Two aspects of optimal diet determination for pig production: efficiency of solution and incorporation of cost variation

For many decades linear programming has been used to find minimum cost diets, notably in the chicken and pig meat industries. More recently, animal growth models together with nonlinear optimisation methods have been used to find feeding schedules which simultaneously minimise feed costs and maximise market return, so maximising gross margin. Genetic algorithms can handle these problems, albeit slowly. In this paper we study the particular nature of the objective function (for pig meat production) and develop a global optimisation algorithm tailored to its discontinuous structure. We also demonstrate the use of stochastic programming to cope with changing feed costs and changing price at slaughter.     

A mathematical programming approach to optimise insurance premium pricing within a data mining framework

In this paper we provide evidence of the benefits of an approach which combines data mining and mathematical programming to determining the premium to charge automobile insurance policy holders in order to arrive at an optimal portfolio. An non-linear integer programming formulation is proposed to determine optimal premiums based on the insurer's need to find a balance between profitability and market share. The non-linear integer programming approach to solving this problem is used within a data mining framework which consists of three components: classifying policy holders into homogenous risk groups and predicting the claim cost of each group using k-means clustering; determining the price sensitivity (propensity to pay) of each group using neural networks; and combining the results of the first two components to determine the optimal premium to charge. We have earlier presented the results of the first two components. In this paper we present the results of the third component. Using our approach, we have been able to increase revenue without affecting termination rates and market share.     

A market-driven transfer price for distributed products using mathematical programming

A distributed product has its manufacturing activities distributed among many locations. These locations could belong to one or more firms in a manufacturing network. Often, components needed to manufacture a distributed product move through different nodes in the network and sometimes across international borders. Hence, a transfer price is needed for the purpose of estimating duties and drawbacks. Being aware of the fact that transfer price can be used to manipulate taxable profits, many countries have instituted rules concerning transfer price estimation. For example, in the United Sates, the Internal Revenue Service (IRS) says that the right price is the market value. But for many components it is difficult to find a free market. Similar products may exist in the market but they may have different attributes. In such cases, it is important to be able to estimate the market-driven transfer price, given other similar products in the open market.We develop a method using a mathematical programming model and providing companies an opportunity to work proactively with the IRS in a cooperative manner in order to avoid costly audit and litigation. This way, companies avoid penalties and also gain certainty regarding tax liability. An example illustrating the method is presented.     

Linear Programming Techniques for the Construction of Palatable Human Diets

Mathematical modelling of the relationship between acceptable diets and satisfactory nutrition is discussed. The classical minimum-cost diet problem is unsuited to most purposes in human dietetics, since the objective of least monetary cost leads to diets which are unpalatable, both generally and to individuals. In this paper a different approach, using an objective function based on the food preferences of individuals, is described. A computational method, based on this analysis, for constructing individually acceptable diets by means of linear programming is described. Careful use of post-optimal analysis provides invaluable insight to individual patients, and their medical advisors, on the changes to their eating habits which will most readily enable them to meet nutrient intake targets. An example, using data appropriate to a patient with chronic renal failure, demonstrates the usefulness to nutritionists and dietitians of this approach to dietary recommendations.     

生猪生产的最优利润

线性规划（ＬＰ）用于确定饲料最低费用配比已有多年的历史。但确定饲料配比仅是农场主所关心的一件事，他们的主要目的是为了从生产中获取最大的利润。本文提出了基于ＬＰ的饲料规划方案，以便使猪是佳地生长，同时也使养猪者获取更大的生产利润。在最大利润模型中，售价和重量的乘积作为收入，饲料费和间接费作为成本。解中包括猪饲料的总量和猪应有的最佳重量。为了把猪的生长模式引入到ＬＰ模型中，我们用数学生长函数来表达生长率和猪龄之间的关系。该函数是经过适当的转化，用回归方法估算出来的。猪在不同生长阶段的营养需求是依据其生长模型和饲料转化率分析的。把上述结果全都引用到ＬＰ模型中，这样可以使利润最优化。实际数据的检验表明：在实践中，ＬＰ模型的结果是合理的。     

A sample-path approach to optimal position liquidation

We consider the problem of optimal position liquidation where the expected cash flow stream due to transactions is maximized in the presence of temporary or permanent market impact. A stochastic programming approach is used to construct trading strategies that differentiate decisions with respect to the observed market conditions, and can accommodate various types of trading constraints. As a scenario model, we use a collection of sample paths representing possible future realizations of state variable processes (price, trading volume etc.), and employ a heuristical technique of sample-path grouping, which can be viewed as a generalization of the standard nonanticipativity constraints.     

Pricing policy in dynamic markets and a generalization of the Ramsey rule

Pricing policy in a regulated monopoly industry is usually based on maximizing welfare or some other measure of utility level of return on investment. Previously, the Ramsey pricing policy which states that the percentage deviation of quasi-optimal price from marginal cost for each product must be inversely proportional to its price elasticity of demand, has been developed for a static market. The Ramsey framework assumes instantaneous demand response to price changes; empirical evidence suggests demand changes occur dynamically through time.In this paper an optimum pricing rule for a profit maximizing firm based on a general time varying demand model in a dynamic market is obtained assuming a single price change at the beginning of the planning period. A dynamic market equivalent of the well known inverse elasticity law of the static market is developed. Defining the concept of average price elasticity for dynamic markets we show that the inverse elasticity law of static markets takes an inequality form in dynamic markets. For demand functions which decrease, increase or are constant with time the optimum price markups are greater than, less than, or equal to the inverse of the average price elasticity, respectively.The results are then generalized to the case of a constrained welfare maximizing firm. This leads to the development of a dynamic market generalization of the well known Ramsey pricing rule. A simple rule for making quantitative arguments about the relative size of the optimum price in static and dynamic markets is also derived.This work was completed when the author was with Bell Laboratories, USA.     

The Effects of Modelling on Log Bucking Solution Techniques

Log bucking is an industrial problem of subdividing longer logs into smaller logs such that potential revenue is maximized. The model has been solved using dynamic programming predominately, and branch-and-bound to a lesser extent. This research considers the effect of modelling assumptions that have been used to accommodate the solution techniques. Three levels of increasing model detail are considered. An examination of the nature and amount of error is made by using a model of lesser detail instead of a more precise model. The main experiment is used to show the differences in solution accuracy for the three dynamic programming strategies when compared with the accuracy of solutions generated by using branch-and-bound. The final analysis considers the cost in terms of time to solution for each solution technique to produce highly comparable solution accuracy.     

Zero-level pricing method with transaction cost

In this research, we extend Luenberger’s (J Econ Dyn Contr 26(10), 1613–1628, 2002) results on zero-level pricing method to the market with transaction cost. We show that the zero-level price exists in this market. Both the zero-level pricing method and the no-arbitrage pricing method produce price intervals, but the zero-level price interval is smaller than the no-arbitrage price interval. Although the zero-level price interval in general depends on the utility function and initial wealth, we show the zero-level price interval is identical for all individuals with different levels of initial wealth and the HARA utility functions in which one parameter is fixed.     

Optimizing pricing and location decisions for competitive service facilities charging uniform price

In this paper, we present the problem of optimizing the location and pricing for a set of new service facilities entering a competitive marketplace. We assume that the new facilities must charge the same (uniform) price and the objective is to optimize the overall profit for the new facilities. Demand for service is assumed to be concentrated at discrete demand points (customer markets); customers in each market patronize the facility providing the highest utility. Customer demand function is assumed to be elastic; the demand is affected by the price, facility attractiveness, and the travel cost for the highest-utility facility. We provide both structural and algorithmic results, as well as some managerial insights for this problem. We show that the optimal price can be selected from a certain finite set of values that can be computed in advance; this fact is used to develop an efficient mathematical programming formulation for our model.     

What price stability? Social welfare in matching markets

In two-sided matching markets, stability can be costly. We define social welfare functions for matching markets and use them to formulate a definition of the price of stability. We then show that it is common to find a price tag attached to stability, and that the price of stability can be substantial. Therefore, when choosing a matching mechanism, a social planner would be well advised to weigh the price of stability against the value of stability, which varies from market to market.     

Competition for cores in remanufacturing

We study competition between an original equipment manufacturer (OEM) and an independently operating remanufacturer (IO). Different from the existing literature, the OEM and IO compete not only for selling their products but also for collecting returned products (cores) through their acquisition prices. We consider a two-period model with manufacturing by the OEM in the first period, and manufacturing as well as remanufacturing in the second period. We find the optimal policies for both players by establishing a Nash equilibrium in the second period, and then determine the optimal manufacturing decision for the OEM in the first period. This leads to a number of managerial insights. One interesting result is that the acquisition price of the OEM only depends on its own cost structure, and not on the acquisition price of the IO. Further insights are obtained from a numerical investigation. We find that when the cost benefits of remanufacturing diminishes and the IO has more chance to collect the available cores, the OEM manufactures less in the first period as the market in the second period gets larger to protect its market share. Finally, we consider the case where consumers have lower willingness to pay for the remanufactured products and find that in that case remanufacturing becomes less profitable overall.     

Competitive Price Equilibrium with Consumer Reservation Utility

This paper studies a competitive price equilibrium in the market of a product category where consumers are homogeneous with a reservation utility below which they will not purchase the product. The impact of the reservation utility on the price equilibrium is of particular interest, because the reservation utility may change according to the business cycle and economic environments. Using multinomial logit model to describe market response, we study the comparative statics of the prices, profits and market shares of firms, each of which produces one brand in the product category, with respect to the reservation utility in the Nash equilibrium. It is shown that, as the reservation utility increases, the prices as well as the profits at Nash equilibrium decrease. Also, in the case of duopoly market, the firm with lower cost structure will increase its market share as the reservation utility increases.     

Optimal ordering and pricing strategies in the presence of a B2B spot market

In the current paper, we examine the effect of a B2B spot market on the strategic behavior and the performance of a reseller who continues to use the traditional channel while participating in a B2B spot market. We analyze the case in which a risk-neutral reseller faces an additive or multiplicative demand function and identify sufficient conditions under which the optimal order quantity and retail price exist and are unique. We then analytically examine the case in which a risk-averse reseller participates in a fully liquid spot market. We also study numerically how varying liquidity, spot price volatility, demand variability, and correlation coefficient affect a firm’s strategies and performance. We find that demand variability significantly affects both pricing and ordering strategies, whereas the spot price volatility has less influence on pricing decisions. Our results also show that for a risk-averse reseller to charge a lower retail price when the spot market liquidity increases is desirable. We further show that a B2B spot market cannot always improve a reseller’s utility. These findings shed light on how resellers can adjust their procurement and pricing strategies to align with the new business environment created by the emergence of B2B spot markets, as well as have obvious implications for the development of a B2B spot market.     

Make-or-buy service capacity decision in a supply chain providing after-sales service

We consider a supply chain comprising a manufacturer and a retailer. The manufacturer supplies a product to the retailer, while the retailer sells the product bundled with after-sales service to consumers in a fully competitive market. The sales volume is affected by the retailer’s service-level commitment. The retailer can build service capacity in-house at a deterministic price before service demand is realized, or buy the service from an outsourcing market at an uncertain price after service demand realization. We find that the outsourcing market encourages the retailer to make a higher level of service commitment, while prompting the manufacturer to reduce the wholesale price, resulting in more demand realization. We analyze how the expected cost of the service in the outsourcing market and the retailer’s risk attitude affect the decisions of both parties. We derive the conditions under which the retailer is willing to build service capacity in-house and under which it will buy the service from the outsourcing market. Moreover, we find that the manufacturer’s sharing with the retailer the cost to build service capacity improves the profits of both parties.     

Pricing and lot-sizing for a deteriorating item in a periodic review inventory system with shortages

We consider a single product that is, subject to continuous decay, a multivariate demand function of price and time, shortages allowed and completely backlogged in a periodic review inventory system in which the selling price is allowed to adjust upward or downward periodically. The objective of this paper is to determine the periodic selling price and lot-size over multiperiod planning horizon so that the total discount profit is maximized. The proposed model can be used as an add-in optimizer like an advanced planning system in an enterprise resource planning system that coordinates distinct functions within a firm. Particular attention is placed on investigating the effect of periodic pricing jointly with shortages on the total discount profit. The problem is formulated as a bivariate optimization model solved by dynamic programming techniques coupled with an iterative search process. An intensive numerical study shows that the periodic pricing is superior to the fixed pricing in profit maximization. It also clarifies that shortages strategy can be an effective cost control mechanism for managing deterioration inventory.     

Heston随机波动率模型下的资产负债管理问题

应用随机最优控制方法研究Heston随机波动率模型下带有负债过程的动态投资组合问题,其中假设股票价格服从Heston随机波动率模型,负债过程由带漂移的布朗运动所驱动.金融市场由一种无风险资产和一种风险资产组成.应用随机动态规划原理和变量替换法得出了上述问题在幂效用和指数效用函数下最优投资策略的显示解,并给出数值算例分别分析了市场参数在幂效用和指数效用函数下对最优投资策略的影响.     

Design of price mechanisms for network resource allocation via price of anarchy

We study the design of price mechanisms for communication network problems in which a user’s utility depends on the amount of flow she sends through the network, and the congestion on each link depends on the total traffic flows over it. The price mechanisms are characterized by a set of axioms that have been adopted in the cost-sharing games, and we search for the price mechanisms that provide the minimum price of anarchy. We show that, given the non-decreasing and concave utilities of users and the convex quadratic congestion costs in each link, if the price mechanism cannot depend on utility functions, the best achievable price of anarchy is ${{4(3-2 \sqrt{2}) \approx 31.4 \% }}$ . Thus, the popular marginal cost pricing with price of anarchy less than 1/3 ≈ 33.3% is nearly optimal. We also investigate the scenario in which the price mechanisms can be made contingent on the users’ preference profile while such information is available.     

疫情环境下价格导向的防护物资最优生产分配机制研究

新型冠状病毒疫情导致防护物资匮乏,增加了医护人员受感染的风险.现构建了防护物资最优生产-分配-定价模型,并在此基础上提出了广义影子价格的概念,以此作为防护物资定价的参考和依据,通过价格引导生产型企业合理扩大生产规模,以解决当前防护物资短缺的困难.广义影子价格反映了企业产能提升成本等因素,相比传统影子价格更适用于为防护物资统一定价.另外,利用广义影子价格与拉格朗日乘子集合之间的联系,提出了一个线性规划模型用以计算广义影子价格.数值仿真结果说明了广义影子价格在防护物资定价上的适用性.     

Analyzing online B2B exchange markets: Asymmetric cost and incomplete information

This research applies the discriminating auction to analyze the online B2B exchange market in which a single buyer requests multiple items and several suppliers having equal capacity and asymmetric cost submit bids to compete for buyer demand. In the present model, we examine the impact of asymmetric cost and incomplete information on the participants in the market. Given the complete cost information, each supplier randomizes its price and the lower bound of the price range is determined by the highest marginal cost. In addition, the supplier with a lower marginal cost has a larger considered pricing space but ultimately has a smaller equilibrium one than others with higher marginal costs. When each supplier’s marginal cost is private information, the lowest possible price is determined by the number of suppliers and the buyer’s reservation price. Comparing these two market settings, we find whether IT is beneficial to buyers or suppliers depends on the scale of the bid process and the highest marginal cost. When the number of suppliers and the difference between the highest marginal cost and the buyer’s reservation price are sufficiently large, each supplier can gain a higher profit if the marginal costs are private information. On the contrary, when the highest marginal cost approaches the buyer’s reservation price, complete cost information benefits the suppliers.