Optimum Settings of Process Mean, Economic Order Quantity, and Commission Fee

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1 Jounal of Applied Science and Engineeing, Vol. 15, No. 4, pp ( Optiu Settings of Pocess Mean, Econoic Ode Quantity, and Coission Fee Chung-Ho Chen 1 *, Chao-Yu Chou 2 and Wei-Chen Lee 1 1 Gaduate Institute of Industial Manageent, Southen Taiwan Univesity of Science and Technology, Tainan 710, Taiwan, R.O.C. 2 Depatent of Finance, National Taichung Univesity of Science and Technology, Taichung, Taiwan, R.O.C. Abstact In the pesent pape we odify Chen and Liu s odel by consideing both consignent policy and quality loss of poducts. Taguchi s quadatic quality loss function is adopted fo evaluating the poduct quality. The optial paaetes of anufactue s pocess ean and coission fee and etaile s econoic ode quantity ae siultaneously deteined by axiizing the expected total pofit of society. A copaative study between the odified odel and the oiginal Chen and Liu s odel is povided fo illustation. Nueical esults show that the selling pice pe unit and the ean of the deand of custoe have ajo effects on the expected pofit of the supply chain syste. Key Wods: Econoic Ode Quantity, Coission Fee, Taguchi s Quadatic Quality Loss Function, Pocess Mean, Consignent Policy 1. Intoduction In a supply chain syste, the tade-off poble between anufactue and etaile to obtain the axiu expected total pofit of society always eceive consideable attention. The anufactue consides the sale, anufactuing, inventoy, shipent, and anageent of poducts fo obtaining the axiu expected pofit. Meanwhile, the etaile consides the ode quantity, the holding cost, and the goodwill loss fo obtaining the axiu expected pofit. Seifet et al. [1] explained the esulting pocueent change and then poposed the benefits of using spot akets fo a supply chain pespective. Consequently, they developed a atheatical odel that deteines the optial ode quantity via fowad contacts and spot akets. Haksoz and Seshadi [2] had a discussion on the supply chain syste with spot akets and pesented a liteatue eview of ecent elated woks. *Coesponding autho. E-ail: chench@ail.stust.edu.tw Chen and Liu [3] pesented the optiu pofit odel between the poduces and the puchases fo the supply chain syste with pue pocueent policy fo the egula supplie and with ixed pocueent policy fo the egula supplie and the spot aket. Chen and Liu [4] futhe poposed an optial consignent policy consideing a fixed fee and a pe-unit coission. Thei odel deteines a highe anufactue s pofit than the taditional poduction syste and coodinates the etaile to obtain a lage supply chain pofit. Li and Liu [5] consideed the poble fo the etaile deteining his optial ode quantity and fo the anufactue deteining his optial eseve capacity. Thei odel is able to enhance pofit fo both sides of the supply chain. Chen and Huang [6] addessed the poble that the etailes puchase thei poducts fo options and online spot akets fo hedging the isk of deand uncetainty. Econoic selection of the ean of pocess chaacteistic is an ipotant poble fo oden statistical pocess contol because it usually has a significant effect

2 344 Chung-Ho Chen et al. on the expected pofit/cost pe ite. Recently, any eseaches have addessed this wok in which both 100% and sapling inspection of poducts ae consideed fo diffeent odels. Taguchi [7] pesented the quadatic quality loss function fo edefining the poduct quality. Hence, based on this quality loss function, the optiu poduct quality should lead to the quality chaacteistic with iniu bias and vaiance. Recently, this quality loss function has been successfully applied in the poble of optiu pocess ean setting. Chen and Liu s odel [4], which is assued that the poduct quality is pefect, with consignent policy did not conside the used cost of custoes. Hence, in the pesent pape, we popose a odified Chen and Liu s odel with quality loss function and 100% inspection fo poduct, whee the quality chaacteistic of poduct is assued to be noally distibuted. Taguchi s [7] quadatic quality loss function is applied in evaluating the poduct quality. The optial etaile s ode quantity and the anufactue s pocess ean and coission fee ae jointly deteined by axiizing the expected total pofit of society including the anufactue and the etaile [8]. The otivation behind this wok stes fo the fact that neglect of the quality loss of poduct ay oveestiate the expected total pofit of society. quadatic quality loss function. 6. The anufactue guaantees that the etaile will be at least as well-off as in the taditional poduction syste. By applying the loss function, the etaile s pofit ay be expessed by (1 Then the etaile s expected pofit can be obtained as follows: (2 whee 2. Modified Chen and Liu s Consignent Policy To siplify atheatical anipulations, soe assuptions ae ade in the odified Chen and Liu s odel as follows: 1. The deand of custoe (X and the quality chaacteistic of poduct (Y ae independent. 2. The vaiable poduction cost pe unit is popotional to the value of quality chaacteistic of poduct. 3. Ipefect poducts occu in the poduction syste and the 100% inspection of poduct is executed befoe shipping to the custoe. 4. The non-confoance of poduct is scapped and sold at a lowe pice. 5. The used cost of poduct is easued using Taguchi s (3 (4 Letting the patial deivative of Eq. (2 with espect to Q be zeo, i.e., E( 0, esults in the optial ode quantity fo the etaile as follows: Q (5

3 Optiu Settings of Pocess Mean, Econoic Ode Quantity, and Coission Fee 345 The pofit of the supply chain syste can be given by (6 (11 Thus, the anufactue s expected pofit can be obtained as follows Theefoe, the supply chain syste s expected pofit can be obtained as follows: (12 (7 Letting the patial deivative of Eq. (7 with espect to Q be zeo, i.e., E( t 0, leads to the optial ode Q quantity fo the supplie chain syste as follows: (8 Thee ae fou decision vaiables in the odified Chen and Liu s odel with consignent policy, including ode quantity (Q, pe-unit coission (, fixed coission (A 1, and pocess ean of poduct ( y. The atheatical odel fo obtaining the optial values of these decision vaiables is as follows: (13 subject to whee (9 whee (14 Setting Q gives the optial pe-unit coission as follows: Q * * t (10 The pofit of the anufactue can be expessed by:

4 346 Chung-Ho Chen et al. (24 (25 (15 (26 (16 (17 (18 (19 (20 The following solution pocedue ay be applied fo obtaining the optial solution of Eqs. (13 and (14: Step 1. Substituting Eqs. (22, (24, and (25 into Eq. (13 esults in the only one unknown decision vaiable, y, in the odel. Step 2. Let L < y < U. One ay use the diect seach * ethod fo obtaining the optial y with the axiu anufactue s expected pofit fo the anufactue. Step 3. Substitute this optial * y fo Step 2 into Eqs. (22, (24, and (25. Then we can obtain the optial values of ode quantity (Q*, pe-unit coission ( *, and fixed coission (A * 1. Step 4. Substituting the cobination of decision vaiables (Q *, *, A * 1, * y into Eqs. (2, (15, and (7 leads to the optial expected pofit of etaile fo consignent policy, the optial expected pofit of etaile fo taditional poduction syste, and the optial expected pofit of supply chain syste fo consignent policy, espectively. 3. Nueical Exaple and Sensitivity Analysis (21 (22 (23 The aket deand is assued to have a unifo distibution with ean 100 and vaiability 200. The nueical exaple eploys financial paaetes of R = 30, S = 0.1, H = 0.36, S p = 0.1, s = 0.1, s = 0.1, s t = 0.2, h t = 0.36, h = 0.18, h = 0.18, b = 0.1, i = 0.02, c = 1 and quality paaetes y 0 =5, y = 0.5, k =4,L =3,U =7.Byapplying the afoeentioned solution pocedue fo solving Eqs. (13 and (14, we can obtain the optial pocess ean * y = 4.79, which esults in the optial ode quantity Q * = 164, the pe unit coission * = and the fixed coission fee A * 1 = with the anufactue s expected pofit , the etaile s

5 Optiu Settings of Pocess Mean, Econoic Ode Quantity, and Coission Fee 347 expected pofit fo taditional syste the etaile s expected pofit fo consignent policy = and the expected pofit of the supply chain syste The optial solution of the oigi- t nal Chen and Liu s odel is: the ode quantity Q * = 191, the pe unit coission * = and the fixed coission fee A 1 * with the anufactue s expected pofit the etaile s expected pofit fo taditional syste the etaile s expected pofit fo consignent policy and the expected pofit of the supply chain syste = Tables 1 2 list 20% agnitude change fo paaete values and show thei effects on the ode quantity, pocess ean, pe unit coission, fixed coission fee, anufactue s expected pofit, etaile s expected pofit, and expected pofit of the supply chain syste fo both odels. A paaete is consideed as a ajo effect on the expected pofit if the change agnitudes of anufactue s, etaile s and supply chain syste s expected pofits ae geate than 10%. Fo Tables 1 2, we ay have the following two obsevations: 1. As the selling pice pe unit, R, inceases, the ode quantity inceases, the pocess ean is alost constant, the pe unit coission inceases, and the fixed coission fee inceases. The selling pice pe unit has a ajo effect on the anufactue s expected pofit, etaile s expected pofit, and expected pofit of the supply-chain syste fo both odels. 2. As the ean of the deand of custoe, x, inceases, the ode quantity inceases, the pocess ean inceases, the pe unit coission deceases, and the fixed coission fee inceases. The ean of the deand of custoe is a ajo effect to the anufactue s expected pofit, etaile s expected pofit, and expected pofit of the supply-chain syste fo both odels. 4. Discussion and Conclusion Copaing Tables 1 2 also gives the following esults: t 1. The odified Chen and Liu s odel always leads to a salle ode quantity, pe unit coission, anufactue s expected pofit, etaile s expected pofit, and supply-chain syste s expected pofit than those of the oiginal Chen and Liu s [4]. 2. The odified Chen and Liu s odel geneally esults in a lage fixed coission fee than that of the oiginal Chen and Liu s [4]. These esults eflect that neglect of the quality loss of poduct causes to oveestiate the expected total pofit of society when the quality loss of poduct and the used cost of custoes exist. The etaile and anufactue should adopt the consevative policy fo pocueent, poduction, and consignent. The sensitivity analysis shows that the selling pice pe unit and the ean of the deand of custoe have a ajo effect on the anufactue s expected pofit, etaile s expected pofit, and expected pofit of the supply-chain syste. Hence, exact estiation on these paaetes is always equied to avoid any incoect outcoe. In the pesent pape, we pesent a odified Chen and Liu s odel with consignent policy and quality loss of poduct, which ay be consideed as a geneation of the oiginal Chen and Liu s odel [4] by consideing the ipefect poduct quality. The ode quantity, pocess ean, pe unit coission, and fixed coission ae siultaneously deteined in the odified odel. The extension to odified Chen and Liu s odel with ulti-attibute quality chaacteistics ay be left fo futhe study. Notations the pe-unit coission paid by the anufactue to the etaile Q the ode quantity by the etaile h s X the pe-unit caying cost fo the etaile the goodwill loss pe unit of stock-out fo the etaile the stochastic deand of custoe having an unifo distibution,

6 348 Chung-Ho Chen et al. Table 1. The effect of paaetes of optial solution fo odified Chen and Liu s [4] odel R Q y A 1 t x Q y A 1 t x Q y A 1 t K Q y A 1 t y Q y A 1 t b Q y A 1 t c Q y A 1 t i Q y A 1 t S p Q y A 1 t s Q y A 1 t s Q y A 1 t

7 Optiu Settings of Pocess Mean, Econoic Ode Quantity, and Coission Fee 349 Table 1. Continued s t Q y A 1 t h Q y A 1 t h Q y A 1 t h t Q y A 1 t S Q y A 1 t H Q y A 1 t Table 2. The effect of paaetes of optial solution fo Chen and Liu s [4] odel R Q A 1 t x Q A 1 t x Q A 1 t B Q A 1 t

8 350 Chung-Ho Chen et al. Table 2. Continued C Q A 1 t I Q A 1 t s Q A 1 t s Q A 1 t s t Q A 1 t h Q A 1 t h Q A 1 t h t Q A 1 t S Q A 1 t H Q A 1 t

9 Optiu Settings of Pocess Mean, Econoic Ode Quantity, and Coission Fee X ~ U x x, x x 2 2 x the ean deand of custoe x the vaiability of custoe deand A 1 the fixed coission fee paid by the anufactue to the etaile R the pe-unit etaile pice c the pe-unit anufactuing cost/the vaiable poduction cost pe unit h t the pe-unit caying cost in the supply chain syste s t the goodwill loss pe unit of stock-out in the supply chain syste h the pe-unit caying cost fo the anufactue s the goodwill loss pe unit of stock-out fo the anufactue H the caying cost pe unit fo the etaile in the taditional poduction syste S the goodwill loss pe unit of stock-out fo the etaile in the taditional poduction syste Y the quality chaacteistic of poduct, which is assued that Y ~ N( y, 2 y y the ean of quality chaacteistic of poduct y the standad deviation of quality chaacteistic of poduct L the lowe specification liit of poduct quality chaacteistic U the uppe specification liit of poduct quality chaacteistic Loss(Y the quality loss pe unit, Loss(Y=k(Y y 0 2 k y 0 ( ( W S p the quality loss coefficient the taget value of poduct quality chaacteistic the cuulative distibution function of standad noal ando vaiable the pobability density function of standad noal ando vaiable the selling pice pe unit fo the confoance poduct in the taditional poduction syste the discounted pice pe unit fo the non-confoance poduct scapped in the taditional poduction syste b the constant poduction cost pe unit i the inspection cost pe unit t t the anufactue s pofit fo consignent policy the anufactue s expected pofit fo consignent policy the etaile s pofit fo consignent policy the etaile s expected pofit fo consignent policy the supply chain syste s pofit fo consignent policy the supply chain syste s expected pofit fo consignent policy the anufactue s pofit in the taditional poduction syste the anufactue s expected pofit in the taditional poduction syste the etaile s pofit in the taditional poduction syste the etaile s expected pofit in the taditional poduction syste Refeences [1] Seifet, R. W., Thoneann, U. W. and Hausan, W. H., Optial Pocueent Stategies fo Online Spot Makets, Euopean Jounal of Opeational Reseach, Vol. 152, pp (2004. [2] Haksoz, C. and Seshadi, S., Supply Chain Opeations in the Pesence of a Spot Maket: A Review with Discussion, Jounal of the Opeational Reseach Society, Vol. 58, pp (2007. [3] Chen, S. L. and Liu, C. L., Pocueent Stategies in the Pesence of the Spot Maket - An Analytical Faewok, Poduction Planning & Contol, Vol. 18, pp (2007. [4] Chen, S. L. and Liu, C. L., The Optial Consignent Policy fo the Manufactue unde Supply Chain Coodination, Intenational Jounal of Poduction Reseach, Vol. 46, pp (2008. [5] Li, J. and Liu, L., Supply Chain Coodination with Manufactue s Liited Reseve Capacity: An Extended Newsboy Poble, Intenational Jounal of

10 352 Chung-Ho Chen et al. Poduction Econoics, Vol. 112, pp (2008. [6] Chen, S. L. and Huang, S. C., Managing Supply Chain Risk with Options and Online Spot Makets, Jounal of Statistics & Manageent Systes, Vol. 13, pp (2010. [7] Taguchi, G., Intoduction to Quality Engineeing, Asian Poductivity Oganization, Tokyo (1986. [8] Chen, C. H., The Tade-Off Model between Retaile s and Manufactue s Pofits, Intenational Jounal of Reliability and Quality Pefoance, Vol. 2, pp (2010. Manuscipt Received: Feb. 20, 2012 Accepted: Jun. 26, 2012

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