Determination of Optimal Supply When Demand Is a Sum of Components

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1 Mathematical Modelling and Application 7; (6: doi:.648/j.mma.76.3 ISSN: (Print; ISSN: (Online Determination of Optimal Supply When Demand I a Sum of Component Vijayakumar Raman, Venkatean Thirunavukkarau, Muthu Chinnathambi Department of Statitic, St. Joeph College, Trichy, Tamil Nadu, India addre: rvijaytat@gmail.com (V. Raman, venkatehmath8@gmail.com (V. Thirunavukkarau, mailtomuthuinbo@gmail.com (M. Chinnathambi To cite thi article: Vijayakumar Raman, Venkatean Thirunavukkarau, Muthu Chinnathambi. Determination of Optimal Supply When Demand I a Sum of Component. Mathematical Modelling and Application. Vol. 3, No. 6, 7, pp doi:.648/j.mma.76.3 Received: September 7, 7; Accepted: November 7, 7; Publihed: December 4, 7 Abtract: Among the variou inventory ytem our method i ued to find the optimal upply ize. To find the optimal upply ize taking in to conideration the apect like inventory holding cot per unit, cot of hortage per unit etc., In many ituation the demand taken to be a random variable. The total demand i in turn a um of three random variable namely (i demand due to conumer (ii demand due to the upply of the product to iter concern or companie. (iii Demand due to replacement of defective item that are not accepted and hence echanged for new unit Under thee aumption the optimal upply ize i derived. Keyword: Demand for the Product, Optimal Supply Size, Sum of Random Variable, Convolution Principle. Introduction In inventory control theory, determination of the optimal order ize i an important apect. Similarly there are many ituation where the optimal ize of the reerve inventory and optimal upply ize are to be determined. The optimal upply ize i found out taking in to conideration the apect like inventory holding cot perunit, cot of hortage perunit etc., There are many ituation where the demand i taken to be a random variable and hence it ha the correponding probability ditribution. Uing the different cot involved and the demand ize, the optimal quantity of upply i determined. A detailed account of uch model i found in Hanmann (96. The o called Newboy problem i one in which the demand i taken to be a random variable and the optimal upply ize i determined taking in to conideration the cot of ece tock and the cot of hortage. Thi baic model ha been dicued in Hanmann (96 Sheik Udumaneet al., (7 have dicued thi model under the aumption that the random variable denoting the demand atifie the o called Setting the Clock Back to Zero (SCBZ property due to Raja Rao and Talwalker (99 The demand for any product or commodity i uually due to the individual conumer and the quantity they conume or ue. But there are ome cae or ituation where the demand may be due to other factor alo and they may influence the quantity demanded. So the total demand may be may be repreented a the um of the demand due to other factor or caue. In thi Chapter two model are dicued. In the cae of the firt model it i aumed that the demand for the product i the um of two component namely, (i The demand due to the purchae by the conumer. (ii The demand due to the tranfer of the product to the iter companie whenever the hortage occur in thoe companie. Hence the demand may be repreented on the um of two independent random variable. An etenion of thi model to the cae of the total demand a the um of three component i dicued. The total demand i the um of three variable of random nature. In thi Chapter it i aumed that the total demand which i a random variable i the um of three component of individual demand which are of random character. The total demand i in turn a um of three random variable namely (i demand due to conumer (ii demand due to the upply of the product to iter concern or companie. (iii demand due to replacement of defective item that are not accepted and hence echanged for new unit Under thee aumption the optimal upply ize i derived.

2 Mathematical Modelling and Application 7; (6: Aumption (i The demand for a given product i the um of two type of demand. (a conumer demand. (b demand due to the tranfer of good to the iter companie. (ii The upply i intantaneou. (iii There are only two cot involved namely the cot of torage and cot of hortage... Notation h Inventory holding cot d Cot of hortage. S Supply ize X The demand for the product which i a random variable, with pdf f ( and cdf F (. Reult. Model I It i a well known reult that the epected cot of overage and hortage i given by E( C h ( S X f ( d + d ( X S f ( d Now ince the random variable X which denote the total demand i the um of three type of demand, we have to find the ditribution of X which i the um of two random variable X, X.. For that purpoe the convolution principle i ued. [ ] [ ] ( E( C h S X f ( d + d X S f ( d Now we conider the cae where w + demand due to conumer purchae, demand due to tranfer to iter companie. f ( f ( y f ( y dy w where w + [by convolution theorem] y e Let ~ ep(, f ( y y ( y fw ( e e dy y y e e e dy ( e e ( e e ( e ( ubtituting ( in (, we get ( de( C [ ] d ( h S X e e d [ ] ( + d X S e e d d h [ S X ] e e d + [ X S ] e e d ( ( ( A ( B h [ S X ] e d [ S X ] e d ( ( C ( D d [ ] [ ] ( + X S e d X S e d (3 (A [ ] S X e d

3 7 Vijayakumar Raman et al.: Determination of Optimal Supply When Demand I a Sum of Component By Leibnitz rule d d ψ ( ψ ( d f (, t dt ψ ( f [ ψ (, ] ϕ ( f [ ϕ(, ] + f (, t dt d φ( φ( ϕ(, ϕ (, ψ (, ψ ( ( [ ] d e + ( S X e d d e da d (4 (B [ ] S X e d e db d (5 (C [ ] By Leibnitz rule dc d X S e d (D [ ] e (6 X S e d e dd e d (7 ubtituting (4, (5, (6 and (7 value in (3., we get h d e + e + e ( ( h h d d + ( e e e e e h d h d h h + e + e ( h + d + ( h + d h ( h d e + + h ( h + d e + e h[ ] Hence h e e h + d h + h + d e [ ] e The optimal S value i determined by auming pecific value for,, h and d.

4 Mathematical Modelling and Application 7; (6: ,.5, h, d 5 LHS LHS.667 By giving different value for S, the optimal value can be determined. S.4 RHS.74 S.5 RHS.547 Hence ^ S lie between.4 and.5. S.43 RHS.663 ^ S Model II In thi model it i aumed that w , where conumer demand, demand due to the iter companie, demand due to replacement of defective unit. 3 3 ( y fw ( e e 3 3e dy ( 3 ( y 3 e e dy ( 3 e e e e + e ( ( ( 3 + ( ( e e + e ( ( 3 + ( E( C h[ S X ] e e + e d ( ( 3 + ( ( 3 + ( d [ X S ] e e e e + d ( h 3 ( 3 + ( 3 + [ S X ] e e + e d ( d 3 ( 3 + ( [ X S] e e + e d ( A B ( 3 + [ S X ] e d [ S X ] e d de( C h 3 d ( C D ( 3 + [ S X ] e d + [ S X ] e d E F ( 3 + [ X S] e d [ X S] e d d 3 + ( G H ( 3 + [ X S] e d + [ X S] e d (8

5 7 Vijayakumar Raman et al.: Determination of Optimal Supply When Demand I a Sum of Component A [ ] S X e d ϕ( ; ϕ ( ; ψ ( ; ψ ( ; da d (9 ( + B [ ] 3 S X e d db d ( + 3 C [ ] ( 3 + S X e d ( dc d ( D [ ] 3 ( + S X e d ϕ( ; ϕ ( ; ψ ( ; ψ ( ; E [ ] X S e d de e d (3 F [ ] 3 ( + X S e d df e d ( + 3 ( 3 + G [ ] X S e d (4 dg e ds (5 ( + H [ ] 3 X S e d dh e ds ( + 3 ( 3 + (6 ubtituting (9, (, (, (, (3, (4, (5 and (6 value in (8, we dd d ( + 3 ( 3 + ( de( C h 3 ( 3 + ( 3 + e + d ( ( 3 ( d 3 ( 3 + ( e + e + e e ( ( 3 + ( 3 + ( 3 + ( 3 + h e + 3 ( 3 ( ( ( 3 + ( d e + e + e e ( 3 ( 3 + +

6 Mathematical Modelling and Application 7; (6: h h h h h h h h e + e + e + e ( + ( + ( + ( + ( 3 + ( d d ( 3 + d d ( 3 + e + e + e e ( 3 + ( 3 + on implification we get ( 3 + ( 3 + ( 3 + e ( 3 + e ( 3 + h ( 3 + ( 3 + e + ( 3 + e ( h d (7 Any value of S which atifie equation (7 for pecific value,, h and d given the optimal value of S..75,.5,.5, h, d e e 7.875e.375e e e e.375 e LHS.8 S 4 R. H. S.53 LHS. (convert to two decimal place S 4. R. H. S.97 LHS. (convert to two decimal place S S RHS.5 ^ S 4.4 Since L. H. S and R. H. S are equal. 4. Concluion On the bai of the model for finding the optimal upply when demand i a um of two component a dicued. The concluion drawn are ( If the demand due to conumer purchae and demand due to tranfer produce to iter companie are both random variable following eponential ^ ditribution then a h increae a higher level of S i uggeted. ( A d the hortage cot increae then a larger ize of inventory i uggeted. Similarly the reult are found to be identical when the demand can be portrayed a the um of three random variable or egment. The validity of thee model for practical ue depend upon the determination of the ditribution of the random vbariable involved in the model. The model will be of practical ue when the ditribution are formulated baed on the data collected from practical ituation. Alo the tet for goodne of fit for the ditribution will improve the accuracy of the model and the optimal olution will be with greater preciion. Reference [] Amanda J. Schmitt, Lawrence V. Snyder, Zuo- Jun Ma Shen (8. Inventory Sytem with Stochatic Demand and upply: Propertie and Aproimation, pp [] Barber. J. H. (95. Economic control Inventory, New York Code Book C. [3] Bellman. R. (956a. On the theory of Dynamic programming A ware houing problem, Management cience, Volume, (No. 3: pp [4] Bellman. R. (956b. Dynamic programming and the moothing problem, Management cience, Volume 3, (No. : pp. -3. [5] Beyer. D and Sethi. S. P. (997. Average cot optimality in inventory Model with Markovian demand, Journal of Optimization theory and Application, Vol. 9, No. 3, pp [6] Bihop. G. R. (957. On a Problem of production cheduling Operation Reearch, Vol. 5, (No. : pp [7] Bowman, E. H., Richard D Irwin and Fetter, R. B. (957. Analyi for production management. Home wood Illinoi. [8] Brill Percy. H. And Ben A Chaouch (995. An EOQ model with Random variable in demand- Management cience, 4, 5 pp [9] T. Venkatean C. Muthu And R. Sathiyamoorthy (. Determination of Optimal Reerve between Two Machine in Serie. Journal of Ultra Scientit of Phyical Science, Vol. (3 M, (, pp

7 74 Vijayakumar Raman et al.: Determination of Optimal Supply When Demand I a Sum of Component [] T. Venkatean C. Muthu And R. Sathiyamoorthy (. Determination of Optimal Reerve of Semi Finihed product between three machine in erie. Journal of Indian Academy of Mathematic, Vl. 34, No. (, pp: [] T. Venkatean C. Muthu And R. Sathiyamoorthy (6. Determination of Optimal Reerve between Three Machine in Serie. International Journal of Advanced Reearch in Mathematic and Application, Volume: Iue: May, 6, ISSN_NO: 35-8X. [] T. Venkatean, G. Arivazhagan And C. Muthu (7. Some Application of Order Statitic in Inventory Control. IOSR Journal of Mathematic (IOSR-JM e-issn: , p- ISSN: X. Volume 3, Iue Ver. IV (Jan. - Feb. 7.