Streamlining inventory flows with time discounts to improve the profits of a decentralized supply chain

We consider a decentralized supply chain, whereby a supplier sells a product to a group of independent buyers, and develop a strategy for the supplier to offer an all-units price discount or cash rebate for orders that are synchronized with its replenishments. As synchronized orders can be met with inventory directly from receiving to shipping without warehousing, the proposed strategy streamlines system inventory flows to minimize inventory and, hence, the related costs. On the other hand, by increasing the replenishment interval of the supplier, the proposed strategy is able to induce buyers to order in large quantities and hence achieve the objectives of quantity discounts. We show that the proposed strategy can achieve nearly optimal (minimum) system cost, and is much more effective than the existing coordination strategies for decentralized supply chains in the literature.     

Optimal ordering policies in response to a discount offer

Sometimes supplier offers a temporary price discount to increase cash flow or decrease the inventory level of certain items. Thus, the manufacturer may be able to improve the effectiveness of his inventory system by ordering a special quantity at this sale period. In this paper, economic order quantity (EOQ) models with a discounted price are developed to obtain the optimal ordering policy during the sale period for five different cases: (a) coincidence of sale period with replenishment time, (b) non-coincidence of sale period with replenishment time, (c) sale period is longer than a cycle, (d) discounted price as a function of the special ordering quantity, and (e) incremental discount. Each case has its own characteristics of the sale period and the discounted price. The objective is to take the maximum possible advantage from the discounted price by ordering a special quantity during the sale period. The optimal ordering policy is obtained by maximizing the difference between the two costs: Regular EOQ cost and special quantity cost during the sale period. Moreover, a comparison of different discount scenarios is developed to sense the effect of different parameters on the ordering policies. The annual gain obtained is linearly related to the discount and the on-hand remnant inventory. Numerical analyses are provided to illustrate and testify the values of the optimal special quantity. The analysis showed an impressive improvement in the effectiveness of the inventory system when a special order is placed during the sale period. The optimal special quantity is driven for each case to visualize real-life problems. Sensitivity analysis is also initiated to study the change in the total savings with respect to the variation of the special optimal quantity.     

Inventory model for an inventory system with time-varying demand rate

The standard inventory problems of the multi-period have been modeled under different situations. Specifically we have considered the demand subjects of a continuous distribution and a discrete distribution, and whether the demand of each period is unchanged or not. A method to get an economic order quantity in inventory systems with discrete and unchanged demand was presented in a previous paper, and this method has been generalized to an inventory model with varying continuous demand. However, it was not achieved due to there being many classified cases in the general situations. In this article the above method is discussed in the case discrete demand to determine whether it increases or decreases from period to period. A theoretical method is presented by using previous results and some examples are given which suggest how the concept can handle on inventory system. In order to make the decision, an algorithm is also presented under some conditions, and examples are shown by using the computer software program, Mathematica, which helps to explain the findings. In general cases, we view the optimal policy in the inventory problems in only a few periods.     

A Comparison of Fuzzy Set and Probabilistic Paradigms for Ranking Vague Economic Investment Information Using a Present Worth Criterion

This article introduces an approach for comparing the fuzzy set and probabilistic paradigms for ranking vague economic investment information when a present worth criterion is used. Comparisons are made between the two paradigms by forming a ratio of fuzzy ranking value to probabilistic ranking value for a set of tested economic decision conditions (vague cash flows and interest rates). A multivariable analysis of the comparison ratios indicates the dominance or spread required in the expected present worths of competing alternatives to guarantee consistent decision ranking among the paradigms. Results of the research suggest that probabilistic methods will not rank vague economic investment information the same as fuzzy set methods. As a result, decision makers who use probabilistic methods to model vagueness in human thought (the condition for which fuzzy theory was developed) may obtain rankings inconsistent with those obtained via the fuzzy models. In such cases, if decisions are based on project rankings, then different choices or recommendations are likely.     

An inventory model with deteriorating items under inflation when a delay in payment is permissible

This study develops an inventory model for initial-stock-dependent consumption rate when a delay in payment is permissible. In the inventory model, shortages are not allowed. The effect of the inflation rate, deterioration rate, initial-stock-dependent consumption rate and delay in payment are discussed. In the study, mathematical models are also derived under two different circumstances, i.e., Case I: The credit period is less than or equal to the cycle time for settling the account; and Case II: The credit period is greater than the cycle time for settling the account. Besides, expressions for an inventory system's total cost are derived for these two cases. Moreover, a computational procedure and GINO (Lasdon et al., ACM Transactions Mathematical Software 4 (1978) 34–50) are proposed to obtain the optimal order size and cycle time. The results can help managers determine the optimal total cost. Finally, a numerical example demonstrates the applicability of the proposed model.     

Die Ökonomik nichtlinearer Tarife am Beispiel Bahnstrom

Nonlinear tariffs such as quantity discounts are common in infrastructure industries. One reason is that nonlinear tariffs enhance efficiency compared to linear pricing schemes. An example is the nonlinear tariff for the all-inclusive 16,7-Hz Traction Power Supply. Linear tariffs would lead to inefficiently high total costs and to inefficiently low utilization of the infrastructure. In fact, both a perfectly informed price regulation and competition would imply nonlinear tariffs. Finally, all market participants, suppliers and consumers, can in principle benefit from nonlinear tariffs.     

A multi-item supply chain with credit periods and weight freight cost discounts

It is possible to realize considerable savings by aggregating the replenishment of a variety of items in a multi-item supply chain. This joint multi-item replenishment policy has already been widely applied in a variety of industries. This type of policy may make it possible for the retailer to take advantage of transport economies of scale by the utilization of freight discounts for greater weight. In addition, a supplier will often extend forward financing to a retailer. In this paper, a multi-item supply chain with a credit period and weight freight cost discounts is considered. The retailer bears the freight costs, but the freight carrier provides freight-transport discounts that are positively related to the weight of the cargo transported. From both the individual and the channel perspectives, we deal with the dual problems of determining the ideal supplier credit period, and of the best way for the retailer to make multi-item replenishment and pricing decisions, while still maximizing profits. We outline the optimal properties and develop algorithms for solving the problems described, as well as discuss the impact of the freight cost discounts, the inventory holding cost, and the interest rate on the behavior of both parties.     

Managing inventory with two suppliers under yield uncertainty and risk aversion

In this paper, we consider a single-product single-period inventory model in which the retailer can source from two suppliers. The primary supplier is cheaper but unreliable in the sense that it generates supply yield uncertainty, whereas the secondary supplier is perfectly reliable but more expensive. The reliable supplier's capacity is fixed and the retailer cannot order more than the quantity reserved in advance. We study the problem in the context of a risk-averse retailer who has to determine the optimal order quantity from the primary supplier and the optimal reserved quantity from the secondary supplier. We develop the model in the perspective of a low risk averse retailer and quantify the risk via an exponential utility function. We show by numerical experiments how the resulting dual sourcing strategies differ from those obtained in the risk-neutral analysis. We also examine the sensitivity of some model-parameters on the optimal decisions.     

Integrated product specifications and productivity decision making in unreliable manufacturing systems

This paper considers joint production control and product specifications decision making in a failure prone manufacturing system. This is with the knowledge that tight process specifications, while leading to a product of more reliable quality and higher market value, are at the same time associated with higher levels of non-conforming parts, a higher rate of parts rejection and thus a lowering of overall plant productivity. The decision making is further complicated by the lack of reliability of the production process, which imposes that an adequate, also to be designed, level of inventory of finished parts be maintained. The overall optimal decision policy is defined here as one that maximizes the long term average per unit time profit of a combined measure of quality and quantity dependent sales revenue, minus inventory and backlog costs, in the presence of random plant failures and random repair durations. Policy optimization is achieved via a revisited model of the Bielecki–Kumar theory for Markovian machines and a simulation and experimental design based methodology for the more general cases.     

Fuzzy Decision Analysis for Alternative Selection Using a Fuzzy Annual Worth Criterion

Very often, in industry, discounted cash flow techniques are applied for analyzing and selecting investment alternatives under consideration. These techniques are usually based on the data under certainty or risk. In reality, however, the decision makers are often facing the situation of vague cash flows and discount rates, or even uncertain durations, when evaluating and selecting potential investments. Fuzzy set theory has the capability of capturing vague data and allows mathematical operations. This article proposes a fuzzy equivalent uniform annual worth (fuzzy EUAW) method to assist practitioners in evaluating investment alternatives utilizing the theory of fuzzy sets. Triangular fuzzy numbers (TFNs) are used throughout the analysis to represent the uncertain cash flows and discount rates. Further, fuzzy capital recovery factors and fuzzy sinking fund factor are derived. Using these two factors, the fuzzy equivalent annual worth of each investment alternative can be found. By ranking these fuzzy numbers with the integral value, the optimal investment alternative is selected. A numerical example is provided to illustrate the results of the alternative selection.     

On the anti-competitive effects of quantity discounts

We analyze the competitive effects of quantity discounts in an asymmetric duopoly. We find that for a sizeable set of parameter values, quantity discounts harm the smaller firm and reduce consumers' surplus. They can even decrease social welfare, i.e. the sum of producers' and consumers' surpluses. However, the circumstances in which quantity discounts may decrease social welfare are limited and difficult to identify in practice.     

A binary solution method for the multi-product newsboy problem with budget constraint

Multi-product newsboy problem (MPNP) with budget constraint is a classical inventory control/management problem. However, solution methods for MPNP under general demand distributions are limited in the current literature. In this paper, by analyzing properties of the optimal solution to the MPNP with a budget constraint, we develop a solution algorithm for the constrained MPNP. The proposed algorithm is binary in nature, and is applicable to general types of demand distribution functions, discrete as well as continuous. For continuous demand distribution function, our approach can obtain the optimal or near optimal solution to the constrained MPNP with polynomial computation complexity of the o(n) order. On the other hand, for discrete demand distribution functions, it can effectively provide good approximate solution. Numerical experiments are presented to show the performance of our method.     

Oligopoly under incomplete information: On the welfare effects of price discrimination

We study competition by firms that simultaneously post (potentially nonlinear) tariffs to consumers who are privately informed about their tastes. Market power stems from informational frictions, in that consumers are heterogeneously informed about firms’ offers. In the absence of regulation, all firms offer quantity discounts. As a result, relative to Bertrand pricing, imperfect competition benefits disproportionately more consumers whose willingness to pay is high, rather than low. Regulation imposing linear pricing hurts the former but benefits the latter consumers. While consumer surplus increases, firms’ profits decrease, enough to drive down utilitarian welfare. By contrast, improvements in market transparency increase utilitarian welfare, and achieve similar gains on consumer surplus as imposing linear pricing, although with limited distributive impact. On normative grounds, our analysis suggests that banning price discrimination is warranted only if its distributive benefits have a weight on the societal objective.     

混合邻域结构的粒子群算法

为了克服标准粒子群算法的早熟、停滞进化或易于陷入局部最优的现象,提出了一种混合模型(简称NSPO)。NSPO将一个粒子映射到无标度网络的多个网络节点上,借助网络结构获得该粒子的邻域拓扑。对粒子的更新,NSPO既考虑种群的最优,又考虑邻域的最优。在3个具有不同难度特点的测试函数上,将NSPO与标准粒子群算法进行了比较。实验结果表明:对于全局最优和梯度信息明显的函数,NSPO具有非常优越的表现;对于具有诸多局部最优的函数,NSPO逃逸局部最优的能力要强于标准粒子群算法;对于具有误导性梯度信息的函数,NSPO偶尔表现优异。     

Concurrent pricing and lot sizing for make-to-order contract production

This paper develops a model for the simultaneous (i.e. concurrent) determination of the inventory and pricing policies of a supplier, which produces and supplies a product to a buyer, on the basis of a contractual agreement, calling for the delivery of a specific quantity periodically. Assuming that the customer is rational, i.e. it follows its own optimal purchasing policy, the objective of the supplier is to determine the product's selling price, in conjunction with an appropriate production/inventory policy, so that a predetermined gross profit level is achieved. It is further assumed that the supplier's production batch size is an integer multiple of the buyer's order quantity. In formulating a mathematical model of this situation, the interactions between the product's price, the buyer's economic order quantity and the supplier's selling price, costs and profit are taken into account. For solving this model, a simple iterative algorithm is proposed, which is illustrated through a numerical example. Sensitivity analysis performed on the model demonstrates that it is relatively robust and quite insensitive to errors in estimating the buyer's ordering to carrying cost ratio on the part of the producer.     

Asymmetry in capacity and the adoption of all-units discounts

In many abuse of dominance antitrust cases, the dominant firm adopts pricing schemes involving all-units discounts, whereas its smaller competitors often use simple linear pricing. We provide a game-theoretic justification for the observed asymmetry in pricing practices by studying a model in which a firm with full capacity faces a capacity-constrained rival. The asymmetry in capacity between the firms, which gives rise to the captive market, allows the dominant firm to take advantage of the quantity commitment through all-units discounts while the capacity-constrained rival is induced to offer simple linear pricing.     

Group purchasing, nonlinear tariffs, and oligopoly

Loyalty discounts are nonlinear tariffs that condition rebates or marginal prices on meeting aggregate purchase or market share targets. These discounts are widespread, and are often the impetus for consumers to form buying groups, or group purchase organizations (GPOs). This paper models the competitive effects of the introduction of a GPO into a market within which the preferences of the GPO's members are horizontally differentiated. While nonlinear tariffs are an effective way for a monopolist to extract consumer surplus, when two suppliers compete using such schedules, the results are far more competitive in comparison to simple Bertrand–Nash competition with linear tariffs. This result holds when the product of each of the suppliers is attractive to a substantial portion of consumers. In our model, the nonlinear schedule puts all customers “in play” to a degree that contrasts sharply with the competition at the margin characteristic of constant per unit prices. Moreover, competing in nonlinear tariffs removes allocative inefficiency that can result from single price competition.     

Replenish-up-to inventory control policy with random replenishment intervals

The multi period inventory problems have been studied under two main assumptions. Continuous review assumption where an order can be made at any time depending on inventory position and periodic review assumption where an order can be initiated only at discrete time epochs. In this study, we analyze a multi period inventory problem that falls under neither of these two assumptions. In the case we consider, there are periodic replenishments but the replenishment intervals are taken to be i.i.d. random variables. This setting represents the real life cases where a supplier visits a retailer with random inter arrival times and the retailer replenishes his inventories based on a replenish-up-to-level inventory control policy. We also assume that only a certain fraction of unmet demand is backordered and the rest of it is lost.In this setting under general distribution between replenishment epochs, we show the concavity of the expected profit function and give the condition that must hold for the optimal replenish-up-to-level. We also present the specific solutions and analysis under two different distributions, namely, uniform and exponential distributions, together with some numerical examples.     

All-Unit Discounts and the Problem of Surplus Division

All-unit discounts (AUD) are non-linear pricing schemes whereby buyers who reach a specific quantity threshold get rebates also retroactively for all units bought before. This sets high incentives for buyers to meet the quantity threshold, and may also have foreclosure effects on potential entrants. In a model where an incumbent faces second-period competition by entrants, we show that AUD can indeed be abused to shift rents from entrants. In contrast to exclusive dealing which is usually seen as very similar to AUD, inefficient quantity distortions may arise even with perfect information if and only if there is sufficiently intense competition among potential entrants.     

Robust exclusion and market division through loyalty discounts

We show that loyalty discounts create an externality among buyers because each buyer who signs a loyalty discount contract softens competition and raises prices for all buyers. This externality can enable an incumbent to use loyalty discounts to effectively divide the market with its rival and raise prices. If loyalty discounts also include a buyer commitment to buy from the incumbent, then loyalty discounts can also deter entry under conditions in which ordinary exclusive dealing cannot. With or without buyer commitment, loyalty discounts will increase profits while reducing consumer welfare and total welfare as long as enough buyers exist and the entrant does not have too large a cost advantage. These propositions are true even if the entrant is more efficient and the loyalty discounts are above cost and cover less than half the market. We also prove that these propositions hold without assuming economies of scale, downstream competition, buyer switching costs, financial constraints, limits on rival expandability, or any intra-product bundle of contestable and incontestable demand.