Session-13 Inventory Model II&III

8/2/2019 Session-13 Inventory Model II&III

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= 1,48,000 + 720 + 1,850 = Rs 1,50,570Since the total cost is minimum at Q =2,000, it represents the optimal order quantity.

Model 3: Build-up Model/EOQ Model for Production RunsIt is assumed in the classical EOQ model that the entire quantity ofthe item ordered forisreceived.lot. We will now relax this assumption and consider situations in which the goods are receivedor'a constant rate over time and they are also being consumed at a constant rate. This is particularlysituations when the item in question is being produced internally rather than being procuredi a nsuppliers. When the production begins, a constant number of units are supposed to be addedtot h e


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Inventory Management 459he production run is completed. Simultaneously, the items would be demanded at aedearlier. Obviously, the rate at which they are produced has to be higher than the ratenly then can we think of the inventory build-up.his model is depicted in Figure 9.4.antityQ is produced over a period, tp' which is defined by the production rate p. Sinceile up in one shot but rather continuously over a time period and is also consumedventory level would be determined not only by the lot size' Q , but also be affectedand depletion (demand) rate d.

Maximum in ve nto ry = tp (p - d)= Q ( 1 - % )d

e-


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Holding cost = h x average inventory = h ; ( 1 - ~ )

460 Quantitative Techniques in ManagementOrdering Cost As before, the ordering cost equals DA/Q. However, as mentioned earlier,nthe production situation, it is referred to as production set-up cost including the cost ofdevel .plans, performing the essential paper work, time and resources employed in preparing prO

Thus, total cost isDA hQ ( d)T(Q) =-+- 1--Q 2 pFrom this, the optimal order quantity (Q*), determined, as shown in the Appendix 9A , wouldbe.

Qh~=F~~P~d

T (Q*) =~2AhD ( 1 - ~ )In Example 9.1, suppose that the delivery of the tubes is not made in a single lot and, instead,theyat a rate of 50 tubes per working day. Under the conditions postulated, we have

E . 0d Quanti Q* ~2 x 150 x 2,000 ~oonomic r er tity, = _ 2.40 50 - 8=545 units

Total cost (of holding and ordering) corresponding to Q* =545 units,

T(Q*) = ~2 x 150 x 2.40 x 2,000 (505~ 8 )=Rs 1,100

Note that this cost is lower than the total cost ofRs 1,200 under the regular EOQ model.Maximum Stock Level In the context of the present problem, tp would be equal to the numberlquired to deliver the entire lot. At the supply rate of 50 units a day, it would take 545/50 = 10.9da)1the lot. Thus, .

Maximum Stock Level =tp (p - d) =10.9(50 - 8) =458 unitsRe-order Level =consumption during the lead time

=consumption rate x lead time= 15 x 8 =120 units.


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Inventory Management 461In t o r y Model with Planned Shortages

situations, a shortage (or stockout) is considered undesirable and is avoided, if possible.agesmay, and are likely to, mean loss of customer goodwill, reduction in future orders,n the market share, and so on. While in some situations the customers shift to otherirements,and so may be lost forever, in some others the customers may not withdraw the

tilthenext shipment arrives. This latter situation is called, as mentioned earlier, the back-. Weshall now develop an inventory model under the assumption that back-ordering ismodelassumes that the inventory is replenished precisely when the inventory level falls offquestionof shortages and, therefore, the cost of shortage is not considered in that model.ionof back-ordering, shortages may in fact be deliberately planned to occur. It may be

iconsiderations, specially when the value of the item in question is very high withholdingcost. Basically, then, it is a question of setting off the cost of shortages against the

ost.ofileof this model is shown in Figure 9.5.

T_DTimeI ,

y Profile-Planned Sh orta ges Modellevelndicatesnegative inventory i.e. the number of units backordered. As soon as the lot ofed,thecustomerswhose orders are pending would be supplied their requirements immedi-tbemaximumnventory level would be Q-S. The inventory cycle Twould be divided in twotimewheninventory is on hand and orders filled as and when they occur, and t2-when therealltheordersare placed on backorder.ecostfunction,we would consider the cost of shortages in addition to the holding and theCostofshortagesor the backordering cost is incurred in terms of the labour and special deliv-theossofcustomer goodwill (which may be taken to be a function of the time a customer has

\ Q \ (variable)cost = ordering cost + holding cost + shortage costA s seenbefore,if the cost of placing an order be A, and the total demand beD, we have,

Annual ordering cost = . ! 2 AQ


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Holding Cost As noted earlier, t, is the period in a given inventory cycle when positive'Since the maximum inventory, M, is Q - S, the average inventory level equals (Q - S) /2 . T h u s ,

Holding cost during a given cycle T= Q-S hi,2

462 Quantitative Techniques in Management

From the Figure 9.5, we observe that the quantity Q - S is sufficient to last aQ - S=t, d , where d is the usage rate. Similarly, a quantity Q is adequate to last a fullcycleT,Q =Td. Dividing the first of these equations by the second, we get

Q-S = t,dQ Td

_T...:..:(Q=--_S~)or t,= QSubstituting the value of t, in this expression for holding cost, we get

Holding cost during a given cycle T= Q- Sh T(Q-S)2 Q(Q-S)2hT

2QThere being N orders, and hence N. cycles, per year, the annual holding cost would be asfoli

(Q_S)2 hNTAnnual holding cost ="""""'--~--2Q(Q- S)2 h (Since NT= Iyear)

2QShortage Cost We shall now develop expressions for the average number of shortages n dwith the help of which we shall determine the annual shortage cost. Since Srepresentstheshortages, the average level of shortages, during the period when there is a shortage (~) shallerepresent the backorder cost, that is to say, the shortage cost per unit of shortage per year,we a

Shortage cost in a given cycle T= 1. bt22As before, we note that d =~ =Qt2 T

TSt2=QrSubstituting this value in the earlier expression,

S rsShortage cost per cycle =- b -' -2 Q2


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Inventory Management 463- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ~ - -

bTNS2Annual shortage cost = .. . : . . . . :: . :: .. c .. : :: . . _2Q= bS22Q

(when there are N cycles)

(TN being equal to one)

is,then,the total cost expression would be as follows:T (Q) = . .Q A + (Q - S)2 h + bS2

Q 2Q 2Qthisexpression,as shown in the Appendix 9A to this chapter, the EOQ, Q*, would be

(h) ~ 2ADhS* = Q* -- or S* = ; andh+ b hb+b2. urnInventoryLevel, M=Q* - S* =Q* ( b! h )

T(Q*) = .hAhD ~ bh+btbattheQ * inthis model is different from Q* in the classical EOQ model by the added square

+ b ) / b . Obviously,as the backordering cost, b, becomes larger relative to the holding cost h ,( h + b ) / b and b/(h + b ) in the earlier stated equations tend to approach unity. Therefore, intheresultsshallbe close to the EOQ model results. Similarly, h/(h + b) would, in such a case,assuchS* would also be quite small. As, however, the holding cost becomes larger int-orderingost, the number of back-orders would be large. That explains why many an item

IRhandledon a back-order basis.theT V tubesexample, let us suppose that the shortage cost is known to be Rs 1.60 per unitKprOduceere the entire information. We have,

D =2,000 units per annum,A =Rs 150 per order,h =Rs 2.40 per unit per annum,b =Rs 1.60 per unit per annum,

No. of working days =250Lead Time =15 days

ation,wehaveEc . 0d Q . Q* J 2 x 150 x 2,000 ~ 2.40 + 1.60onormc r er uantity, = 2.40 1.60

=500 x 1.58 =790 unitsOptimallevel of shortages, S*=790 ( 2.40 )2.40+ 1.60

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Session-13 Inventory Model II&III