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Economic analysis of inventory systemsCHANDAN GURNANI aa 11177 W28th Street, Los Angeles, California, CA 90007, U.S.APublished online: 27 Apr 2007.

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11'T..J. PROD. RES., 1983, \'01.. 21, ;>0. 2,261-277

Economic analysis of inventory systems

CHANDAN GURNANlt

Traditional inventory analysis is based on undiseounted constant cycle costs.Often it has been argued that incorporating the time value of money will not havea significant effect on the lot size. This paper presents discounted cost analysis ofafew inventory models. Effect of discounting rate on the economic order quantityis studied. Alternative discounted profit functions are also given.

NotationListed below are the symbols used for the terms mentioned against each.

cl = set-up or ordering costc2 = production or purchase (invoice) cost per unit.C3 = average inventory holding cost per unit per unit. time, includes cost of

space, deterioration, obsolescence, etc.C4 = back ordering cost per unit per unit timeCs = sale price per unitd = uniform demand rate

f(l) = continuous cash flow function of timef(n) = discrete cash flow function of timeF(s) = Laplace transform of f(l)F(z) = Z-transform of f(n)

i=discrete annual discounting ratej = nominal annual inflation ratek = number of discrete cash flows in the first periodL = lag time between placement. of an order and its receiptn = an index coefficient; discrete time variablep = period (inventory cycle) lengthq=inventory level at the beginning of a cycle

ql =maximum inventory level at any point during a cycleQ=replenishment size (lot size produced or ordered)

=q, if no shortages allowedr = nominal annual discounting rate, the rate of return desireds=a complex variable, Laplace frequency1= continuous time variable

T=life (planning horizon) of the inventory system1l = uniform production ratez=Z-transform operator

EOQ = economic order quantityEUAV = equivalent uniform annual value

Ie= undiscounted inventory cycle costIC'['=total undiscounted inventory cost for time T

Received i April 1982.t 1177 W28th Street, Los Angeles, California. CA 90007. U.S.A.

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2U2 C. Curl/ani

PW = present worth, present worth of inventory cycle costPW I = present worth cost of the first inventory cyclePW 1 = present worth for a finite planning horizon T: present worth cost of

the inventory system for T yearsr=tax rate

naT= before tax profitnAT = after tax profit

*=superscript to denote the optimum value.

IntroductionIn inventory analysis literature, the economic order quantity (EOQ) is computed

by optimizing the average cycle costs ignoring the time value ofmoney . Perhaps this'is because ofdifficulty ofdiscounting the periodic cash flows that occur in inventorysystems. Though, in general, discounted cash flow analysis is strongly recommendedand often used in captial budgeting and investment analysis (Gurnani), yetinventory analysis is treated differently. Van Horne (1!J74, p. 508) justifies this onthe basis of an artificial" difference between a capital asset and inventory ... theformer involves a discrete investment, while the latter represents a continuum ofpossible investments".

Thompson (1975) questioning "why inventory management be treated anydifferently than the management of fixed assets?" finds economic order quantity bymaximizing present worth of the after tax profits for an "indefinite future" planninghorizon. '

Often, it has been claimed that optimizing the average cycle cost (or profit) willyield the same result as obtained by optimizing the discounted cost (or profit),particularly if the interest rates are "low" and the planning horizon is "short" (forexample, Hadley and Whitin (1963, p. 23), Hillier and Lieberman (1975, p. 475)).

Interest rates currently are not 'low' and the planning horizon mayor may not be'short'. Inventory policies obviously affect profitability and any particular policychoice depends upon relative profitability. A measure ofprofitability that disregardsthe time value of money has "the advantage of simplicity, but can yield verymisleading results" (Holland et al, 1973). Therefore, it is preferable to use discountedcash flows to improve validity of the analysis. In addition, it is desirable to knowwhether discounting makes any difference or not. Grubbstrom (1979) argues infavour of discounting irrespective of length of the planning horizon. However, heassumes that the inventory process continues ad infinitum and obtains anapproximation to equivalent uniform annual cost (EUAC) by Taylor's expansion.An infinite horizon inventory process is ofrare occurrence because the costs are likelyto vary disproportionately and because of change in product specifications anddesign or its abandonment or substitution by another product.

Grubbstrom's (1979) analysis differs from that of Thompson (1975) in that heuses cost minimization objective function, whereas the latter maximizes after-taxprofits. The traditional undiscounted cost inventory analysis is also based on costminimization criterion. Schroeder and Krishnan (1976) present a 'return oninvestment inventorv model arguing that maximization of return on investmentrepresents the shareholder's objective function. This has rekindled the controversyregarding the appropriate objective function for a firm (Schroeder and Krishnan1978) and is not pursued further here. However, Schroeder and Krishnan (1976)ignore the time value ofmoney and hence their analysis is theoretically questionable.

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Economic analysis of inventory syslems 263

Present Worth of periodic cash flowsPresent Worth (PW) is the figure of merit in this paper. A general consensus

appears to have been reached on the use ofP"V as an economic decision criterion asevidenced from its more frequent use in the published literature and its increasingadoption in industry (Gurnani). PW is defined as the current (present.) value of afuture cash flow stream or continuum discounted at an appropriate rate.For discrete cash flows

T

PW= L C.(l+W·rl=Q

and for a continuous flow function,

PW= J:C(I)exp(-rl)dl

where

(la

(1 b)

C. = cash flow at the end of nth yearC(t) = continuous cash flow function

i = discrete annual discounting raten.= discrete time variabler = continuous annual discounting rateI=~ontinuous time variable

T = cash flow function life (years)

If desired, equivalent uniform annual value (EUAV), also known as theannuity--defined as a series of uniform annual cash flows for a specified number ofyears (T)-may be obtained by a linear transformation of PW.

EUAV =PW exp (rT) [exp (rl) -1][exp(rl)-I]

(1 c)

(2 c)

(2a)

(2b)

The cash flows occurring in an inventory system are periodic, that is, thecontinuum or sequence of cash flows repeats every period. Using elementary algebraand calculus, one can obtain present worth (PW) of such cash flow functions bycomputing PW of the cash flows in each period and adding them. However, closedform expressions for PW of periodic cash flow profiles are developed in (Gurnani) byusing the transform calculus (Churchill 1972), and are given below.

For continuous periodic cash flow functions (Fig. 1 (a))

l-exp (-rl) IPWT = F,(s). l-exp (-rp) s=,

1- exp ( - rT) fPI, (I)exp (-rl) dl

l-exp (-rp) 0

l-exp (-rT)= PW,

l-exp( -rp)

and for discrete periodic cash flows (Fig. 1 (b))

PW T l-Z=:F , (Z)[l-z z=exp(r)

(2d)

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C. Cumani

>0 tz

(a)

r( n)

1,

(b)

Figure I. Periodic cash flow functions. (a) Continuous; (b) discrete.

where

I-z- rk I

1 - z p "~Of 1(n)z -" ,= exp In

I-exp (rT)---=---1"W(I-exp( -rp)

(2 e)

(2j)

fdn) = discrete first. period cash flow function=f(n) O~t<p

= 0 otherwisefdt) = continuous first period cash flow function

=f(t) O~t<p

=0 otherwisef\ (8) = Laplace transform of fdt)f\ (z) =Z-transform of fdn)

p = period length in yearsr = continuous annual discounting rate8 = Laplace transform operator

T = life (years) of the cash flow functionz =Z-transform operator

PW ( = PW of the first period cash flowPW r = PW of the cash flow function for planning horizon T years.

PW of some typical cash flow functions (extracted from Gumani (1980) are given inTable I).

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Economic analysis of inventory systems 265

Functional form PW k={l-exp( - Tr)}j{l-exp -prj)

(I) Saw tooth (ramp or linear growth)

fdx, t) = a(tjp)a>O

O,,:;t<pa

k-z {1-(1+pr)exp(-pr))pr

(2) Saw tooth (linear decay)

f,(x,t)=a(1-tjp) O,,:;t<pa>O a

k-z {exp(-pr)+pr-I}pr

lJ[Y~Obcp 2p t

(3) Linear growthfdx, t) =a+b(tjp)

a>O, b>O

(4) Triangularfdx, t) =a(tjp)·

a(t-p)

b-p

a>O, b>O

(5) Trapezoidalfdx, t) =a(tjb)

=aa(t-p)

c-p

a>O, b>O, c>O

O,,:;t<p

k-z[(apr+b){l-exp (-pr)) -bprexp (-pr)]pr

O,,:;t<b

ak z {b-p+bexp(-pr)-pexp(-br)}

b(b-p)r

c,,:;t<pa

lc z {c-p+bexp (-cr) -cexp (-pr)b(c-p)r

+bcrexp (-cr)}

Table 1. PW of some typical periodic cash flow functions.

A note on discounting rateContinuing discounting is used here. It can be argued that continuous discount­

ing is theoretically correct as it posits that the time value of money changescontinuously with respect to time, rather than at discrete time points (say year end).However, if desired, discrete discounting can be easily accommodated since thecontinuous and discrete discounting rates are related by

orr=ln(l+i)

i=exp (i)-1

(3a)

(3b)

Traditional inventory analysis implicitly assumes that the costs entering theanalysis do not vary during the planning horizon, that is, that the inventory cyclecosts are invariant. This limitation is not necessary for the analysis advanced here. If

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266 C. Gurnani

an inflation ratej can be estimated, it can be combined with the rate ofreturn desired(r') to get a compostite discounting rate r given by (Davidson 1975)

or

(l+r)=(I+r')(l+j)

r=j+r' +jr'

(4a)

(4 b)

Using r as the discounting rate allows the various costs to be held as constant(thereby simplifying the analysis), any variation in them being accommodated bythe inflation ratej. Note thatj is conceived to be a single aggregate variation in costadjustment rate for all the inventory cycle costs. This assumption, however, is notnecessary as different inflation rates specific to various costs entering the analysiscan be incorporated at the cost of simplicity.

Present Worth analysis of inventory systemsInventory models may be classified on the basis of nature of demand (de­

terministic or random) or inventory review policy (continuous or periodic). Demandis an important component of inventory control system and is generally beyond thefirm's control. In the absence of further information, a simple and intuitively logicalapproximation is to assume the demand to be uniform. Inventory models treated inthis chapter are single product, uniform demand, deterministic models. Such modelsdo not differentiate between continuous and periodic review policy because of thedeterministic constant demand rate. Other elements influencing inventory controland management are various costs. Only variable costs enter inventory analysis forcost minimization as the fixed costs do not affect the inventory policy. Other costs,such as taxes and insurance, mayor may not be included depending upon theirrelevance. For example, insurance cost may be included in' the inventory carryingcost and the tax effect can be accounted for as in eqn. (12).

Inventory items can be classified according to the hierarchy given below.

End-product

Sub-assembly

I

Sub-components

Raw-materials

The analysis below applies to all the above levels of production. Also, it does notdifferentiate between the parts made in-house and those supplied by a vendor.Various costs referred above will take correspondingly appropriate meaning. Forexample, for raw materials or the parts supplied by a vendor, the order costcomprises the cost of inviting and scrutinizing bids, placing an order and follow up orexpediting, etc., whereas for the in-house manufactured items, it includes cost ofmaking a job order, set-up cost, etc. Manufacturing cost in the latter case will includethe cost of materials or parts required, the labour costs and the associated overhead.Thus, for the end-product, cost of sub-assemblies will be a part of its manufacturingcost. In such a case, holding cost for the SUb-components may be zero if they are

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Economic analysis of inventory systems 267

made and transferred to the final assembly work station at the rate commensuratewith their demand for it, and may be non-zero ifthey are manufactured at a differentschedule to accommodate other products.

The following hypothetical data are used for the purpose of illustration.

c, = $12 000C2 =$10c3=SO'30C4 = S1-10d=8000 units/monthu=30 000 units/month£=0'5 month

Uniform demand, infinite replenishment rate, zero lag, no siockouis inventory system(Case I)

This is the most common system encountered in inventory theory. Also known asthe economic lot size model, it postulates that:

(a) Demand for the item occurs uniformly at a constant rate per unit time, andquantity demanded is a continuous function of time. Under this hypothesis,inventory review policy does not have any influence on the system, fordeterministic uniform demand implies perfect information about the in­ventory level at any time, whence not requiring any need for a continuousreview of inventory position.

(b) Replenishment rate is infinite-the ordered quantity arrives in a single lot.(c) Replenishment orders are placed whenever inventory levels reach a pre­

scribed reorder point.(d) Replenishment size is constant-the items are produced or ordered in equal

numbers.(e) Demand orders are instantly filled.(j) Payments are received (due to sale of the products) or incurred (due to

placement of the orders) at the instant corresponding activity occurs.(g) The inventory process starts just after the lot ordered previously has been

received-starting inventory level equals the lot size.(h) All unit costs are independent of time and the demand or production rates.

Figure 2 shows the relationship between inventory level and time for the case ofuniform continuous demand, 'single lot, instantaneous delivery inventory systemwith no shortages. Corresponding associated cash flows are also shown.

For the first inventory cycle (O~t<p)

PW(set-up cost) =c[PW(production cost) =QC2PW(inventory holding cost}=QC3{ exp (-pr) +pr-l }/pr2

hence PW cost of the first cycle is

PW, =(c, +QC2)+QC3{ exp( -pr)+pr-l}/pr2

and PW for the planning horizon T is

PWr=PW, x {1- exp (-Tr)}/{l-exp (-pr)}

(6a)

(6 b)

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268 C. (Turnruii

Gl1p 3p 4, I

Q'1

Cost of Ifnits Ordered

'1

Ordering Cost

131. •

Inventory Carrying Cost

TQ

1 •Inventory Level

Figure 2. Uniform demand, infinite replenishment rate, zero lag, no stockouts inventorysystem (Case I).

The undiscounted inventory cycle cost

IC=c, +C2Q+C3Q2/2d (7a)

and

(7b)

where

T'[p =number of cycles in the planning horizon T.

Both the PW cost and the IC cost can be minimized by differentiating equations(6 b) and (7 b) with respect to Q. Equation (7 b) yields the well-known square rootformula:

(8)

Equation (6b), when differentiated, gives a nonlinear equation in Q.

iJPW T~=F= -[(c,rjd) +C2+ h/r)J exp (-Q*r/d)+ (C2 +C3/r)

- [(C2r/d) +C3/d)JQ* exp ( - Q*r/d) =0 (9)

Equation (9) may be solved by numerical methods since an explicit expression forQ* in terms ofthe other parameters is not obtainable. However, Table 2 values were

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Economic analysis of inventory systems 269

Case I Case II Case III Case IV Case V

r Q* p*(days) Q* p*(days) Q* q* p*(days) Q* p*(days) Q* p*(days)

0 25298 95 25298 98 28540 22424 107 29542 111 48990 490'05 15246 57 15435 58 18575 18380 70 17933 67 49666 500'1 11853 44 12146 46 13450 13412 50 13994 53 50360 500·2 8802 33 9236 35 9387 9248 35 10447 40 51809 520'3 7281 27 7820 29 7590 7320 29 8674 33 53343 530'4 6330 24 6957 26 6518 6099 24 7565 28 54956 550'5 5664 21 6368 24 5785 5203 22 6787 25 56667 57

Table 2. EOQ as a function of r.

obtained by optimizing eqn. (6 b) using the ZXGSN routine ofIMSL (1979) based onthe Golden Section Search Method (Brent 1973, Fletcher 1972, Gottfried andWeisman 1973), a dichotomous sequential search method in which the ratio of thesuccessive extremal intervals is constant.

Note that the optimum Q* from eqn. (8) does not depend upon C2, the productcost, whereas it is not independent of C2, when discounting is allowed. This can beseen from eqn. (9), which also shows that Q* depends upon the value of thediscounting rate (r). The relationship between Q* and r is plotted in Fig. 3. Q* and thecorresponding p* are seen to decrease with an increase in r. This is due to theinteraction between the various costs. Inventory holding cost decreases with adecrease in the period length, whereas the production or purchase cost and the order

-----CaseIV•••.•••••.•• Case III--- Case II_._._._. Case I

30. 000

25,000

0 20,000~

e,0. 15,000

0

j 10,000

5,000

0.1 0.2 0.3 0.4 0.5

pf scoun t tnq Rate(r)

Figure 3. Effect of discounting rate on economic order quantity (EOQ).

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270

cost increase with it. Note that the optimum order quantity is independent of theplanning horizon, T (eqn. 9), though the total inventory cost does depend upon it.

It may be observed that the profit maximization objective function can be easilyconstructed if the cash inflows due to sale of the inventory items are included in theanalysis above. Since the demand is uniform, the cash inflows may be considered tooccur at a uniform rate ifit is assumed that the payment is received at the instant thedemand is met. The time between dispatch of the items and receipt of payment can,however, be accounted for. PW of the cash inflows for the first cycle (O,,;;t <p) is

Qcs{exp ( -pr)+ pr -I }/pr 2

Hence before-tax profits for T years is

(10)

and if the tax rate is T, the corresponding after-tax profit is

I-exp (Tr) [Q(Cs -C3) ]llAT=(I-T) 2 {exp(-pr)+pr-l}-(Cl+QC2) (12)

l-exp (-pr) pr

Uniform demand, infinite replenishment rate, nonzero lag, no siockouis inventory system(Case 11)

The (lag) time between placing an order and its receipt is generally beyond afirm's control and is in practice not zero. As a first approximation, one may take it tobe a constant, say L. Figure 4 gives the inventory level and the associated cash flowstructure due to various costs as a function of time. For the first cycle (O,,;;t<p)

PW (set-up coSt)=CI exp {-<p-L)r}

PW (production coSt=QC2

PW (inventory holding cost) =QC3{exp (-pr)+pr-l}/pr 2

hence

PW (total cycle cost) =PW 1 = CI exp {- (p - L)r}+QC2 +Qc3{exp (-prj +pr-l }/pr 2 (13)

and

PW,.=PWI x {1-exp (- Tr)}/{l-exp (-pr)} (14)

arw,~= - [(clr/d) exp (Lr)+ (Q*rc2/d)]{exp (-Q*r/d)} + [C2 + (C3/r)]

- [(<;2 +C3/r) + (C3Q*/d)]{ exp ( - Q*r/d} =0 (15)

Since undiscounted cycle cost is not affected by timing of the cash flows, the deliverylag does not influence it, and is thus given by eqn. (7).

Table 2 lists Q* for different values of r and Fig. 3 plots Q* as a function of r. Acomparison with Case I shows that introduction of the lag time results in a higher Q*due to delayed payments of the production 01' purchasing (C3) costs, as it has beenassumed that the costs are incurred at the time ofreceipt of the ordered items. Theincrease in Q* due to the lag varies from 1'2 to 12 per cent depending upon the valueof r.

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Economic analysis of inventory systems

':-_--'__--JL-_--JL-_--'......E:lt

271

Figure 4. Uniform demand, infinite replenishment rate, nonzero lag, no stockouts inventorysystem (Case II).

Uniform demand, infinite replenishment rate, zero lay, shortaqes with back-orderinginventory system (Case III)

Sometimes, it may be economical to permit shortages ifloss of revenue due to theunsatisifed demand is offset by the reduced inventory cost due to fewer orders placedbecause of the increased cycle length. Two distinct cases arise: (a) the unsatisfieddemand is not met and is lost; and (b) it is back-ordered until procurement arrives.The essential difference between the two is in the costs associated with the back­ordering or the loss of sales. The back-ordering case is analysed below. It. is assumedthat the back-orders are filled before any new demand is satisfied. Zero lag isassumed, though nonzero lag can be easily accommodated as demonstrated inCase n.

An important question relates to the shortage costs. The cost of unsatisfieddemand varies, depending upon the system characteristics. Hit is too high, one willnever want to incur any shortages; on the other hand, it cannot be zero, for then theproduct may never be inventoried. In the absence of further information and for thepurpose ofanalysis, one may hypothesize that the back-ordering cost is composed oftwo components: a constant part when a shortage occurs, and a variable partproportional to the length of time for which the shortage lasts. However, the fixedpart of the back-ordering cost is ignored here for the sake of simplicity.

Shortage per cyde=Q-q (Refer to Fig. 5)

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272 C. Gurnani

El':------l.----!---i,----,l---< t

Cost of Units Ordered

Order Cost

13

1 ..........~---->......I...-....Jo...I----' ..........ir---_Inventory Holc1ing Cost

•t

ITQ q

1 1 I-"<+-----'rlf-----''rl--'<-t-f---

Inventory Level

Figure 5. Uniform demand, infinite replenishment rate, shortages with back ordering, zerolag inventory system (Case III).

Undiscounted cycle cost (Hadley and Whitin 1963, p. 479)

to= c, +czQ +C3qz/2d +C4(Q _q)z /2d (16a)

and

(16 b)

(17)

Differentiating eqn. (16 b) with respect to Qand q and equating to zero, one obtains

J(2dC1 C3+(4) J(2dCI (4).Q*= --x--- and q*= --x---

C3 C4 C3 C3 +C4

For O~t<p,

PW (set-up costj l e- c.

PW (production cost) =qcz

PW (inventory holding cost) =qC3{ exp (-br) +br-l}/brz

PW (shortage COKt) =c4(Q -q)r exp (-br) - (b,r+ 1) exp (-pr)]x e~p (-br)/blr z

where

b=q/d,b , =(Q-q)/dand p=Q/d.

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Economic analysis of inventory systems 273

Therefore, PW (total first cycle cost) is

PW 1 = (Cl +czq) +C3q{exp (-br) +br-l }/brz +C4(Q -q)

[exp (-br) - (b1r+ 1)exp (-pr)] exp (-br)b1r z (18)

and

PWT=PW 1 x {1- exp (- Tr)}/{I-exp (-pr)} (19)

Note that eqn. (19) is a function oftwo variables Q and q. By differentiating itwith respect to Q and q, one will get two nonlinear equations which, when solvedsimultaneously, will give the optimum value of Q and q. However, direct optimiz­ation ofeqn. (19) using the ZXMIN routine ofIMSL (1979) using a quasi-Newtonianmethod (Brent 1973, Fletcher 1972, Gottfried and Weisman 1973) was used to obtainthe values of Q* and q* listed in Table 2.

The profit function for this case can be obtained in the same way as for the Case Iin eqns. (11) and (12). Assuming that the cash inflows due to sale of the inventoryitems take place at the instant the demand is met, PW of the cash inflows for Tyearsis given by

[Qcs Qcs(P-b) ]1-eXP(-Tr)-z {exp(-br)+br-l}+ exp(-pr)br p l-exp (-pr)

(20)

Hence the before-tax profit function is obtained by subtracting the discounted cost(eqn. 19) from the discounted income (eqn. 20). Multiplying the before-tax profit by(1- 0) yields the after-tax profit.

Uniform demand, finite replenishment rate, zero lag, no stockouis inventory system(Case IV)

The assumption of an infinite replenishment rate increases the inventorycarrying cost and may give erroneous results if the replenishment rate is not infinite.This section deals with an inventory system with a finite uniform production rate of1l per unit time. It is assumed that the items go directly to the warehouse as they areproduced, making the production process an integral part of the inventory system.As before, it is supposed that the demand is deterministic, at a constant rate d perunit time, and that no shortages are permitted.

During the time the items are being produced, there is a net outflow of (u-d)units per unit time into the warehouse, while the outflow from the warehouse iscontinuous at the rate d. Figure 6 portrays the inventory level and the paymentstructure of the system as a function of time. It can be easily shown that

ql =Q(u-d)/u=b(u-d)

b=Q/u=qd(u-d)

(p-b)=q(u-d)/ud

and

p=Q/d

Then

(21a)

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274 C. Curnani

~':-_--'-__-:'-_---:.I-_--'---<~

Set Up Cos t

Production Cost

•tInventory Holding cost

J, A1I~ /\ .l~,.t-,_-_

I;~1V\I\IVb P I

Inventory Level

Figure 6. Uniform demand, finite replenishment rate, zero lag, no stockouts inventorysystem (Case IV).

and

(21 b)

Differentiating with respect to Q and equating to zero, one obtains

J2dcj

Q*=. c3(l-d/1l)

PW for the first cycle (0:;;; t < p) is

PW (set-up cost')=G1

PW (production cost) =Qc2{l-exp (-br)}/br

PW (inventory holding cost)=qjG3{b-p+bexp(-pr)-p

x exp (-br)}/br 2(b-p)

(22)

hence total PW cost of the first cycle is

PW t =G j +QG2{I -exp (-br)}/br+qtC3

x {b-p +bexp (-pr) -pexp (-br)}/br2(b-p) (23a)

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Economic analysis of inventory systems 275

PWT=PW I x {1-exp (- Tr)}/{I-exp (-pr)} (23 b)

oPWT--aQ= [C2 + (c,/r)] exp (-Q*r/u) - [(clr/d) + (C2u/d) + (c,u/dr)]

x exp ( -Q*r/d) + [(c,u/dr) + (C2u/d) - (c2+c,/r)]

xexp(-Q*r/d)exp(-Q*r/1tl=O (24)

The optimum values ofQ for different values of the discounting rate r, given inTable 2 are obtained by using the ZBRENT routine of IMSL (1979) based on analgorithm by Brent (1973) for finding a zero of a function which changes sign in agiven interval. Itcombines linear interpolation and inverse quadrature interpolationwith bisection, and is guaranteed to converge. Figure 3 shows that Q* decreasesmonotonically with increase in r, the discounting rate.

It may be noted that the before tax profit function for this case is given by eqn.(23 b) ifc, is replaced by (c, -(5) and the resulting expression is multiplied by (-1).

Batch demand, finite uniform production rate, zero lag, no stockouts inventory system(Case V)

Further refinement to the inventory models treated above is introduced bytreating demand as discrete and impulsive, instead of being continuous. In thefollowing, items are considered to be produced at a continuous finite uniform rateand are in ven toried as produced, but demand (deterministic and uniform) is discreteand equal to the total production up to the time demand occurs. Figure 7 sketchesthe inventory level versus time relationship and gives the cash flow structure as well.Referring to Table 1, PW cost for the first cycle is

PW =c1 +Qc,{I- (1 +pr)exp (-pr)}/pr2+Qc2{I-exp (-pr}}/pr (25a)

and

PW1·=PW x {1-exp (-Tr}}/{I-exp (-pr)} (25 b)

Noting thatp =Q/u and differentiating eqn. (25 b) with respect to Q, one obtainseqn. (26), given below. When solved numerically for a specific value of r, this givesthe optimum value of Q.

oPWT--aQ= - [cdr/u) + (C3/r)] exp (-Q*r/u) + (Q*c,/u) exp (-Q*r/u) + (c,/r)

exp (-2Q*r/u}=O (26)

Undiscounted cycle cost

(27b)

and

Hence, the optimum value of Q is

(27 b)

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276 C. Gurnani

~!-----',------,L--.."...--.,.4P,.-{'~

Set Up Cost

•t

Production Cost

~£IIIIJ]--------lt---__

Inventory Holding Cost

TQ

1 ..lnven tnr-y level

Figure 7. Batch demand, finite uniform production rate, zero lag, no stockouts inventorysystem (Case V).

Equation (26) was solved numerically by using the ZBRENT routine ofIMSL(1979). The optimum values ofQ so obtained are tabulated in Table 2 for differentvalues of r, the discounting rate. Discounting is seen to increase the period length,indicating that the consequent decrease in the set-up cost outweighs the increase inthe inventory holding cost.

The before tax profit function for this inventory model is obtained by subtractingthe PW cost from the present worth of the cash inflows due to sale of the items, thatis,

l-exp(-Tr)Qcs exp (-pr)

l-exp (-pr)

ConclusionsThe inventory models analysed in this paper demonstrate that the inventory

policy derived from the discounted (PW) cost differs substantially from that givenby the traditional undiscounted inventory analysis. The difference between the twodepends upon the discounting rate. The discounting causes the optimum orderquantity and correspondingly the period length to vary ,monotonically with thediscounting rate. However, it is not influenced by the inventory system planninghorizon.

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Economic analysis of inventory systems 277

L'analyse d'inventaire traditionelle se base sur les couts sans rabais du cycleconstant. Souvent. a-t-on soutenu que Ie fait d'incorporer la valeur temporelle del'argent n'aura pas un effet important sur la taille du lot. Cet article presente uneanalyse de cout avec rabais de quelques modeles d'inventaire. L'effet du taux durabais sur la quantite de l'ordre economique est etudie. Des fonctions alternativesdu profit sans rabais sont. aussi donnees.

Die herkornmliche Bestandsaufnahme basiert auf konstanten Zykluskostenohne Diskont. Es ist. oft argumentiert worden, dall die Beriicksichtigung desGeld-Zeitwerts keinen bedeutenden Einflull aufdie Losgrosse haben wiirde , DieseArbeit zeigt eine mit Diskont Kostenrechnung von einigen Bestands­kontrollmodellen. Der Einflull der Diskontrate auf die Grosse der wirtschaftlicheBestellung wird untersucht. Auch alternative Gewinnfunktionen mit Diskontwerden angegeben.

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