NEWSVENDOR PROBLEM WITH PRICING: PROPERTIES, ALGORITHMS, AND SIMULATION


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1 Proceedings of the 2005 Winter Simulation Conference M. E. Kuhl, N. M. Steiger, F. B. rmstrong, and J.. Joines, eds. NEWSVENDOR PROBLEM WITH PRICING: PROPERTIES, LGORITHMS, ND SIMULTION Roger L. Zhan ISE Deartment University of Florida Gainesville, FL , U.S.. ZuoJun Max Shen IEOR Deartment University of California Berkeley, C , U.S.. BSTRCT When a newsvendor faces stochastic ricesensitive demands, he/she has to make the ricing and inventory decisions before the demand is realied. In the literature, this roblem is tyically solved by reducing it to an otimiation roblem over a single variable. In contrast, we treat this roblem as a nonlinear system of two variables and rovide some solution roerties (e.g. existence, uniqueness) of the system. We also develo an iterative algorithm and a simulation based algorithm for this roblem. INTRODTION The classic newsvendor roblem is to make a singleeriod rocurement decision of a single roduct under stochastic demands. This roblem, because of its simle but elegant structure as well as its rich managerial insights, is a crucial building block of the stochastic inventory theory. It has been extensively studied over decades with extensions including different objectives and utility functions, multile roducts with substitution, multile locations, and different ricing strategies. For extensive reviews see Khouja (999) and Porteus (990). In this aer, we study the extension that incororates ricing decisions into the classic newsvendor roblem. In this scenario, the newsvendor faces stochastic demands which are influenced by the rice. The roblem is to determine the rocurement and ricing strategies that maximie the exected rofit over a single eriod. We call this roblem the Newsvendor Problem with Pricing (NPP). The research on NPP started in 950s with the work of Whitin (955) and Mills (959). Petrui and Dada (999) rovide a good review on this roblem. Recent studies incororate more market features such as rice markdown and inventory sharing among multile retailers (e.g., Sošić 2004 and Karakul and Chan 2004). To the best of our knowledge, literature on NPP mostly focuses on transforming this roblem of two decision variables (rice and order quantity) to a roblem of a single variable, either rice () or quantity (Q), then solving the roblem by using the first order conditions or an exhaustive search method. In this aer, we aroach the roblem by directly dealing with a system of two equations that are derived from the first order conditions. Using this aroach, roerties of the solutions to the roblem, as well as a good geometric exlanation of the roblem, can be obtained. Furthermore, solution algorithms are develoed based on this aroach. In articular, we roose an efficient iterative algorithm and a simulation based algorithm. In the simulation based algorithm, random samles of demands are generated and the Samle verage roximation (S) scheme and IP (Infinitesimal Perturbation nalysis) gradients are used. The simulation based algorithm can be extended to more general cases, which involve multile newsvendors. The rest of the aer is organied as follows. Section 2 resents the model and the existence and uniqueness roerties of the solution. Section 3 rooses the algorithms for the NPP roblem. Finally, Section 4 summaries the results and discusses future work. 2 MODEL ND ITS PROPERTIES 2. Notation = retail rice. Q = order quantity. v, s, c = er unit salvage value, shortage cost, and urchase cost, resectively. D(, ɛ) = a b + ɛ. ɛ = stochastic term defined on the range [, B] with mean μ. f( ), F ( ) = df and cdf of the distribution of ɛ. = Q (a b). (, ) = exected rofit function., = otimal solution. 0 = a+bc+μ. 743
2 2.2 Model In this aer, the stochastic ricesensitive demand D is modeled in an additive demand form, i.e., D(, ɛ) = a b + ɛ(a > 0,b > 0). ɛ is a random variable defined on [, B] with mean μ, cumulative distribution function (cdf) F( ) and robability density function (df) f( ). Before the stochastic term ɛ is realied, the newsvendor determines simultaneously an order quantity, Q, and a retail rice,, to maximie the exected rofit. We also assume that v<c to avoid the trivial solution. Denote x + = max{x,0}, the newsvendor s rofit can be exressed as the difference between the revenue and the total cost: π(q, ) = min{q, D(, ɛ)} cq + v(q D(, ɛ)) + s(d(,ɛ) Q) +. () This rofit function can be rewritten by substituting D(, ɛ) with a b + ɛ and defining = Q (a b): π(,) = (a b + min{, ɛ}) c(a b + ) +v( ɛ) + s(ɛ ) +. (2) This variable transformation simlifies the comutations and its economic meaning is rovided in Petrui and Dada (999). Noticing that (x y) + = x min{x,y}, the rofit function can be further exressed as: π(,) = ( c)(a b) (c v) sɛ +( + s v)min{, ɛ}, (3) with the exected value (, ) = E[π(,)]=( c)(a b) (c v) sμ + ( + s v)e(min{, ɛ}). (4) The objective of the newsvendor is to maximie the exected rofit (, ) by choosing and. However, the otimal solution is not necessarily an interior solution, in articular, the value of can be on the boundary, i.e., or B. In the next subsection, we exlore the conditions that guarantee the existence of the interior otimal solution. The geometric exlanation of these condition is also rovided. 2.3 Solution Proerties Define () = B (u )f (u)du. Notice that B E(min{, ɛ}) = uf (u)du + f (u)du = μ (), and (E(min{, ɛ})) = f () + (( F ())) The first artial derivatives of (, ) are (, ) (, ) = F(). = (c v) + ( + s v)[ F()], (5) = a + bc + μ () = ( 0 ) (). (6) Equations (5) and (6) imly the relationshi between and as following: () = F ( + s c ), (7) + s v () = 0 (). (8) Equation (7) is simly the solution formulation for the classic newsvendor roblem. It naturally requires that c s. nd Equation (8) imlies that 0. Thus, the boundary condition for is c s 0. In the literature, researchers usually utilie Equation (8) to reduce the original roblem to an otimiation roblem over a single variable. Our aroach, however, works on these two first order conditions directly, and rovides more insights into the roblem. Let r() = f() F(),wehave () () = r()( + s v), (9) = F(). (0) Equations (9) and (0) imly that the functions () and () increase in and, resectively. nd the second artial derivatives are dr() d 2 () r() 2 = + r() ( + s v) 2 r 2 (), () 2 () 2 = f() < 0. (2) Thus function () is concave in and based on Equation () the following lemma is established. Lemma If F( ) is a distribution function satisfying the condition r 2 () + dr() d > 0 for [, B], () is concave in. This condition is satisfied by all nondecreasing haard rate distributions, which include PF 2 distributions and the 744
3 lognormal distribution (Barlow and Proschan 975). In articular, uniform, normal, logistic, chisquared and exonential distributions all belong to this category (Bagnoli and Bergstrom 989). The next result follows from Lemma and the fact that if functions () and () are increasing and concave in and, resectively, then the system of the equations = () and = () has at most two solutions. Lemma 2 If the condition for Lemma is satisfied, then the system of Equations (7) and (8) has at most two solutions. Geometrically, the lemma states that the two curves in Figure have at most two intersections. We denote the uer intersection as " and the lower intersection as ". Zhan and Shen B () () (a+bc+)/() c s 0 Figure 3: Two Solutions B () () 0 Figure : Two Solutions By considering the boundary conditions, B and c s 0, roerties of the NPP roblem can be established. Proosition If the condition for Lemma is satisfied and a b(c 2s) + >0, then equations (7) and (8) have a unique solution, which is also otimal for the roblem. The condition a b(c 2s)+ >0, i.e., a+bc+ >c s is to ensure there is only one intersection of the two curves in the feasible region. This case is geometrically shown in Figure 2. B () () Figure 4: No Solution If the condition a b(c 2s)+ >0 is not satisfied, then there are two scenarios. The first scenario is where Line in Figure 2 moves lower than oint ", and the system has two solutions (see Figure 3); the other scenario is where Line moves higher than oint ", and the system then has no solution, which imlies NPP roblem has no interior solution (see Figure 4). The following roositions characterie these two cases. Proosition 2 If the condition for Lemma is satisfied, a b(c 2s)+ <0 and f () < a+b(c+2s 2v)+, then equations (7) and (8) have two solutions. The one with the larger value is the otimal solution for the NPP roblem. Proosition 3 If the condition for Lemma is satisfied, a b(c 2s)+ <0 and f () > a+b(c+2s 2v)+, then equations (7) and (8) have no solution. The NPP roblem has the otimal solution on the boundary. ll the above roerties are derived under the condition that Lemma holds. If Lemma does not hold, an exhaustive search method is not avoidable. In the next section, we develo the algorithms that are based on the roerties derived in this section. c s (a+bc+)/() 0 Figure 2: Unique Solution 745
4 3 LGORITHMS Proositions and 2 state the conditions for the NPP roblem to have an interior solution. Past research usually limited the analysis on theoretical results, identified some simle cases that can be solved analytically and did not resent algorithms that could be used to solve more general roblems. In contrast, we develo an iterative algorithm, as well as a simulation based algorithm, to solve the system of Equations (7) and (8), which leads to the solution for the NPP roblem under the conditions given by Proositions and Iterative lgorithm Recall the relationshi between and : () = F ( + s c + s v ), () = 0 (). s shown later in Proosition 4 of this section, the above system can be solved iteratively and the solution converges to the otimal one. The algorithm works in this fashion: Start with an initial rice, 0, the value of can be calculated using Equation (7), and with Equation (8), a new rice can be uniquely determined. This new rice can then be used to udate in the next iteration. The algorithm reeats this rocess till the otimal solution is found. The rocedure is summaried in a seudocode format in the following, where δ reresents the recision that is used in the stoing criterion. In ractice, we can also simly secify the number of the iterations as the stoing criterion. Iterative lgorithm Set δ; Set 0 0 ; begin loo Calculate using equation (7); Calculate using equation (8), denote the value by ; if 0 <δthen sto the loo; else set 0, continue the loo; end if; end loo If the initial rice starts from 0, the above algorithm finds the solution to the system of Equations (7) and (8), which has a larger value of if the system has two solutions as shown in Proosition 2. The following roosition shows that the algorithm converges to the otimal solution, and. Proosition 4 Starting from 0, the iterative algorithm converges to the otimal solution, and. Proof. From Equation (8) and () 0, we know that 0. To rove the convergence result, we examine the two sequences: 0 = 0, = ( 0 ), 2 = ( )... and 0 = ( 0 ), = ( ), 2 = ( 2 )..., where ( ) and ( ) are the functions given in Equations (7) and (8). Since 0 = 0, due to the monotonicity of the function ( ), 0. Similarly, due to the monotonicity of the function ( ), we get 0. The same analysis carries on while the algorithm kees roducing the elements in the sequences. That is, we have and Therefor these two sequence are monotonically decreasing and bounded. Thus they will converge to and resectively. 3.2 Simulation Based lgorithm The iterative algorithm will work well and converge to the otimal solution very fast, if it is not difficult to solve Equation (7) analytically and easy to comute () in Equation (8). To byass these two otential difficulties, we need to resort to a different aroach. Therefore, arallel to the iterative algorithm, we roose a simulation based algorithm to solve the NPP roblem, which will work articularly well if the random instance can be easily generated. The simulation based algorithm is more romising if we want to extend the algorithm to deal with more comlicated cases which may involve multile newsvendors. From Equation (4), the first order conditions can be written as (, ) (, ) = (c v) +( + s v) E(min(ɛ, )), (3) = + bc + a + E(min(ɛ, )). (4) The information of Equation (3) can be used to udate the value of by the gradient search method. ( However, ) we need to estimate E(min(ɛ,)) by E (min(ɛ,)). This estimation is justified by the unbiased IP estimator for the gradient because min(ɛ, ) is almost surely continuous with resect to. detailed discussion of the alication of IP can be found in Fu (994). Furthermore, the exectation is aroximated by the samle average value, which is usually called the Samle verage roximation (S) scheme. The rocedure is summaried in a seudocode format in the following, where U reresents the number of regenerative cycles, and a k reresents the ste sie at iteration k, and δ reresents the recision that is used in the stoing criterion. In ractice, we can use other stoing criteria such as a reset number of iterations. 746
5 Simulation Based lgorithm Set δ; Set U; Set 0 0 ; begin loo Set u ; begin loo i. Generate an instance of the random variable ɛ; ii. ccumulate the desired gradients, dmin; iii. u u + ; end loo until u = U; Calculate the desired gradient, dmin/u; Udate the value, + a k (dmin/u); k k + ; Find using the new, denote the value by ; if 0 <δthen sto the loo; else set 0, continue the loo; end if; end loo In ste (ii) of the algorithm, we use IP to comute the gradient. In the comuter imlementation, it works as follows: if the instance of ɛ is greater than, then dmin = dmin++s c; otherwise dmin = dmin c+v. These equalities are derived from Equation (3). Starting with dmin = 0 and dividing dmin by U yield the derivative estimation. 3.3 Numerical Results In this section, we use two small numerical examles to illustrate the quality of the solutions from the the simulation based algorithm. Examle In this examle, the arameters are set as v = 0.5,s = c =,a = 200,b = 35, and ɛ follows the normal distribution with mean of ero and standard deviation of 20. Starting from 0 = 3.357, the iterative algorithm gives the otimal solution = and = We set U = 00 in the simulation based algorithm. The rogram is coded in Matlab and results from 0 runs are reorted in Table. Examle 2 In this examle, the arameters are set as v = 0.5,s = c =,a = 200,b = 35, and ɛ is exonentially distributed with mean value of 0. The iterative algorithm gives the otimal solution = and = The simulation solutions are shown in Table 2. Tables and 2 clearly indicate that we can find the high quality solutions that are very close to the otimal solutions using the simulation based algorithm. lthough this aer mainly considers the single newsvendor roblem, we believe the simulation based algorithm would have more advantage over the analytical algorithms in scenarios with multile newsvendors. 4 CONCLUSIONS Table : Simulation lgorithm: Normal Distribution Run No Mean Std. Dev Otimal Table 2: Simulation lgorithm: Exonential Distribution Run No Mean Std. Dev Otimal In this aer, we analye, from a different ersective, the solution roerties of the newsvendor roblem with ricing. The analysis focuses on the system of the two equations derived from the first order conditions. The derived roerties guide us to develo both the iterative algorithm and the simulation based algorithms. There are several extensions to this aer. The simulation based algorithm can be generalied to deal with more comlicated scenarios which may involve multile roducts and multile newsvendors. The model analyed in this aer uses the additive demand form. Extending the model to other demand forms, such as multilicative demand form, is also an interesting toic for future research. 747
6 REFERENCES Bagnoli, M., and T. Bergstrom Logconcave robability and its alications. University of Michigan. Working Paer. vailable at <htt:// ucsb.edu/ tedb/theory/delta.df>. Barlow, R. E., and F. Proschan Statistical theory of reliability and life testing: Probability models. New York: Holt, Rinehart, and Winston. Fu, M. C Otimiation via simulation: review. nnals of Oerations Research 53: Karakul, M., and L. M.. Chan Newsvendor roblem of a monoolist with clearance markets. YEM vailable at <htt://yaem2004. cukurova.edu.tr/>. Khouja, M The singleeriod (newsvendor) roblem: literature review and suggestions for future research. Omega, the International Journal of Management Science 27: Mills, E. S Uncertainty and rice theory. Quarterly Journal of Economics 73:6 30. Petrui, N. C., and M. Dada Pricing and the newsvendor roblem: a review with extensions. Oerations Research 47(2): Porteus, E. L Stochastic inventory theory. In Handooks in OR & MS, ed. D. P. Heyman and M. J. Sobel, Elsevier, NorthHolland. Sošić, G Pricesetting newsvendors: Effects of rice markdowns and inventory sharing. University of Southern California. Working aer. vailable at <htt://wwwrcf.usc.edu/ sosic/ ricesetting.df>. Whitin, T. M Inventory control and rice theory. Management Science 2:6 68. and his web age is <htt:// shen/>. UTHOR BIOGRPHIES ROGER LEZHOU ZHN is a Ph.D. candidate in the Deartment of Industrial and Systems Engineering at the University of Florida. His research interests include auction design, decisions under uncertainty and global otimiation. He is a member of IIE, INFORMS, and SIM. His address is ZUOJUN MX SHEN is an assistant rofessor in the Deartment of Industrial Engineering and Oerations Research at University of CaliforniaBerkeley. He obtained his Ph.D. in Industrial Engineering and Management Sciences from Northwestern University. Before joining Berkeley, he taught in the Deartment of Industrial and Systems Engineering at the University of Florida. Dr. Shen s rimary research interests are in the general area of integrated suly chain design and management, and ractical mechanism design. His address is 748
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