setp cost crves slope downward with increased size of the order while the interest crves slant pward. The sm of the two elements gives total costs the crve of which is the pper one and shows a minimm opposite where the two lower crves cross.
4 950 / HARRIS of 100, or abot three cents each. This is not a large amont, bt at the rate of consmption given, it will involve a loss of $10.80 a year, which is avoidable and, therefore, worth considering. So it is seen, the more valable the article, the more "worth while" it is to apply the formla. There are not many men who nderstand the theory nderlying the economic size of lots, and so a knowledge of it shold be of considerable vale. For example, having once determined that it is wise to pt in orders for lots of one hndred, based on a certain consmption, it is of vale to know that this consmption mst increase for fold to warrant dobling the manfactring qantities. It is frther gratifying to know that the effect on profits from an error is so small as shown by the crves. In conclsion, it may be well to say that the method given is not rigorosly accrate, for many minor factors have prposely been left ot of the consideration. It may be objected that interest and depreciation shold be figred, not only on original cost, bt also on the setp cost, since that has to be incrred before the parts can be stocked. Sch refinements, however, while interesting, are too fine spn to be practical. The general theory as developed here is reasonably correct and will be fond to give good reslts.
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