Excel-Based Solution Method For The Optimal Policy Of The Hadley And Whittin s Exact Model With Arma Demand

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1 Excel-Based Soluion Mehod For The Opimal Policy Of The Hadley And Whiin s Exac Model Wih Arma Demand Kal Nami School of Business and Economics Winson Salem Sae Universiy Winson Salem, NC Phone: (336) namik@wssu.edu Fidelis Ikem School of Business Virginia Sae Universiy P. O. Box 9038 Peersburg, VA Phone: (804) fikem@vsu.edu ABSTRACT In he deerminaion of he opimal policy of an invenory model wih a sochasic demand which includes he calculaion of he reorder poin and he order size, one has o deal wih mean rae of demand, sandard deviaion, safey facor, forecas and lead-ime. The calculaion of he re-order poin is ypically based on he assumpion ha he mean rae demand is deerminisic as a funcion of ime. This assumpion is by far removed from realiy. A more appropriae assumpion would involve he use some sor of probabiliy disribuions o represen he unis demanded including he lead-ime o accoun for he increasing uncerainy in he marke environmen. Under he sochasic environmen, Hadley and Whien developed wo ypes of backorder invenory formulas; an approximae and exac formulae for Poisson and Normal lead-ime demand disribuions wih he assumpion ha here is no correlaion beween wo period demands. In many pracical siuaions, he period demands are no independen, bu exhibi a serially correlaed process. (An, Foopolo, and Wang 1989); (Charles, Marmorsen, and Zinn 1995). In his research paper, we will develop he formula for he calculaion of reorder poin, safey sock and order quaniy of Hadley and Whiin s (1963) exac invenory model when he unis demanded are generaed by a serially correlaed process and can be represened by ARMA Box- Jenkins ime series model for when he lead-ime is boh deerminisic and probabilisic. ARMA ime series process generaing demand wih deerminisic and sochasic discree lead-ime. The disribuion of forecas errors from he calculaion process in Box-Jenkins (1976) ARMA analysis will be used as he measuremen of he esimaion wih which he reorder poin and safey sock are deermined. In he firs par of his research, he deerminaion of he model s reorder poin is based on he assumpion ha he procuremen lead-ime is a random variable generaed by an ARMA process wih consan lead imes. Laer on, we would invesigae he problem in aemping o accoun for he ARMA sysem wih he probabilisic discree lead-imes. 493

2 I. INTRODUCTION The conrol and mainenance of invenories of physical goods is a problem common o all enerprises in any given economy. Two fundamenal quesions ha mus be answered in conrolling he invenory of any physical goods are when o replenish he invenory and how much o order for replenishmen. EOQ models answers he quesion of how much o order, bu no he quesion of when o order. The laer is he funcion of models ha idenify he reorder in erms of a quaniy: he reorder poin occurs when he quaniy on hand drops o a predeermined amoun. The amouns generally includes expeced demand during lead ime and perhaps an exra cushion of sock, which serve o reduce he risk of experience a sock-ou during lead ime especially in he environmen when variabiliy is presen in demand or lead ime or boh. The following four facors are being used in deermining he reorder poin quaniy 1. The rae of demand (usually based on a forecas value). 2. The lengh of lead-ime. 3. The variabiliy of demand/or lead-ime. 4. The degree of accepable sock-ou risk. Taking ino he consideraion of hese four facors, Hadley and Whiin (1963) suggesed boh approximae and exac <Q,r> models wih backorder which aemps o answer boh wo fundamenal quesions menioned above. Their expeced coss included in he model are, he expeced annual seup, holding, and he shorage coss. Under he normal disribuion environmen, he average annual cos is where D Q K = A + IC[ + r µ ] πe ( Q, + ( π + IC) B( Q, Q 2 D = Average annual unis demanded Q = Order quaniy A = Cos per order I C = Carrying charge in dollars per dollar per year = Uni cos of he invenory r = Reorder poin µ = Average lead-ime demand π = Backorder cos in dollars per backorder π = Shorage cos in dollars per uni year of shorage 494

3 E( Q, = The expeced number of backorder incurred per year D = [ α ( α( r + Q)] Q v µ v µ α( v ) = σφ ( v µ ) Φ σ σ φ (*) = The normal densiy funcion Φ(*) = The complemenary cumulaive of he normal disribuion B( Q, = The expeced number of backorders a any ime 1 = [ β ( β ( r + Q)] Q 2 2 v µ v µ β ( v) = 0.5[ σ + ( v µ ) ] Φ 0.5σ ( v µ ) φ σ σ II. DETERMINING THE MEAN AND VARIANCE OF THE LEAD-TIME DEMAND In order o compue he reorder poin wih a safey sock ha will mee a specific service level, we have o know he probabiliy densiy of he lead ime demand, he sum demand during he lead ime period, and he variance of he oal lead ime demand. When he demand can be represened by an ARMA process [Box e al, 1976], he condiional probabiliy disribuion p ( z z, z,... z ) of he fuure value z of he process will + l l ) be Normal wih mean (l) - he forecas of he fuure from he origin, and variance z l 1 2 { and hen is a mulivariae wih mean j = 1ψ j } σ a p( z + l, z + l 1,... z + 1 z, z 1,... z1) ) z (1) Ζˆ =, where zˆ ( l) is he forecas value of z + l provided ha ) z ( l) z, z 1,... z1 values are available, and he covariance marix z + l 495

4 G = σ 2 a g g g l1 g1l g 2l gll where l 1 2 and l 1 = ψ ψ where ψ 1. j = 1 g = {1 + ψ }, jj j g l, l + j i = 0 i j + i The oal amoun of demand during he lead-ime period is S = UZ, where U = [ 1,1,.... 1,1] and = z + z + + l + l... z 1 +1 z+ l zˆ ( l) z + l 1 Z = and E ( Z ) = = Zˆ z + 2 z + 1 zˆ (1) Ε S ) = UE ( Z ) = U Zˆ = z ˆ ( l ) + zˆ ( l 1) zˆ (1) ( T T 2 ( S ) UZ Z U = a Var = l σ l i = 1 j = 1 g ij As we can see from he above analysis ha, for Gaussian demand like ARMA process, he problem reduces o idenifying he firs wo momens of he disribuion of he demand rae for each period during he lead-ime period. The following seps will be used o compue he variance of a given lead-ime. 1. Calculaing of he ψ j weighs using he following equaions: ψ = ϕ j ψ ψ 1 1 θ 1 0 = 2 = ϕ 1ψ 1 + ϕ 2 θ 2 j = ϕ 1 ψ j ϕ p + dψ j p d θ where ψ 0 = 1, ψ j = 0 for j < 0 and θ j = 0 for j > q. and ϕ j and θ are he coefficiens of he auoregressive and moving average in ARMA j 2. Calculaing g and g. ij ii 3. Compue zˆ ( i), for i = 1,... l, he forecas values using he difference equaion forms and hen compue E S ) = z ˆ ( l) + zˆ ( l 1) zˆ (1) l l 2 a i= 1j= 1 ( 4. Compue Var( S ) = σ g ij See Appendix I he Excel emplae for he compuaion of he mean and variance of he forecas error disribuion. j 496

5 Example. Suppose ha he lead-ime demand can be represened by an ARMA(2,2) model as Z 1.6Z Z 2 = a 0.82a a 2 Using he Excel Templae in Exhibi I, he value of sandard deviaion of he lead-ime demand = , for σ = a III. SOLUTION COMPUTATION METHOD BY SOLVER ( SEE APPENDIX II) According o Hadley and Whien s (1963) analysis, he erms α ( r + Q) and β ( r + Q) are negligible in he usual case. Thus for a given value of reorder poin r, he opimal value of Q can be deermined from he following formula 2DA( Q = IC where π + IC A( = A + πα( + 2( ) β ( D K ( = 2DA( IC + IC ( r µ ). and he average oal cos for a given value of r is If lead-ime periods are reaed as discree random variables as suggesed by Boone e al (2000), hen our expeced oal cos of he model can easily modified o incorporae he probabiliies of he ime periods as follows. K = 2DA ( IC + IC( r µ ), where A p( p p p π+ IC) ( = παp( + 2( ) βp( D α p( v) = M v µ M L v µ L σ Lφ pl ( v µ L) Φ L= 1 σ L L= 1 σ L p β p( v) = 0.5 M M 2 2 v µ L v µ P [ σ L + ( v µ L) ] Φ pl 0.5 σ L( v µ L) φ( L 1 σ = L L= 1 σ L L ) p L and p L here is he probabiliy ha here are L periods in he lead-ime of he model. 497

6 The above figure shows he average oal cos curve of he following parameers. D = 700 unis per year C = $50.00 per uni of he invenory I = $0.20 per dollar per year A = $15.00 per order π = $1.00 per backorder π = $15.00 per uni year of shorage Using he Solver, he opimal soluion is Appendix II. r * * = , Q = , K( r * ) = See REFERENCES. Box, George E. P. Box and Jenkins, Jenkins, Gwilym, M., (1976). Time Series Forecasing and Conrol., Holden-Day, San Francisco. Hadley, G., and Whien, T.,M. (1963). Analysis of Invenory Sysems, Prenice-Hall., Englewood Cliffs, N.J. Boone, Tonya, and Ganeesham, Ram, (2000). Models and Mehods o Suppor a New Type of Invenory Performance Measure: The ESWSO, Decision Science, Vol. 31, No. 1, Winer

7 Appendix Ia 499

8 Appendix Ib 500

9 Appendix Ic 501

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