single period multiproduct inventory models with substitution we study a single period multiproduct inventory problem with substitution and proportional costs and revenues we consider n products and n demand classes with full downward substitution i e excess demand for class i can be satisfied using product j for i j we first discuss a two stage profit maximization formulation for the multiproduct substitution problem we show that a greedy allocation policy is optimal we use this to write the expected profits and its first partials explicitly this in turn enables us to prove additional properties of the profit function and several interesting properties of the optimal solution in a limited computational study using two products we illustrate the benefits of solving for the optimal quantities when substitution is considered at the ordering stage over similar computations without considering substitution while ordering specifically we show that the benefits are higher with high demand variability low substitution cost low profit margins or low price to cost ratio high salvage values and similarity of products in terms of prices and costs

time deposits dimensions and fraud we stipulate arguendo that fractional reserve demand deposit banking is per se fraudulent we ask whether or not time deposit banking can also be illicit and answer in the positive if there is a mismatch between the time dimensions of deposits and loans to wit if an intermediary borrows short and lends long

mixed forests and finance methodological approaches abstract mixed forest ecosystems show effects of risk compensation similar to diversified portfolios of stocks other than portfolios of stocks the composition of long lived forests cannot be adjusted at short notice to current market developments thus the financial robustness of tree species portfolios is very important however it is still an open question which financial approaches are appropriate to evaluate the financial robustness of mixed forests compared to single species forests this paper analysed various possibilities to assess portfolios of tree species namely the mean variance mv approach applied for the selection of optimal portfolios the second order stochastic dominance ssd criterion and the information gap approach for the numerical analysis existing financial data on norway spruce picea abies l karst and european beech fagus sylvatica l were used which were generated by monte carlo simulations mcs under risk regardless of the analysed approach in all cases pure forests did not outperform mixed forests the mv approach was restricted by the assumption of normally distributed financial returns it had the tendency to favour larger proportions of the high risk species norway spruce ssd was only able to distinguish between the efficient and the inefficient portfolios it showed that forests with less than 20 financial returns generated by mcs or less than 30 normally distributed financial returns of norway spruce were inefficient under information gap modelling the advantages of risk compensation in mixed forests became clearest under the given assumptions the financial robustness of a mixed forest 50 european beech and 50 norway spruce was between 4 and 18 times greater than that of pure norway spruce modern information gap modelling is very similar to the well known safety first rules and as such it is not that novel to financial optimisation as it seems at the first glance although the information gap theory resulted in the same rankings as the minimisation of the estimated probability of failure it nevertheless proved to be useful as a supplement to the other methods against the background of severe uncertainty inherent in long term decision making in forestry information gap modelling is based on appropriate assumptions it favours a decision which promises the greatest tolerance to possible errors in the available information while still guaranteeing a minimal acceptable financial result in principle classical safety first rules do the same however the information gap model is able to quantify the immunity of a decision against errors in the estimated financial data from a theoretical point of view this approach provides a supplementary and methodologically sound opportunity to incorporate severe uncertainty in decision making

fractional reserve banking client collaboration and fraud this paper traces the recent debate over the legitimacy of maturity mismatching and fractional reserve banking it shows that there is common ground between bagus and howden journal of business ethics 90 3 399 406 2009 106 295 300 2012 on the one hand and evans journal of business ethics 2013 on the other regarding contractual arrangements that lead to fractional reserve banking while both agree that fractional reserve banking that arises out of a bailment or storage contract constitutes fraud block and barnett journal of business ethics 88 4 711 716 2009 100 229 238 2011 stress the illegitimacy of fractional reserve banking for creating more money substitutes than there is actual money while it is true that fractional reserve banks are capable of creating money this activity still cannot be regarded as fraudulent using a common law definition of fraud for it can only take place with requisite client collaboration that makes it impossible to identify a victim

capacitated two stage multi item production inventory model with joint setup costs we analyze a continuous time two stage production inventory system in the first stage a common intermediate product is produced in batches and possibly stored in the second phase the intermediate product is fabricated into n distinct finished products several finished products may be included in a single production batch of limited capacity to exploit economies of scale we propose a planning methodology to address the combined problem of joint setup costs and capacity limits per setup we restrict ourselves to a class of replenishment strategies with the following properties a replenishment strategy specifies a collection of families subsets of items covering all end items if an item belongs to several families a specific fraction of its sales is assigned to each family each time the inventory of one item in a family is replenished the inventories of all other items in the family are replenished as well we derive a simple roughly 0n log n algorithm that results in a strategy whose long run average cost comes within a few percentage points of a lower bound for the minimum achievable cost within the above class of strategies

inventory pooling under heavy tailed demand risk pooling has been studied extensively in the operations management literature as the basic driver behind strategies such as transshipment manufacturing flexibility component commonality and drop shipping this paper explores the benefit of risk pooling in the context of inventory management using the canonical model first studied in eppen eppen gd 1979 effects of centralization on expected costs in a multi location newsboy problem management sci 25 5 498 501 specifically we consider a single period multilocation newsvendor model where n different locations face independent and identically distributed demands and linear holding and backorder costs we show that eppen s celebrated result i e that the expected cost savings from centralized inventory management scale with the square root of the number of locations depends critically on the light tailed nature of the demand uncertainty in particular we establish that the benefit from pooling relative to the decentralized case in term

evaluation of stock allocation policies in a divergent inventory system with shipment consolidation in this paper we consider a one warehouse n retailer inventory system characterized by access to real time point of sale data and a time based dispatching and shipment consolidation policy at the warehouse more precisely inventory is reviewed continuously while a consolidated shipment for example a truck to all retailers is dispatched from the warehouse at regular time intervals the focus is on investigating the cost benefits of using state dependent myopic allocation policies instead of a simple fcfs first come first serve rule to allocate shipped goods to the retailers the analysis aims to shed some light on when if ever fcfs is a reasonable policy to use in this type of system the fcfs allocations of items to retailers are determined by the sequence in which retailer orders or equivalently customer demands arrive to the warehouse applying the myopic policy enables the warehouse to postpone the allocation decision to the moment of shipment from the warehouse or the moments of delivery to the different retailers and to base it on the inventory information available at those times the myopic allocation method we study is often used in the literature on periodic review systems

exact and approximate cost functions for product aggregates we show that certain large multiproduct problems can be well approximated by smaller models of the same form representing only one aggregate product or a few such products this reduction follows from a new technique to approximate the minimal cost of a special allocation or disaggregation problem by a simple closed form aggregate cost function this function moreover has the same form as the cost functions for the original individual products for special cases the approximation is exact this latter result is related to economic theoretic research on aggregation with homothetic preferences the results are applied to a one stage multiperiod multi item production smoothing model and then to a similar model of a two stage system yielding a reduction in the number of items to one or to relatively few

capacitated multi item inventory systems with random and seasonally fluctuating demands implications for postponement strategies we address multi item inventory systems with random and seasonally fluctuating and possibly correlated demands the items are produced in two stages each with its own lead time in the first stage a common intermediate product is manufactured the production volumes in the first stage are bounded by given capacity limits we develop an accurate lower bound and close to optimal heuristic strategies of simple structure the gap between them evaluated in an extensive numerical study is on average only 0 45 we use the model to investigate the benefits of various delayed product differentiation postponement strategies as well as other strategic questions including i the benefits of flexible versus dedicated production facilities ii the trade off between capacity and inventory investments and iii the trade off between capacity investments and service levels

94 effective policies for a two stage serial inventory system with stochastic demand a two stage inventory system is considered where poisson demand occurs at stage 1 and stage 1 replenishes its inventory from stage 2 which in turn orders from an outside supplier with unlimited stock each shipment either to stage 2 or to stage 1 incurs a fixed setup cost under the assumption that the supply leadtime at stage 2 is zero we characterize a simple heuristic policy whose long run average cost is guaranteed to be within 6 of optimality i e a 94 effective policy the paper also provides heuristic policies for more general inventory systems and reports computational results

oil and water do not mix or aliud est credere aliud deponere the financial crisis has led to new interest in the ethics of financial markets in this article we further the debate on the nature of banking contracts by showing that the fundamental subjective purposes of loan and deposit contracts are irreconcilable any resultant mixture of the two contracts is a legal aberration we consider a mutual fund as an important and legitimate alternative to the common demand deposit to provide high liquidity and some yield without offering full availability of a nominal sum besides being a close substitute for how many deposit accounts function today the mutual fund has the additional benefit of satisfying all legal and ethical requirements loan and investment contracts such as money market mutual funds allow for the bank to make use of their clients funds while the intents of money owners are clearly classified without running into legal or ethical problems

lower bounds for multi echelon stochastic inventory systems we establish lower bounds on the minimum cost of managing certain production distribution networks with setup costs at all stages and stochastic demands these networks include serial assembly and one warehouse multi retailer systems we obtain the bounds through novel cost allocation schemes we evaluate the bounds performance for one warehouse multi retailer systems by comparing them with simple heuristic policies the bounds are quite tight for systems with a small number of retailers we also present simplified proof of known optimality results for serial and assembly systems

option pricing in a lognormal securities market with discrete trading abstract this paper derives a call option valuation equation assuming discrete trading in securities markets where the underlying asset and market returns are bivariate lognormally distributed and investors have increasing concave utility functions exhibiting skewness preference since the valuation does not require the continouus time riskfree hedging of black and scholes nor the discrete time riskfree hedging of cox ross and rubinstein market effects are introduced into the option valuation relation the new option valuation seems to correct for the systematic mispricing of well in and well out of the money options by the black and scholes option pricing formula

a standard measure of risk and risk value models in this paper we propose a standard measure of risk that is based on the converted expected utility of normalized lotteries with zero expected values this measure of risk has many desirable properties that characterize the notion of risk it is very general and includes many previously proposed measures of risk as special cases moreover our standard measure of risk provides a preference based and unified method for risk studies since the standard measure of risk is compatible with the measure of expected utility it can be used explicitly or implicitly in an expected utility model under a condition called risk independence a decision could be made by explicitly trading off between risk and value which offers an alternative representation of the expected utility model named the standard risk value model finally we discuss some other applications of the standard measure of risk and extensions of our risk value tradeoff framework for descriptive decision making

the case against fiduciary media ethics is the key salerno http mises org daily 4389 2010a asserts that in the debate between those who favor and oppose fractional reserve banking the important issue is not whether or not this institution is inherently fraudulent but rather does it or does it not cause the business cycle we join this author in thinking both important however we reverse this order

optimal control of a divergent multi echelon inventory system consider a divergent multi echelon inventory system such as a distribution system or a production system at every facility in the system orders are placed or production is initiated periodically the order arrives after a fixed lead time at the end of each period linear costs are incurred penalty costs are incurred at the most downstream facilities for back orders the objective is to minimize the expected holding and penalty costs per period we prove that under the balance assumption it is cost optimal to control every facility by an order up to policy the optimal replenishment policy i e the order up to level and the allocation functions at each facility can be determined by system decomposition this decomposition reduces complex multi dimensional control problems to simple one dimensional problems

optimizing the shares of native tree species in forest plantations with biased financial parameters abstract addressing uncertainty is a key requirement to follow the principle of precaution in sustainable ecosystem management the maximization of worst case outcomes according to the maximin decision rule based on the two parameters mean and variance of a financial indicator is a prominent approach to integrate uncertainty in decision making in forestry the problem of selecting the optimum tree species combination for a forest plantation investment can be seen as a problem of optimal portfolio selection to be solved according to the maximin decision rule yet it is well known that portfolios computed from expected means and variances are highly sensitive to changes in the estimated parameters the financial results may be poor if we rely too much on the historical data this paper tests an extended worst case model that considers a lower bound for the expected mean net present value npv of a tree species portfolio and an upper bound for its variance biased expected mean npvs variances and correlations for the tree species picea abies l karst spruce and fagus sylvatica l beech were used to test the variability of the resulting tree species portfolios 27 scenarios a comprehensive simulated data set which was adopted from an existing study and defined as the independent reference served to evaluate the financial performance of the tree species portfolios obtained from optimization with the biased data compared with the results of classical worst case optimization instances it was feasible to reduce the variability of tree species shares effectively when the optimization was carried out with the extended worst case approach furthermore the financial performance of this approach was better when tested with the independent data the worst case forest npvs achieved with the extended approach were on average 10 statistical confidence 0 95 or 147 statistical confidence 0 99 greater in comparison to the results of the classical approach the influence of the uncertainty parameter selection was tested and the results were discussed against the controversial viewpoints on the usefulness of the information gap decision theory finally the significance of our results for sustainable ecosystem management is pointed out

entrepreneurial error does not equal market failure barnett and block 2015 claim that bagus and howden 2012b support indirectly the concept of market failure in this paper we show that maturity mismatching in an unhampered market may imply entrepreneurial error but cannot be considered a market failure we demonstrate why fractional reserve banking leads to business cycles even if there is no central bank and why maturity mismatching does not per se lead to clusters of errors in a free market finally in contrast to the examples provided by barnett and block we assure that maturity mismatching does not imply the creation of two incompatible contracts due to the fungible nature of money

the multiperiod multilocation inventory problem this paper studies the structure of optimal policies in general multiperiod multilocation inventory problems with proportional costs it is shown that the multiperiod problem has the same generic form in each period as a one period problem hence properties of the one period problem are inherited by the multiperiod formulation in particular the form of the optimal policy can be characterized in the special case of a leontief system structure and for a somewhat more general class it is shown that base stock policies are optimal for certain initial states for the stationary problem some results are derived about the monotonicity of the sequence of base stock vectors

approximations of dynamic multilocation production and inventory problems consider a central depot or plant which supplies several locations experiencing random demands orders are placed or production is initiated periodically by the depot the order arrives after a fixed lead time and is then allocated among the several locations the depot itself does not hold inventory the allocations are finally received at the demand points after another lag unfilled demand at each location is backordered linear costs are incurred at each location for holding inventory and for backorders also costs are assessed for orders placed by the depot the object is to minimize the expected total cost of the system over a finite number of time periods this system gives rise to a dynamic program with a state space of very large dimension we show that this model can be systematically approximated by a single location inventory problem all the qualitative and quantitative results for such problems can then be applied

an integrated supply chain problem with environmental considerations in this paper a novel formulation for a multiple product capacitated inventory location supply chain model with risk pooling and carbon emission considerations is proposed various products are shipped from a single plant to retailers with stochastic demand via a network of capacitated warehouses in the presence of environmental regulations the locations and inventory policies of the warehouses are chosen so as to minimize the sum of fixed facility location transportation inventory carrying costs and emissions related costs the warehouses retain safety stock so as to maintain appropriate service levels in the face of uncertain demand at the retailers for multiple products computational results are also presented

optimal policy for a muti product multi period non stationary stochastic production and customization problem abstract this paper is concerned with a multi product multi period non stationary stochastic production and customization problem in which a firm wishes to satisfy multi period stochastic demands for m products the firm produces a generic product that can be customized into the m products the lead time to produce the generic product is one period the generic product can be customized into any of the m products immediately assume that the costs of production customization storage and shortage are linear also assume that the unsatisfied demands for the m products at the firm in each period are lost the objective of the firm is to find a production customization policy that minimizes the expected cost over a planning horizon we obtain the conditions that ensure that a myopic production customization policy is optimal

design for postponement a comprehensive characterization of its benefits under unknown demand distributions recent papers have developed analytical models to explain and quantify the benefits of delayed differentiation and quick response programs these models assume that while demands in each period are random they are independent across time and their distribution is perfectly known i e sales forecasts do not need to be updated as time progresses in this paper we characterize these benefits in more general settings where parameters of the demand distributions fail to be known with accuracy or where consecutive demands are correlated here it is necessary to revise estimates of the parameters of the demand distributions on the basis of observed demand data we analyze these systems in a bayesian framework assuming that our initial information about the parameters of the demand distributions is characterized via prior distributions we also characterize the structure of close to optimal ordering rules in these systems for a variety of types of order cost functions

the strategic perils of delayed differentiation the value of delayed differentiation also known as postponement for a monopolist has been extensively studied in the operations literature we analyze the case of imperfectly competitive markets with demand uncertainty wherein the choice of supply chain configuration i e early or delayed differentiation is endogenous to the competing firms we characterize firms choices in equilibrium and analyze the effects of these choices on quantities sold profits consumer surplus and welfare we demonstrate that purely strategic considerations not previously identified in the literature play a pivotal role in determining the value of delayed differentiation in the face of either entry threats or competition these strategic effects can significantly diminish the value of delayed differentiation in fact under plausible conditions these effects dominate the traditional risk pooling benefits associated with delayed differentiation in which case early differentiation is the dominant strategy for firms even under cost parity with delayed differentiation we extend the main model to study the effects of alternate market structures asymmetric markets and inventory holdback our results in particular that for a broad range of parameter values early differentiation is a dominant strategy even under cost parity with delayed differentiation are robust to these relaxations

stochastic dominance and expected utility survey and analysis while stochastic dominance has been employed in various forms as early as 1932 it has only been since 1969 1970 that the notion has been developed and extensively employed in the area of economics finance agriculture statistics marketing and operations research in this survey the first second and third order stochastic dominance rules are discussed with an emphasis on the development in the area since the 1980s

coordinated replenishments in a multi item inventory system with compound poisson demands in many practical applications or multi item inventory systems significant economies of scale can be exploited when coordinating replenishment orders for groups of items this paper considers a continuous review multi item inventory system with compound poisson demand processes excess demands are backlogged and each replenishment requires a lead time there is a major setup cost associated with any replenishment of the family of items and a minor item dependent setup cost when including a particular item in this replenishment moreover there are holding and penalty costs we present an algorithm which searches for a simple coordinated control rule minimizing the long run average cost per unit time subject to a service level constraint per item on the fraction of demand satisfied directly from on hand inventory this algorithm is based on a heuristic decomposition procedure and a specialized policy iteration method to solve the single item subproblems generated by the decomposition procedure the model applies to multi location inventory systems with similar cost structures for coordinated deliveries

on the necessary and sufficient conditions for legitimate banking contracts what role do demand deposits serve in the financial system the answer to this simple question has great implications in keeping the legal terms of the contract consistent with the demands of the financial system demand deposits are a perfect monetary substitute since money is only held to hedge against perceived uncertainty in both the timing and magnitude of future expenditures demand deposits are demanded for the same reason from this we derive three main conclusions first a financial contract similar to a demand deposit e g very short term bonds money market mutual funds etc cannot substitute for money second full agreement to a financial contract does not create a perfect substitute for money unless it provides money s two key characteristics on demand and par value redemption finally the demand for fractional reserve demand deposits is fostered by an exogenous source deposit insurance and that demand for a good or service is not a sufficient condition to justify its legality or ethicality

coordinated replenishment strategies in inventory distribution systems in this paper we study the impact of coordinated replenishment and shipment in inventory distribution systems we analyze a system with multiple retailers and one outside supplier random demand occurs at each retailer and the supplier replenishes all the retailers in traditional inventory models each retailer orders directly from the supplier whenever the need arises we present a new centralized ordering policy that orders for all retailers simultaneously the new policy is equivalent to the introduction of a warehouse with no inventory that is in charge of the ordering allocation and distribution of inventory to the retailers under such a policy orders for some retailers may be postponed or expedited so that they can be batched with other retailers orders which results in savings in ordering and shipping costs in addition to the policy we propose for supplying inventory to the retailers we also consider three other policies that are based on these well known policies in the literature a can order policy b echelon inventory policy and c fixed replenishment interval policy furthermore we create a framework for simultaneously making inventory and transportation decisions by incorporating the transportation costs or limited truck capacities we numerically compare the performance of our proposed policy with these policies to identify the settings in which each policy would perform well

a model for analyzing multi channel distribution systems to capture customers in different market segments many firms distribute their products using multiple channels involving in store mail order and electronic mediums nevertheless the business literature is replete with examples where the management of multiple distribution channels becomes unwieldy as a way to evaluate different distribution strategies arising from multiple channels we develop a model of a general multi channel distribution system subject to stochastic demand since the exact analysis of our model is intractable we propose a decomposition scheme that enables us to develop a near optimal distribution policy with minimum total expected distribution cost we utilize our analysis of this multi channel system in two specific applications arising from multi channel distribution the first application deals with a store based retailer expanding into a direct sales channel the retailer can fill the orders generated from this additional sales channel by using the existing stores or the existing depots for this application we evaluate the trade off between these two options the second application intends to find a cost effective way to distribute products by utilizing the combined distribution system of two retailers after a potential merger and acquisition m a for this application we quantify the benefit of an m a in terms of distribution efficiency

a two echelon allocation model and the value of information under correlated forecasts and demands abstract in this article we explore the effects of incorporating forecasts explicitly in a two echelon allocation model which consists of a central depot and several retailers in particular we investigate the possible benefits on the system costs and inventory levels the depot does not hold any inventory and the demand is observed only at the retailers under a general correlated demand forecast structure we obtain the approximate system wide order up to level and the expected system cost in order to assess the value of information provided by keeping track of the forecasts we construct a comparable inventory system operating under a standard demand model we demonstrate that the standard demand model results in higher order up to levels and higher system costs

determining near optimal base stock levels in two stage general inventory systems abstract this paper proposes an easily implementable scalable decomposition heuristic for determining near optimal base stocks in two level general inventory systems in this heuristic the general system is decomposed into assembly systems one for each end product for these assembly systems the base stock levels are calculated separately taking into account risk pooling effects for the common components our numerical analyses yield two main insights first the base stock levels determined by the heuristic are close to optimal second considerable improvements can be obtained compared to common sense heuristics

mean lower partial moment valuation and lognormally distributed returns in this paper we develop a capital asset pricing model in a mean lower partial moment framework specifically we show that when partial moments are computed about the expected risky portfolio return optimal portfolio choice in a mean lower partial framework permits a two fund portfolio separation between a riskless asset and a market portfolio of risky assets in this new framework risk is measured as semideviation for second degree stochastic dominance and semivariance for third degree stochastic dominance further when security returns are lognormally distributed and small risk this new mean lower partial moment valuation specializes to the mean logarithmic variance capital asset pricing model

optimal centralized ordering policies in multi echelon inventory systems with correlated demands this paper treats a depot warehouse system in which demand occurs at the warehouse or retail level this work differs from a number of other studies in that we allow item demands to be correlated both across warehouses and also correlated in time our motivation for this generalization arises from our experience with an actual system of this type used by a major national producer and distributor of consumer products we observed both high correlations between successive monthly demands around 0 7 and correlations between demands for an item at different locations also about 0 7 in a given time period we derive an explicit expression for the optimal safety stock as a function of the level of correlation through time the analysis requires two assumptions 1 the allocation assumption and 2 the equal coefficient of variation assumption similar assumptions have been used by other researchers finally numerical evaluations are included to illustrate the impact of the various magnitudes of correlation

optimal and approximate policies in multiperiod multilocation inventory models with transshipments this paper examines the effects on the optimal ordering policy and its cost of allowing for interactions among retail outlets specifically transshipments are allowed as recourse actions occurring after demands are realized but before they must be satisfied the resultant savings in holding and shortage costs are balanced against the costs of transshipment a base stock ordering policy is optimal for this model if the final period base stock order up to point is nonnegative then it will be the base stock order up to point for all periods unfortunately it can be found analytically only for two special cases where either the cost parameters are equal at every outlet or there are only two outlets these two special cases are used to validate a heuristic solution technique employing monte carlo integration which is then compared to an easily calculated base policy in order to gauge the contribution of this model the additional savings from using this heuristic policy are significant particularly for problems with many retail outlets and low transshipment costs

variance vs downside risk is there really that much difference abstract the popularity of downside risk among investors is growing and mean return downside risk portfolio selection models seem to oppress the familiar mean variance approach the reason for the success of the former models is that they separate return fluctuations into downside risk and upside potential this is especially relevant for asymmetrical return distributions for which mean variance models punish the upside potential in the same fashion as the downside risk the paper focuses on the differences and similarities between using variance or a downside risk measure both from a theoretical and an empirical point of view we first discuss the theoretical properties of different downside risk measures and the corresponding mean downside risk models against common beliefs we show that from the large family of downside risk measures only a few possess better theoretical properties within a return risk framework than the variance on the empirical side we analyze the differences between some us asset allocation portfolios based on variances and downside risk measures among other things we find that the downside risk approach tends to produce on average slightly higher bond allocations than the mean variance approach furthermore we take a closer look at estimation risk viz the effect of sampling error in expected returns and risk measures on portfolio composition on the basis of simulation analyses we find that there are marked differences in the degree of estimation accuracy which calls for further research

balancing retailer inventories this paper examines the optimality of inventory balancing in a one warehouse n retailer distribution system facing stochastic demand for a single product over t successive time intervals in particular we consider the division of predetermined quantities of warehouse stock among retailers in each interval balancing attempts to bring the retailer inventories to the same normalized with respect to first interval demand net inventory level when the demand distribution over the t periods is symmetric with respect to all pairs of retailers and retailer cost over the t intervals is a convex function of the t shipments we show that balancing divisions are optimal the required convexity condition is shown to be satisfied for some familiar cost functions when shortages are backordered and lost however for other cost functions the convexity condition is satisfied under backordering but not under lost sales we also consider the nonidentical retailer demands are not symmetric case and provide examples