MODELING A RANDOM YIELD IN- HOUSE PRODUCTION SET UP IN A NEWSVENDOR PROBLEM

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1 MODEING A RANDOM YIED IN- HOSE PRODCTION SET P IN A NEWSVENDOR PROBEM Krishna Solanki 1 & Ravi Gor 1.D.R.P. Institute of Technology and Research, Gandhinagar, India Dr. Babasaheb Ambedkar Oen niversity, Gujarat, India ABSTRACT In suly uncertainty caused by random yield, the firm receives a random ortion of an order laced with a sulier. Yield uncertainty exists in many industries, including semiconductors, electronic fabrication and assembly, food rocessing, bio-harmaceuticals, and resource based industries such a mining and agriculture. The existence of yield / suly rocess uncertainty often results in lower suly chain erformance. We consider this issue in the context of a newsboy roblem where before the selling season; the firm makes joint ricing and inventory decisions. We consider the in-house roduction case in this model and derive analytical exressions therein. 1. INTRODCTION Our objective is to investigate the effect of suly uncertainty on otimal decisions and exected rofit. A fundamental roblem in suly chain management is how to match suly and demand so that the system oerates effectively. However, manufacturers and retailers often suffer suly rocess uncertainty in addition to demand uncertainty. When the suly rocess is random, the quantity received may be different from the quantity ordered. We consider a rice-setting newsvendor model in which a firm needs to make joint inventory and ricing decisions before the selling season. The suly rocess is uncertain such that the received quantity is the roduct of the order quantity and a random yield rate.it is well known that in the classical newsvendor roblem without a random yield, the otimal rice in the stochastic system is greater than the otimal rice in the deterministic system if the demand model is in the form of multilicative. We rove that this is also true in the random yield model.. ITERATRE REVIEW This aer is related to several streams of literature. The first stream is on the rice setting newsvendor roblem. For a comrehensive review, see the work of Petruzzi and Dada (1999), Yano and Gilbert (003), and Yao et al.(006) and the references therein. Additive and multilicative forms are two commonly used demand models in the newsvendor ricing literature (see, e.g., Petruzzi and Dada 1999, Agrawal and Seshadri 000, Chen et al.009).we assume that the demand model is multilicative, which can be reresented as the roduct of a deterministic, rice-deendent demand function and a random noise term (see, e.g., Granot and Yin 005 and 008, Song et al.009, Chen and Bell 009).Studies in this stream do not usually consider suly uncertainty. The second stream cares otimal inventory decisions under suly uncertainty due to random yield. Yano and ee (1995) rovide an excellent review of this literature. Recent works include those of Bollaragada and Morton (1999), Khouja(1999),Guta and Cooer(005),and Guler and Bilgic (009).The stochastically roortional yield model is the most widely studied (see,e.g.,shih 1980,Henig and Gerchak 1990,i and Zheng 006,and iu et al.010),and is also used in this aer. For other classes of yield models, lease refer to Dada et al. (007).Most aers in this line of research focus on determining otimal lot size or characterizing the structure of the otimal inventory olicy. In the literature on roduction lanning under suly uncertainty, selling rice is commonly assumed to be exogenously given. However, there are a few excetions. Tang and Yin (007) study the ricing olicy under suly uncertainty and examine the imact of that uncertainty on otimal decisions with a known, rice-deendent demand. Kazaz (004) studies roduction lanning for olive oil with random yield and stochastic demand where both the selling rice and urchasing cost of fruit are yield deendent. Recently, Kazaz and Webster (011) consider an agricultural firm that oerates under suly uncertainty and yield-deendent trading costs. They extend Kazaz (004) by considering deterministic, rice-deendent demand and offering a risk-averse analysis, and considering the influence of fruit futures. Kazaz and Webster (011) focus on the imact of the yield-deendent trading cost on the otimal selling rice and roduction quantity. Kazaz (008) studies a similar model with two model settings: an early ricing model and a ostoned ricing model. Kazaz (008) focuses on identifying the otimal selling rice and roduction decisions for the two models. Our aer is close to those of Kazaz and Webster (011) and Kazaz (008) because we also model the roblem by using an emergent urchasing otion and rice-deendent stochastic demand. However, our focus is on the imact of suly uncertainty on both otimal decisions and exected rofits. In this aer, we assume that the firm has an emergency order/roduction otion for excessive demand. With the emergency suly otion, in additional to a normal order before the start of the selling season, the inventory 395

2 manager may use an emergency order to fulfil shortages after demand realization, usually at a higher roduction/urchase cost. The existence of an emergency suly otion may have more advantages in enhancing suly chain erformance, esecially in random yield settings. 3. MATHEMATICA MODE In the In-house roduction model a firm has to ay for the quantity inut instead of quantity received. Before the selling season the firm decides the roduction quantity q and the unit selling rice to maximize its exected rofit. During the selling season demand is realized and satisfied by firm s on-hand inventory. Here Є is the random yield rate with a suort Є, Є ( Є > Є 0)and a mean µ.then qє is the yield from roduction. If the realized demand exceeds the quantity received,qє,then remaining demand is satisfied by emergency urchase from sot market at unit rice e. Salvaged value of leftover inventory is denoted as unit rice s(s 0).The unit order cost is denoted as c. We assume that s<c<e. et D(,)be the random demand.it is a function of the unit selling rice and market noise that is indeendent of. We use multilicative demand model in which demand is reresented as the roduct of a deterministic, rice deendent demand function and a random noise term, D(,) = d(), where is defined on Є, Є ( Є > Є 0). We assume that d() is the continuous, strictly decreasing, nonnegative, twice differentiable function defined on c,, where is the maximum admissible rice. We can take = min { d() = 0}.If d()> 0 for all > 0, we define = +. d() has an IPE, i.e. dη() 0, where η()=d () d(). The Profit function The firm s exected rofit as a function of inut quantity q and unit selling rice can be written as π q, = E D, + se, q D, + ee, D, q + c (1) =( e)e D(, ) E, q D, + + e c q() =( s)e D(, ) E, D, q + c s q (3) 4. ANAYTICA RESTS emma 1 If d() has IPE, then π d is quasi Concave in in the interval c,. Hence the otimal rice d can be uniquely determined by dπ d () = 0, i.e.,( d c )d ( d )+ d( d ) = 0 Proof: et z = q be the stocking factor from () and (3); π z, = e d + eµ c z d z u F x dx dg u d() (4) ( E D(, ) = d () for = 1 and relace q = z d() ) = d () e + eµ c z F x dx dg u Relacing q = z d () and E D(, ) = d () in (3) ; π z, = d () s c sµ z z df x G u du (5) For fixed rice the following lemma gives the otimal stocking factor. z u emma : Suose that eµ c. Then for any given, π z, is concave in z and the otimal stocking factor z () satisfy µ - F u d G u = E (1 F z ) = cs (6) or x z udg u df x du = cs e c = µ (7) Hence, the corresonding roduction inut quantity is q = d z (). Proof: Taking first and second derivatives of π z, w.r.t. z in (4), we have 396

3 π(z, ) z = d () uf dg(u) + eµ c + s s ) = d () uf dg u + µ (c s) = d eµ c () uf dg(u) = d () uf dg u + u d G(u) (c s) ( µ = u d G(u) = d u 1 F dg u (c s) (8) π(z,) = d u (F()) dg(u) z z = d u f() dg(u) < 0 π z, is concave in z for any given. Taking (8) equal to zero and rearranging, µ - u F dg u = cs on the other hand from (5), we get π (z,) z u 1 F dg u = u dg u u F dg u c s = ) ( µ = u d G(u) c s π z, = d s c s z z G u du df z = d c sµ () G u dudf z z x z x z x z x z = d c sµ () G u dudf z z G = d c sµ () G u dudf z z G = d c sµ () G u dudf z + G d dz ( d G u dudf(z) df(z). x z df(z). df(z) = d c sµ + () u dg u df x (by integration of arts) (9) Taking (9) equal to zero, udg u df x du = c s e c = µ emma 3: The otimal selling rice is greater than the otimal riskless rice d. Proof: For (condition) any c < < d, We have ( - c ) d () + d() 0 i.e. d() c d () we have for any > c dπ z, = d () e + xdf x dg u + d (10) = d () d () e + e + ( c) c d () 397

4 = d () = ()d () = ()d () = ()d () Define A(z) = 1 (e + + e c ( z u uf dg(u) (by 6) uf dg(u) 1 uf dg (u). z u uf dg (u) Now we show that A(z) > 0 for all 0 < z <. Note that lim z A z = 0, there fore it suffices to show that A z < 0 w. r.t z for all 0 < z <. Differentiating A(z) w.r.t z and rearranging items we get, A z = = uf dg u. uf dg u. d dz df dg (u) uf dg (u) uf dg (u). d dz (. ( u d F dg (u)) dz uf dg (u)) = uf dg u. f dg (u) uf dg (u). ( u f dg (u)) = u f dg (u) ( F tdf t ) uf dg(u) For all z, 0 < z < ; f() - tdf t > 0 So for all c <, we have d dπ z, > 0. Since d < 0 dg (u) So, the otimal selling rice is greater than the otimal riskless rice d. Theorem: Suose that the mean demand d() has IPE. Then π z, is a quasi concave function of over the interval ( c µ, ) and hence there exists a unique maximizer of π z,. Proof: To show that π z, is quasi concave in over ( c µ, ), we only need to show that d π z, From (10) we know that at dπ z, (-e) + (e-s) dπ z, = 0. tdf t dg(u) = d() With the assumtion of IPE roerty, =0 < 0. d () (11) d () d d () 1 (1) d() From (10) we get that; differentiate (10) w.r.t ; d π z, dπ z, dη() d () 0 d + d d d () d () 0 =0 = d e + tdf t dg(u) + d + d() 398

5 = d e + tdf t dg(u) From (11) relace the value; = d() d () d + d d(). d () d () = d(). d () d() d () d() + d() + d 1 + d (relacing from (11)); + d = d + d() Relacing value of d() from (11); d π z, = d () = d () µ < d () µ dπ z, = d () =0 e + () eµ + µ () (eµc) < 0 ( by A z > 0 ) e + tdf t dg(u) tdf t dg(u) + µ tdf t dg(u) tdf t dg(u) ( eµ < -eµ+c) + d 5. CONCSION AND FTRE RESEARCH We consider a rice-setting newsvendor model in which a firm needs to make joint inventory and ricing decisions before the selling season. The suly rocess is uncertain such that the received quantity is the roduct of the order quantity and a random yield rate.we consider the in-house roduction case in this model and derive analytical exressions therein. We rove that in the rice deendent newsvendor model with random yield (suly), when the in house roduction set u model is considered, and the demand is modeled in the multilicative form, the otimal rice in the stochastic system is greater than the otimal rice in the deterministic system. For further research we may consider the rocurement model wherein the firm ays for the quantity received only, unlike in this case where the firm ays for the order/roduction quantity. Further, we may consider the additive demand model which is also commonly used in literature. REFERENCES [1]. Agrawal, V. and S. Seshadri Imact of uncertainty and risk aversion on rice and order quantity in the newsvendor roblem. Manufacturing & Service Oerations Management (4), []. Bollaragada, S. and T.E.Morton Myoic heuristics for the random yield roblem. Oerations research, 47(5), [3]. Chen, Y., M. Xu and Z. G. Zhang A risk averse newsvendor model under the CVaR criterion. Oerations Research, 57(4), [4]. Dada, M., N. C. Petruzzi and.b.schwarz.007. A newsvendor s rocurement roblem when suliers are unreliable.manufacturing&serviceoerationsmanagement,9(1),9-3. [5]. Granot, D. and s. Yin On the Effectiveness of returns olicies in the rice-deendent news vendor model. Naval Research ogistics, 5(8), [6]. Granot, D. and S. Yin Price and order ostonement in a decentralized newsvendor model with multilicative and rice-deendent demand. Oerations Research, 56(1), [7]. Guta, D. and W.. Cooer Stochastic comarisions in roduction yield management. Oerations Research, 53(), [8]. Guler, M. and T. Bilgic.009. On coordinating an assembly system under random yield and random demand. Euroean Journal of Oerational Research, 196(1), [9]. Henig, M. and Y. Gerchak The structure of eriodic review olicies in the resence of random yield. Oerations research, 38(4), [10]. Kazaz, B Pricing and Production Planning nder Suly ncertainty. Working aer. Whitman School of Management, Syracuse niversity, 71 university Ave., Syracuse, NY

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