* Ravi Gor 1 and Ashok Patel 2. * Author for Correspondence

Transcription

1 A SINGLE PERIOD MODEL WHERE THE LOST SALES RECAPTURE IS A FUNCTION OF log m 1 r Ravi Gor 1 and Ashok Patel 2 1 Dr. Baba Saheb Ambedkar Oen University, Gujarat 2 P.S. Science and H. D Patel Arts College, Kadi Author for Corresondence ABSTRACT We consider a lost sale recature model in a newsvendor framework. In this aer we analyse how to recature lost customers in which easier to win back old customers than it is to acquire new customers. We consider a single-eriod decision of a retailer facing uncertain and rice deendent demand. The tyical modeling of the roblem in a newsvendor framework assumes the unfulfilled demand to be lost once and for all. However, in reality, there may be an oortunity to backlog the lost sales, by offering some incentive for waiting. Nevertheless, the retailers rocurement rice may be higher, due to the likely cost increase of the emergency urchase. Further, not all the customers that could not buy in the first instance may avail the rebate offer and buy. The backlog fill rate is modeled as a function of the roortion of the rebate to the rice. Then the retailer has to decide ahead of the realization of the demand the quantity to be ordered, the rice and the rebate to be offered for backlogged sales that will maximize its exected rofit. Numerical examles are resented to highlight model sensitivities to arametric changes. The back log fill rate is modelled as a log function of adding one to the roortion of rebate relative to the rice. Sensitivities of otimal rebate to demand errors are carried out with uniform distribution. Keywords: Newsvendor Problem, Lost Sales, Rebates, Price Deendent Demand INTRODUCTION This aer considers the buying and ordering olicies of a newsvendor-tye retailer, faced with the ossibility of backordering at least some of the shortages incurred from demand underestimation. The backordering occurs through an emergency urchase of the items in question at some remium over the regular urchasing cost. In turn, the retailer offers to the end-customers left out of the initial sale a rebate incentive uon urchase of each item backordered. The roblem of backordering shortage items has been considered recently by Weng (2004) and Zhou and Wang (2009). Both generalize the newsvendor roblem (heretofore NVP) into a two-ste decision rocess. In the first stage, the retailer laces the initial order that equates the costs of over- and underestimation of the demand, as corresonds to the traditional NVP. In the second, the retailer may lace a secial order from the manufacturer at the end of the selling season. The basic difference between the two models lies in whether the manufacturer (Weng, 2004) or both arties (Zhou and Wang, 2009) ay for the setu costs of the secial order. Our model differs from these two in five fundamental ways. First, we consider a rice-deendent demand, with the selling rice,, a decision variable, more in accordance with the main tenets of microeconomic theory (e.g. Arcelus and Srinivasan, 1987). Second, we introduce a rebate-deendent fill rate, Ω, reresenting the robability of the end-customers returning to satisfy the unfilled demand. This fill rate is a function of the size of the rebate, r, offered relative to the selling rice. Third, the olicy decisions on the emergency order and on the rebate olicy occur u front, along with the remaining ordering and ricing olicies, rather than at the end of the season, thereby rendering the resulting formulation into a more traditional one-stage, rather than a two-stage, NVP. Fourth, the decision variables are the selling rice, the order size and the rebate offered as an incentive to satisfy at least a ortion of the unfulfilled demand. Our model yields a unique rofit-maximizing solution, for a family of deterministic mean Coyright 2014 Centre for Info Bio Technology (CIBTech) 12

2 demand functions and of robability distributions of the demand error that encomasses the vast majority of the models in the existing literature. The organization of the aer is as follows. The next section resents the formulation of the model, based uon that of Zhou and Wang (2009), to which we add the offering of a rice rebate er backordered unit urchased. This aer is similar in lines of Arcelus et al., (2012), and Patel and Gor (2013). Here, we use an entirely different fill rate function than discussed in Patel and Gor (2013). We describe the characteristics of the model, develo the objective function and derive the rofit-maximizing otimality conditions that are shown to be unique. Section 3 resents a numerical examle. In addition to illustrating the main features of the model and discussing some comarative statics of interest, this section attemts to conjecture the behavioural relationshi between various arameters and variables. A conclusions section comletes the aer. Table 1 lists the notations used throughout the aer. Table 1: Notation The selling rice er unit (decision variable) v The salvage value er unsold unit q The order quantity (decision variable) r The rebate er backordered item (decision variable) c The acquisition cost er unit s The shortage enalty er unsold unit D The total demand rate er unit of time g, ε The deterministic and stochastic comonents, resectively, of D a,b The uer and lower values, resectively, of ε μ, ζ The mean and standard deviation, resectively, of ε f, F The density function and the cumulative distribution function, resectively, of ε δ 0,δ 1 The intercet and sloe, resectively, of the deterministic linear demand function The intercet and the demand elasticity, resectively, of the iso-elastic deterministic γ 0, γ 1 demand function Ω The fill rate of backlogged demand d The remium on the urchase rice of each backlogged unit acquired z The stocking factor Λ, Φ The exected number of leftovers and shortages, resectively e The rice elasticity of demand I ε The generalized failure rate function π(,q,r) The retailer s rofit function E(,q,r) The retailer s exected rofit function Model Formulation In this section, we describe the key characteristics of the model, formulate the retailer s rofit-maximizing objective function and derive the otimality conditions. Observe that, in the develoment of the models, the arguments of the functions are omitted whenever ossible, to simlify notation. Characteristics of the model Characteristic 1: Key roerties of the demand function. The random single-eriod total demand, D (,ε), is of the form: D(, ) g( ), if additive error (1) g( ), if multilicative error g() has an IPE or increasing rice elasticity, e, which satisfies the following condition: e g e 0, where e g Coyright 2014 Centre for Info Bio Technology (CIBTech) 13

3 ε has a GSIFR or generalized strictly increasing failure rate, I ε, since I I / 0, where I f /(1 F ) Observe in (1) that the total demand includes a deterministic comonent of g units, denoted as the mean demand; and a stochastic element, denoted by units. Following the customary conventions of the literature on the subject, the relationshi between g and ε is assumed to be either additive (Mills, 1958) or multilicative (Karlin and Carr, 1962), with the former (latter) exhibiting a constant (variable) error variance and a variable (constant) coefficient of variation. Chan, et al., (2004), Lau, et al., (2007), Petruzzi and Dada (1999), Yao (2002) and Yao, et al., (2006) discuss the imlications of these assumtions and rovide a review of the extant works on the field. Furthermore, unless otherwise stated, there is no need to identify a functional form of the mean demand, g(). The results resented here are alicable to all the demand distributions normally used in the salesromotion field, i.e. linear, iso-elastic, log-concave or concave in and the like (Yao, 2002; Yao, et al., 2006). Detailed discussions can be found in, Arcelus et al., (2012) and Patel and Gor (2013). Characteristic 2: A fill rate, Ω, given by the following exression: r log m 1, where 0 r, 0 1, 2 m, r m 1 (2) The fill rate, Ω, measures the fraction of end-customers who wish to fulfill their demand from the emergency order. Its functional form in (2) is rooted on the emirical literature on the subject and satisfies several roerties of interest. First, it is a function of the value of the rebate relative to the selling rice, log m 1 r 0<r<. Also, as m. Second, the value of falls between 0 and 1, but does not aroach either value as r 1, 1 and as m, 0. Only in the absence of the rebate i.e. r=0, =0. This reflects emirical findings imlying that, if there is no rebate, buying of lost sales will not take lace, unless the roduct enjoys a monooly. Arcelus, Gor and Srinivasan (2012) have develoed a lost sale recature model validating Bawa and Shoemaker (1989) that there is still some exosure effect to the original sale that leads some end-customers to urchase, even in the absence of a couon, i.e. even when r=0, =0. On the other hand, in this model, as r 1 m, 1 indicating the ossibility of every lost sale converting if the roduct is offered at a rebate equal to the selling rice i.e almost absolutely free. Characteristic 3: The stocking factor, z z q g, if additive q / g, if multilicative B ( z ) f ( )d z (3) z ( z ) f ( )d z A In (3), Φ and Λ reresent the exected number of shortages and leftovers, resectively, as a result of demand fluctuations. The shortage level is exected to decrease with the rebate incentive. With resect to the stocking variable, z, it was introduced by Petruzzi and Dada (1999) and subsequently used by Arcelus, et al., (2005), among many others, as a relacement for another decision variable, namely the order quantity. It reresents the exected level of leftover and shortages, generated by the demand uncertainty and by the retailer s otimal olicies. Its inclusion simlifies the interretation of the findings of the current study and the derivations of the otimality conditions. Coyright 2014 Centre for Info Bio Technology (CIBTech) 14

4 The retailer s rofit-maximizing objective The retailer rofit function is decomosable into two arts, deending uon whether the retailer order quantity exceeds or understates the demand for the roduct. If the first, then q exceeds D and the retailer sells D units at er unit, disoses of the rest at a salvage value of v er unit and incurs an acquisition cost of c for each of the q units ordered. If the second, q is below D, in which case the retailer buys and sells the q units at a rofit margin of (-c) er unit, acquires a fraction Ω of the shortage demand at a remium d er unit, sells it at (-r), the regular selling rice,, net of the er unit rebate offered, r, and ays a shortage enalty of s er unit on the rest of the merchandise. Formally, the functional form of the retailer s rofit function, π(,q,r), is as follows: (, q, r) D cq v( q D), if q D (4) ( c) q ( r) ( c d) ( D q) s(1 )( D q), if q D The objective is to find the levels of, q and r that maximizes E(,q,r), the retailer s exected rofit. Using (3) and (4), it can be readily seen that E may be written as follows: E(,q,r ) ( c )( g ) ( c v ) ( c s )(1 ) ( r d ), if additive (5) ( c )g g( c v ) g ( c s )(1 ) ( r d ), if multilica tive First-order otimality conditions: To simlify the exlanation, only the additive-error/linear-demand case will be discussed. The multilicative case can be develoed along the same lines. Let E i E / i, i, r, Q be the first derivative of the exected rofit with resect to each of the decision variables. Setting these derivatives to zero, we obtain the following first-order otimality conditions. E 0 ( g ) g ( c ) (1 ) ( c s r d ) E r 0 r( c s r d ) E z 0 ( c v ) z ( v s ) ( c s r d ) (6) Where and are defined in (3). The otimality conditions in (6) have straightforward economic r interretations. All reresent tradeoffs between rofit gains and losses associated with unit changes in, r and q, resectively. With resect to the first, a one-dollar increase in rice generates (i) a rofit increase of $(g+μ) from the units sold: (ii) minus a loss of $ g (-c), from the decrease in demand caused by the rice increase; (iii) minus an oortunity cost of the shortages not sold even with the emergency order; and (iv) oortunity cost on the decrease of the fill rate due to the rice increase. As for the second, a one-dollar increase in the in the shortage rebate, r, results in (i) an increase in rofits from the associated rise in the fill rate, r >0, from (3); and in (ii) an increase in the rebate costs from the back-logged end-customers urchasing from the emergency order. The third condition indicates that a one-dollar increase in the stocking factor results from the marginal rofit changes in the exected leftovers, together with the oosite weighted marginal rofits in the exected shortages, with the weights reresenting the ercentage of returning and not returning customers. Numerical Analysis This section resents a numerical illustration of key roerties of the model just described, to highlight the main features of the various solutions roosed in the aer. Given the central objective of the aer, our numerical analysis centers on the imact of fluctuations in base m of the fill rate function, uon the fill rate, Ω, and through it, uon the retailer s rofit-maximizing ricing, ordering, rebate olicies. All comutations were carried out with MAPLE s Otimization toolbox. Coyright 2014 Centre for Info Bio Technology (CIBTech) 15

5 Base-case numerical structure The starting oint consists of two sets of examles that serve as the base-case for the analysis of this section. The first (second) set, denoted by AL (MI), assumes the deterministic demand, g, to be linear (isoelastic) and its error, additive (multilicative), i.e. D( ) 0 1, 0 0, 1 0, for AL total demand (7) 1, 0, 0 1, for MI total demand For comarability uroses, this section oerates with the arameter values of Patel and Gor (2013) to which suitable values for the remaining arameters have been added. These values aear in Tables 2. In this way, any sensitivity analysis can be carried out by adroit maniulation of the aroriate arameter values for any of the comonents of the base-case. Further for maximum comarability among robability distributions, all cases are related to a random variable uniformly distributed over the interval (-3,500, 1,500), for the AL demand model and (0.7, 1.1), for its MI counterart. Either suort interval describes the uniform distribution comletely. Base-case numerical results Having described the nature of the numerical structure that gives rise to the arameter values of the AL and MI comonents of the base case, we now discuss the numerical results. Unless otherwise stated, we concentrate our remarks on the AL demand case. As mentioned latter on in this section, the results for the MI case can be interreted in similar fashion. Table 2.Numerical Analysis: Base Case Otimal Policies Distribution Suort, mean and Standard deviation Uniform Distribution Additive Error and Linear Demand A > -a suort [A,B] [-3500, 1500], Mean = -1000, SD = 1440 Multilicative Error and Iso-elastic demand [0.7, 1.1], Mean = 0.9, SD = 0.06 A>0 Additive Error Linear Demand Parameter values: γ 0 = ; γ 1 =1500 ; c = 35; d = 3; v = 10; s = 3 Profit q Λ Φ Multilicative Error Iso-Elastic Demand Parameter values: γ 0 = ; γ 1 = 2.5; c = 35; d = 3; v = 10; s = 3 Profit q Λ Φ Numerical Examle and Interretations The otimal results using MAPLE for the fill rate model with varied bases on m 1 r log are shown in Table 3. Both the cases Additive Error Linear Demand and Multilicative Error Iso-elastic Demand are showcased to highlight the variations in the otimal solutions too. The following observations and interretations are made: (a) The otimal olicy for the fill rate model with m=2, as shown in row 1 of Table 3 in Additive Error Linear Demand case, consists of the retailer acquiring q=23125 units at a unit cost of c=$35 and selling them at a unit rice of =$ With resect to the fill rate, aroximately Ω = 4% of the shortages are recatured at an extra urchasing cost of d=$3.00 to the retailer, who allows a rebate of r=$7.36 er unit backlogged. Afterwards, all unsold units, i.e. of s=$3., will be assigned a unit shortage enalty Coyright 2014 Centre for Info Bio Technology (CIBTech) 16

6 On the other hand, when demand falls below the q=23,125 units ordered and all urchased at the cost of c=$35 er unit, D units are sold at the regular unit rice of =$50.25 and the remaining, at the salvage value of v=$10.00 er unit. The resulting otimal olicy is π[, q, r] = $ [50.26, 22975, 5.08]. As show in Table 2, these results contrast with the otimal solution for the AL certainty case of π[; q] =$333,909 [$50.22; 23,276] (b) Similar interretation follows for the other models in the Additive Error Linear Demand case, where the base on m 1 r log increases as shown in Table 2. The increase in the ower of the fill rate function tends to increase the otimal order quantity and the rebate, whereas decreases the selling rice and rofits. (c) Table 3 also gives results for the MI case. Observe though that unlike its Additive Error Linear Demand counterart, in this case, increase in the base of the fill rate function, tends to increase the order quantity and the rebate and also the selling rice. Profits decrease with the increase in the base of the fill rate function. Table 3: Otimal Policies for lost sale recature model with fill rate Ω= log m 1 r Additive Error Linear Demand m Profit q r Ω Λ Φ Multilicative Error Iso-Elastic Demand m Profit q r Ω Λ Φ Sensitivity Analysis Table 4 describes the sensitivities of the otimal olicies to the change in the salvage and shortage costs in the Additive Error and Linear Demand case. Corresonding results for the Iso-elastic demand and multilicative error case can be easily comuted. The rimary objective to carry out the sensitivity analysis is to observe the directional change in the shortages and the leftover values. Observe that, even though, in all of the examles of Table 3, the exected number of leftovers, Λ, never exceeds the exected shortages, Φ, the relationshi between these two is arameter secific, since sensitivity analysis shows that we can construct numerous examles, where Λ > Φ. Table 4: Sensitivities to the salvage and shortage costs in Additive Error Linear Demand Case for m=1 Linear Demand Additive Error Case for m=2 v s q π Coyright 2014 Centre for Info Bio Technology (CIBTech) 17 r Ω Λ Φ

7 Linear Demand Additive Error Case for m= Next, we erform sensitivity analysis to the change in the suort values [A,B] for the Uniform distribution for the fill rate model with base m=2. Similar sensitivities can be erformed for various other values of m, as well as suort structures. Table 5: Sensitivities to the Uniform Distribution Suort Changes: CASE m=2 Linear Demand and Additive Error SUPPORT Mean π q r Ω Λ Φ -5500, , , , , Iso-elastic Demand and Multilicative Error.6, , , , , Some Concluding Comments The rimary contribution of this aer has been to consider a comletely new lost sale recature function than discussed in Patel and Gor (2013), the imact uon the ordering and ricing olicies of a newsvendor-tye, rofit-maximizing retailer, faced with the ossibility of backordering at least some of the shortages incurred from demand underestimation, by offering some rebate incentives for waiting. The backordering occurs through an emergency urchase of the items in question at some remium over the regular urchasing cost. In turn, the retailer offers to the end-customers left out of the initial sale a rebate incentive uon urchase of each item backordered, quite aware that not all the customers that could not buy in the first instant may avail themselves of the rebate offer and buy. The backlog fill rate, reresenting the robability of the end-customers returning to satisfy their unfilled demand, is modelled as a function of the size of the rebate offered relative to the selling rice. Further, the olicy decisions on the emergency order and on the rebate olicy occur u front, along with the remaining ordering and ricing olicies, rather than at the end of the season. Then the retailer has to decide, ahead of the realization of the demand, the rofit-maximizing ordering, ricing and rebate olicies. The decision variables are the selling rice, the order size and the rebate offered as an incentive to satisfy at least a ortion of the unfulfilled demand. REFERENCES Alford BL and Biswas A (2002). The effects of discount level, rice consciousness and sale roneness on consumers rice ercetion and behavioral intentions. Journal of Business Research Arcelus FJ and Srinivasan G (1987). Inventory olicies under various otimizing criteria and variable marku rates. Management Science Coyright 2014 Centre for Info Bio Technology (CIBTech) 18

8 Arcelus FJ, Kumar S and Srinivasan G (2005). Retailer s resonse to alternate manufacturer s incentives under a single eriod, rice-deendent, stochastic-demand framework. Decision Sciences Arcelus FJ, Ravi Gor and Srinivasan G (2012). Price, rebate and order quantity decisions in a newsvendor framework with rebate deendent recature of lost sales. International Journal of Production Economics 140(1) Aydin G and Porteus EL (2009). Manufacturer-to-Retailer versus Manufacturer-to-Consumer Rebates in a Suly Chain. In N. with the left-skewed (right-skewed) the next lowest (highest) and the symmetrical, Agrawal and S. Smith (Eds.), Retail Suly Chain Management (New York: Kluwer). Bawa K and Shoemaker RS (1989). Analyzing Incremental Sales from a Direct Mail Couon Promotion. Journal of Marketing Chan LMA, Shen ZJM, Simchi-Levi D and Swan JL (2004). Coordination of ricing and inventory decisions: A survey and classification. In: D. Simchi-Levi, S.D. Wu and Z.J.M. Shen, (Eds.), Handbook of Quantitative Analysis: Modeling in the E-Business Era (New York: Kluwer) Karlin S and Carr CR (1962). Prices and otimal inventory olicy. In Studies in Alied Probability and Management Science edited by KJ Arrow and H Scarf (Palo Alto, CA: Stanford University Press) Lau AHL, Lau HS and Wang JC (2007). Some roerties of buyback and other related schemes in a newsvendor roduct suly-chain with rice-sensitive demand. Journal of the Oerational Research Society Mills ES (1958). Uncertainty and rice theory. Quarterly Journal of Economics Patel A and Gor Ravi (2013). A single eriod model where the lost sales recature is a function of (r/). Alied Mathematical Sciences 7(103) Petruzzi NC and Dada M (1999). Pricing and the newsboy roblem: A review with extensions. Oerations Research Weng ZK (2004). Coordinating order quantities between the manufacturer and the buyer: A generalized newsvendor model. Euroean Journal of Oerational Research Yao L, Chen Y and Yan H (2006). The newsvendor roblem with ricing: Extensions. International Journal of Management Science and Engineering Management Yao L (2002). Suly chain modeling: Pricing, contracts and coordination. Ph.D. Thesis, Chinese University of Hong Kong. Zhou YW and Wang SD (2009). Manufacturer-buyer coordination for newsvendor-tye roducts with two ordering oortunities and artial backorders. Euroean Journal of Oerational Research Coyright 2014 Centre for Info Bio Technology (CIBTech) 19