Economic planning with institutional price constraints for a decentralized economy

This paper describes the relationship between market prices and shadow prices when the economy has general types of institutional price constraints. We consider a decentralized linear economy where market prices quide the decentralized behavior of each activity and the shadow prices measure the social values of resources. To measure the social values, we introduce a social objective criterion. Hence, our approach could be regarded as a central economic price control with institutional price constraints for a decentralized economy. A simple example is employed to graphically illustrate the wedges between market prices and shadow prices. It has been shown that our problem can be solved through mixed integer linear programming techniques.     

Exact algorithms for procurement problems under a total quantity discount structure

In this paper, we study the procurement problem faced by a buyer who needs to purchase a variety of goods from suppliers applying a so-called total quantity discount policy. This policy implies that every supplier announces a number of volume intervals and that the volume interval in which the total amount ordered lies determines the discount. Moreover, the discounted prices apply to all goods bought from the supplier, not only to those goods exceeding the volume threshold. We refer to this cost-minimization problem as the total quantity discount (TQD) problem. We give a mathematical formulation for this problem and argue that not only it is NP-hard, but also that there exists no polynomial-time approximation algorithm with a constant ratio (unless P = NP). Apart from the basic form of the TQD problem, we describe four variants. In a first variant, the market share that one or more suppliers can obtain is constrained. Another variant allows the buyer to procure more goods than strictly needed, in order to reach a lower total cost. We also consider a setting where the buyer needs to pay a disposal cost for the extra goods bought. In a third variant, the number of winning suppliers is limited, both in general and per product. Finally, we investigate a multi-period variant, where the buyer not only needs to decide what goods to buy from what supplier, but also when to do this, while considering the inventory costs. We show that the TQD problem and its variants can be solved by solving a series of min-cost flow problems. Finally, we investigate the performance of three exact algorithms (min-cost flow based branch-and-bound, linear programming based branch-and-bound, and branch-and-cut) on randomly generated instances involving 50 suppliers and 100 goods. It turns out that even the large instances of the basic problem are solved to optimality within a limited amount of time. However, we find that different algorithms perform best in terms of computation time for different variants.     

Inverse Problems of Submodular Functions on Digraphs

In this paper, we study the inverse problem of submodular functions on digraphs. Given a feasible solution x* for a linear program generated by a submodular function defined on digraphs, we try to modify the coefficient vector c of the objective function, optimally and within bounds, such that x* becomes an optimal solution of the linear program. It is shown that the problem can be formulated as a combinatorial linear program and can be transformed further into a minimum cost circulation problem. Hence, it can be solved in strongly polynomial time. We also give a necessary and sufficient condition for the feasibility of the problem. Finally, we extend the discussion to the version of the inverse problem with multiple feasible solutions.     

A variance minimization problem for a Markov decision process

In the steady state of a discrete time Markov decision process, we consider the problem to find an optimal randomized policy that minimizes the variance of the reward in a transition among the policies which give the mean not less than a specified value. The problem is solved by introducing a parametric Markov decision process with average cost criterion. It is shown that there exists an optimal policy which is a mixture of at most two pure policies. As an application, the toymaker's problem is discussed.     

An optimization-based conjectured supply function equilibrium model for network constrained electricity markets

This paper proposes a model to compute nodal prices in oligopolistic markets. The model generalizes a previous model aimed at solving the single-bus problem by applying an optimization procedure. Both models can be classified as conjectured supply function models. The conjectured supply functions are assumed to be linear with constant slopes. The conjectured price responses (price sensitivity as seen for each generating unit), however, are assumed to be dependent on the system line's status (congested or not congested). The consideration of such a dependence is one of the main contributions of this paper. Market equilibrium is defined in this framework. A procedure based on solving an optimization problem is proposed. It only requires convexity of cost functions. Existence of equilibrium, however, is not guaranteed in this multi-nodal situation and an iterative search is required to find it if it exists. A two-area multi-period case study is analysed. The model reaches equilibrium for some cases, mainly depending on the number of periods considered and on the value of conjectured supply function slopes. Some oscillation patterns are observed that can be interpreted as quasi-equilibria. This methodology can be applied to the study of the future Iberian electricity market.     

Reflections about pseudo-dual prices in combinatorial auctions

Combinatorial auctions permitting bids on bundles of items have been developed to remedy the exposure problem associated with single-item auctions. Given winning bundle prices, a set of item prices is called market clearing or equilibrium if all the winning (losing) bids are greater (less) than or equal to the total price of the bundle items. However, the prices for individual items are not readily computed once the winner determination problem is solved. This is due to the duality gap of integer programming caused by the indivisibility of the items. In this paper, we reflect on the calculation of approximate or pseudo-dual item prices. In particular, we present a novel scheme based on the aggregation of winning bids. Our analysis is illustrated by means of numerical examples.     

Event risk, contingent claims and the temporal resolution of uncertainty

We study the pricing and hedging of contingent claims that are subject to Event Risk which we define as rare and unpredictable events whose occurrence may be correlated to, but cannot be hedged perfectly with standard marketed instruments. The super-replication costs of such event sensitive contingent claims (ESCC), in general, provide little guidance for the pricing of these claims. Instead, we study utility based prices under two scenarios of resolution of uncertainty for event risk: when the event is continuously monitored, or when it is revealed only at the payment date. In both cases, we transform the incomplete market optimal portfolio choice problem of an agent endowed with an ESCC into a complete market problem with a state and possibly path-dependent utility function. For negative exponential utility, we obtain an explicit representation of the utility based prices under both information resolution scenarios and this in turn leads us to a simple characterization of the early resolution premium. For constant relative risk aversion utility functions we propose a simple numerical scheme and study the impact of size of the position, wealth and expected return on these prices.     

A Lagrangian decomposition approach for the pump scheduling problem in water networks

Dynamic pricing has become a common form of electricity tariff, where the price of electricity varies in real time based on the realized electricity supply and demand. Hence, optimizing industrial operations to benefit from periods with low electricity prices is vital to maximizing the benefits of dynamic pricing. In the case of water networks, energy consumed by pumping is a substantial cost for water utilities, and optimizing pump schedules to accommodate for the changing price of energy while ensuring a continuous supply of water is essential. In this paper, a Mixed-Integer Non-linear Programming (MINLP) formulation of the optimal pump scheduling problem is presented. Due to the non-linearities, the typical size of water networks, and the discretization of the planning horizon, the problem is not solvable within reasonable time using standard optimization software. We present a Lagrangian decomposition approach that exploits the structure of the problem leading to smaller problems that are solved independently. The Lagrangian decomposition is coupled with a simulation-based, improved limited discrepancy search algorithm that is capable of finding high quality feasible solutions. The proposed approach finds solutions with guaranteed upper and lower bounds. These solutions are compared to those found by a mixed-integer linear programming approach, which uses a piecewise-linearization of the non-linear constraints to find a global optimal solution of the relaxation. Numerical testing is conducted on two real water networks and the results illustrate the significant costs savings due to optimizing pump schedules.     

二供应商经济批量问题的多项式时间算法

为了从采购费用结构不同的供应商中找到最佳补货策略,考虑一个零售商从两个供应商补货的二供应商经济批量问题.零售商在两个供应商处的采购费用结构分别为复合安装费用和全单位数量折扣费用结构.通过对问题结构性质的分析论证,将问题的可行解转化为一个有向网络,降低问题求解的计算复杂性.综合动态规划和Dijkstra最短路算法证明了该问题是多项式时间可解的.     

Solving nonlinear multi-objective optimization problems with fuzzy relation inequality constraints regarding Archimedean triangular norm compositions

We propose an approach to solve a nonlinear multi-objective problem subject to fuzzy relation inequalities with max-Archimedean-t-norm composition by a genetic algorithm. The additive generator of Archimedean t-norms is utilized to reform the existent genetic algorithm to solve the constrained nonlinear multi-objective optimization problems. We consider thoroughly the feasible set of the fuzzy relation inequality systems in three possible cases, namely “≤”, “≥” and the combination of them. In general, their feasible sets are nonconvex which are completely determined by one vector as their maximum solution and a finite number of minimal solutions. The maximum and minimal solutions are formulated by using the additive generator. Additionally, we present some conditions for each case under which the problem can be reduced. Finally, each reduced problem is solved by the genetic algorithm and the efficiency of the proposed method is shown by some numerical examples.     

On computational complexity of the electric power flow optimization problem in market environment

Under consideration is the electric power flow optimization problem for an electric power system which typically arises in calculation of electrical power auctions in the “day-ahead” and balancing markets. It was established that the problem of finding a feasible flow in the balancing market is NP-hard in the strong sense even in case of one generator. The problem of finding an optimal flow in the day-ahead market is proved to be NP-hard even with one generator and without controlled cuts.     

On Multiple Criteria Decision Support for Suppliers on the Competitive Electric Power Market

As an active participant of a competitive energy market, the generator (the energy supplier) challenges new management decisions being exposed to the financial risk environment. There is a strong need for the decision support models and tools for energy market participants. This paper shows that the stochastic short-term planning model can be effectively used as a key analytical tool within the decision support process for relatively small energy suppliers (price-takers). A self-scheduling method for the thermal units on the energy market is addressed. A schedule acquired for given preferences can be used as a desired pattern for bidding process. The uncertainty of the market prices is modeled by a set of possible scenarios with assigned probabilities. Several risk criteria are introduced leading to a multiple criteria optimization problem. The risk criteria are well appealing and easily computable (by means of linear programming) but they meet the formal risk aversion standards. The aspiration/reservation based interactive analysis applied to the multiple criteria problem allows us to find an efficient solution (generation scheme) well adjusted to the generator preferences (risk attitude).     

Capacitated dynamic lot sizing problems in closed-loop supply chain

In this paper, we address the capacitated dynamic lot sizing problem arising in closed-loop supply chain where returned products are collected from customers. These returned products can either be disposed or be remanufactured to be sold as new ones again; hence the market demands can be satisfied by either newly produced products or remanufactured ones. The capacities of production, disposal and remanufacturing are limited, and backlogging is not allowed. A general model of this problem is formulated, and several useful properties of the problem are characterized when cost functions are concave. Moreover, this problem is analyzed and solved to optimality using dynamic programming algorithms under different scenarios. It is shown that the problem with only disposal or remanufacturing can be converted into a traditional capacitated lot sizing problem and be solved by a polynomial algorithm if the capacities are constant. A pseudo-polynomial algorithm is proposed for the problem with both capacitated disposal and remanufacturing. The problem with capacitated production and remanufacturing and the problem with uncapacitated production and capacitated remanufacturing are also analyzed and solved. Through numerical experiments we show that the proposed algorithms perform well when solving problems of practical sizes. From the experimental results also indicates that it is worthwhile to expand the remanufacturing capacity only when returned products exist in a relatively long planning horizon, and production capacities have little effect on the remanufacturing plan when the demand is mainly satisfied by the production.     

HARA frontiers of optimal portfolios in stochastic markets

In this paper, we consider the optimal portfolio selection problem in continuous-time settings where the investor maximizes the expected utility of the terminal wealth in a stochastic market. The utility function has the structure of the HARA family and the market states change according to a Markov process. The states of the market describe the prevailing economic, financial, social and other conditions that affect the deterministic and probabilistic parameters of the model. This includes the distributions of the random asset returns as well as the utility function. We analyzed Black–Scholes type continuous-time models where the market parameters are driven by Markov processes. The Markov process that affects the state of the market is independent of the underlying Brownian motion that drives the stock prices. The problem of maximizing the expected utility of the terminal wealth is investigated and solved by stochastic optimal control methods for exponential, logarithmic and power utility functions. We found explicit solutions for optimal policy and the associated value functions. We also constructed the optimal wealth process explicitly and discussed some of its properties. In particular, it is shown that the optimal policy provides linear frontiers.     

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.     

A GENERATOR AND A SIMPLEX SOLVER FOR NETWORK PIECEWISE LINEAR PROGRAMS

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Relaxation techniques applied to some loading problems

In this paper a two-stage loading problem, dealing with allocation of jobs to machines, is studied. The outer problem is to choose a subset among a number of available machines such that a feasible assigment exists and the total cost price is minimized. The inner problem, is then to find the optimal allocation, given the subset of machines and some assigment criterion at this lower level. It is shown that the choice of problem formulation can be crucial for the strength of the continuous relaxation. Computational results are also presented     

Transportation Problems with Inadmissible Routes

Transportation problems with inadmissible routes (routes that cannot be used) need not have a feasible solution, even if the total supply exceeds the total demand. A method for adjusting the demands equitably to obtain a feasible solution is presented. The algorithm used generates a feasible minimum cost solution in roughly twice the amount of time that it would take to solve a transportation problem of the same size. It is shown how the solution time is reduced by aggregating the adjustment of demands and decomposing the cost minimization.     

Impulsive Control of Portfolios

In this paper a general model of a market with asset prices and economical factors of Markovian structure is considered. The problem is to find optimal portfolio strategies maximizing a discounted infinite horizon reward functional consisting of an integral term measuring the quality of the portfolio at each moment and a discrete term measuring the reward from consumption. There are general transaction costs which, in particular, cover fixed plus proportional costs. It is shown, under general conditions, that there exists an optimal impulse strategy and the value function is a solution to the Bellman equation which corresponds to suitable quasi-variational inequalities.     

A theory for the optimal synthesis of heat-exchanger systems

The problem of the optimal synthesis of heat exchanger systems is treated here with an entirely new approach. It is formulated, using the heat spectrum diagram, as a problem of finding the combination of heat donors and receivers that will minimize a given criterion function. This function is the cost of the heat exchanger system, assumed to be proportional to the total heat-transfer area.The problem is a combinatorial problem for continuous elements. It is solved in general by applying the maximum principle of Pontryagin. The solution for a specific problem is shown to be obtainable by a simple graphical operation.