A portfolio approach to managing procurement risk using multi-stage stochastic programming

Procurement is a critical supply chain management function that is susceptible to risk, due mainly to uncertain customer demand and purchase price volatility. A procurement approach in the form of a portfolio that incorporates the common procurement means is proposed. Such means include long-term contracts, spot procurements and option-based supply contracts. The objective is to explore possible synergies among the various procurement means, and so be able to produce optimal or near optimal results in profit while mitigating risk. The implementation of the portfolio approach is based on a multi-stage stochastic programming model in which replenishment decisions are made at various stages along a time horizon, with replenishment quantities being determined by simultaneously considering the stochastic demand and the price volatility of the spot market. The model attempts to minimise the risk exposure of procurement decisions measured as conditional value-at-risk. Numerical experiments to test the effectiveness of the proposed model are performed using demand data from a large air conditioner manufacturer in China and price volatility data from the Shanghai steel market. The results indicate that the proposed model can fairly reliably outperform other approaches, especially when either the demand and/or prices exhibit significant variability.     

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.     

Combining guaranteed and spot markets in display advertising: Selling guaranteed page views with stochastic demand

While page views are often sold instantly through real-time auctions when users visit websites, they can also be sold in advance via guaranteed contracts. In this paper, we present a dynamic programming model to study how an online publisher should optimally allocate and price page views between guaranteed and spot markets. The problem is challenging because the allocation and pricing of guaranteed contracts affect how advertisers split their purchases between the two markets, and the terminal value of the model is endogenously determined by the updated dual force of supply and demand in auctions. We take the advertisers’ purchasing behaviour into consideration, i.e., risk aversion and stochastic demand arrivals, and present a scalable and efficient algorithm for the optimal solution. The model is also empirically validated with a commercial dataset. The experimental results show that selling page views via both channels can increase the publisher’s expected total revenue, and the optimal pricing and allocation strategies are robust to different market and advertiser types.     

Supply chain operations in the presence of a spot market: a review with discussion

We survey the recent literature on the use of spot market operations to manage procurement in supply chains. We present results in two categories: work that deals with optimal procurement strategies and work related to the valuation of procurement contracts. As an example of the latter, we provide new results on valuation of a supply contract with abandonment option. Based on our review, we also discuss the scope for doing further work.     

Bid Evaluation in Procurement Auctions with Piecewise Linear Supply Curves

Consider a marketplace operated by a buyer who wishes to procure large quantities of several heterogeneous products. Suppliers submit price curves for each of the commodities indicating the price charged as a function of the supplied quantity. The total amount paid to a supplier is the sum of the prices charged for the individual commodities. It is assumed that the submitted supply curves are piecewise linear as they often are in practice. The bid evaluation problem faced by the procurer is to determine how much of each commodity to buy from each of the suppliers so as to minimize the total purchase price. In addition to meeting the demand, the buyer may impose additional business requirements that restrict which contracts suppliers may be awarded. These requirements may result in interdependencies between the commodities which lead to suboptimal results if the commodities are traded in independent auctions rather than simultaneously. Even without the additional business constraints the bid evaluation problem is NP-hard. The main contribution of our study is a flexible column generation based heuristics that provides near-optimal solutions to the procurer’s bid evaluation problem. Our method scales very well due to the Branch-and-Price technology it is built on. We employ sophisticated rounding and local improvement heuristics to obtain quality solutions. We also developed a test data generator that produces realistic problems and allows control over the difficulty level of the problems using parameters.     

Mitigating supply risk: an approach with quantity flexibility procurement

This paper considers a buyer that procures from its major supplier whose production is subject to random yield risk. To mitigate supply risk, the buyer can procure from another reliable supplier who provides quantity flexibility (QF) contract. Under both deterministic and stochastic demand, we study the buyer’s optimal procurement decisions. We analyze the structural properties of optimal solutions and identify the conditions under which the quantity flexibility procurement policy should be used. We also examine the effect of supply risk, flexibility, wholesale price and demand risk on the procurement decisions. We find that the higher supply risk and demand risk reduce the buyer’s profit but have different impact on the buyer’s order policy. For the QF supplier, it may not obtain more orders by providing larger flexibility to the buyer, on the contrary, doing this may benefit the risky supplier. For the QF supplier or risky supplier, given its competitor’s wholesale price, it can increase its order share by lower wholesale price.     

Management Adopts Operational Research: Session Report

Quantitative methods are derived to assist buyers purchasing commodities in fluctuating price markets. Demand is known whilst price is a stochastic variable which may contain trends or seasonal fluctuations. The essential feature of the problem is that the buyer has many opportunities to make a purchase.Mathematical models are formulated to describe particular commodity buying problems. The optimal purchasing policy is derived by using dynamic programming. It consists of a set of discrete price breaks at each buying opportunity together with the associated stock levels the buyer should aim to achieve at each price break with his purchase at this opportunity. The price breaks are dependent on the probability density functions of future prices and the number of future buying opportunities. Recurrence relations are derived to calculate these price breaks. The case of restrictions on the purchase quantity at each price offer, either because of supply limitations or by the buyer as a policy decision, and price discounts are also considered.A case study illustrating the techniques is given and the methods are extended to purchasing for a blending problem with substitutable commodities.     

Asymmetric Supply Function Equilibria with Forward Contracts

We consider markets in which firms offer supply functions, rather than a quantity or price alone: the most important examples are wholesale electricity markets. The equilibria in such markets can be hard to characterize. In many cases, whole families of supply function equilibria occur so there are difficulties in determining which equilibrium will be chosen. In this paper, we consider supply function equilibria, when firms hold forward contracts, which is common in electricity markets. Under the assumption that contract positions have been fixed in advance, we characterize the families of supply function equilibria in a duopoly. The existence of forward contracts implies a tightening of the conditions for an equilibrium, and a greater likelihood that no equilibrium solution exists. In the case of three firms, there can be at most one supply function equilibrium, provided that the lowest demand be small enough.     

考虑海外代购的品牌商定价模型研究

随着海外代购体量的日趋增大,代购带来的低价威胁对于在不同国家不同市场销售产品的公司来说已经成为一个日益严重的问题。同时,代购渠道中假货的问题也愈发严重。因此,在海外代购背景下探究产品定价模型具有必要性。以往研究普遍认为这种未经授权的销售会削减品牌方的利润,但实则不然。基于这一发现,本研究为在两个不同市场销售相同产品但面临代购低价威胁的公司制定考虑代购的市场定价模型。由公司制定两个市场的价格,消费者选择是否从包括代购在内的三个渠道购买产品。推出两个授权市场的最优价格,分析各参数变化对最优价格的影响,并校验最优价格对消费者需求和总利润的影响。模型分析表明,高价市场中有部分消费者需求转向海外代购,同时低价市场的消费者需求也受到了影响,且在一定条件下,提高高价市场的产品定价能够扩大低价市场的需求,从代购的角度解释了现实中需求曲线向上倾斜的现象。此外,两个独立市场之间的价格差距对代购市场的销售也产生了积极影响,并且在某些条件下,增大价格差距可以提高公司的收益水平。随后讨论了一种极端模型和三种扩展模型,通过模型分析表示,扩展后的定价模型也显示出与基础市场模型相似的灵敏度分析结果,同样得到两个市场的价差扩大会导致代购市场的销售额增加的结论,并且在一定条件下,公司的利润更高,增加了结论的可信度。     

Dual sourcing using capacity reservation and spot market: Optimal procurement policy and heuristic parameter determination

This contribution focuses on the cost-effective management of the combined use of two procurement options: the short-term option is given by a spot market with random price, whereas the long-term alternative is characterized by a multi period capacity reservation contract with fixed purchase price and reservation level. A reservation cost, proportional with the reservation level, has to be paid for the option of receiving any amount per period up to the reservation level. A long-term decision has to be made regarding the reserved capacity level, and then it has to be decided – period by period – which quantities to procure from the two sources. Considering the multi-period problem with stochastic demand and spot price, the structure of the optimal combined purchasing policy is derived using stochastic dynamic programming. Exploiting these structural properties, an advanced heuristic is developed to determine the respective policy parameters. This heuristic is compared with two rolling-horizon approaches which use the one-period and two-period optimal solution. A comprehensive numerical study reveals that the approaches based on one-period and two-period solutions have considerable drawbacks, while the advanced heuristic performs very well compared to the optimal solution. Finally, by exploiting our numerical results we give some insights into the system’s behavior under problem parameter variations.     

Pricing and clearing combinatorial markets with singleton and swap orders

In this article we consider combinatorial markets with valuations only for singletons and pairs of buy/sell-orders for swapping two items in equal quantity. We provide an algorithm that permits polynomial time market-clearing and -pricing. The results are presented in the context of our main application: the futures opening auction problem. Futures contracts are an important tool to mitigate market risk and counterparty credit risk. In futures markets these contracts can be traded with varying expiration dates and underlyings. A common hedging strategy is to roll positions forward into the next expiration date, however this strategy comes with significant operational risk. To address this risk, exchanges started to offer so-called futures contract combinations, which allow the traders for swapping two futures contracts with different expiration dates or for swapping two futures contracts with different underlyings. In theory, the price is in both cases the difference of the two involved futures contracts. However, in particular in the opening auctions price inefficiencies often occur due to suboptimal clearing, leading to potential arbitrage opportunities. We present a minimum cost flow formulation of the futures opening auction problem that guarantees consistent prices. The core ideas are to model orders as arcs in a network, to enforce the equilibrium conditions with the help of two hierarchical objectives, and to combine these objectives into a single weighted objective while preserving the price information of dual optimal solutions. The resulting optimization problem can be solved in polynomial time and computational tests establish an empirical performance suitable for production environments.     

Capacity expansion and forward contracting in a duopolistic power sector

The surge in demand for electricity in recent years requires that power companies expand generation capacity sufficiently. Yet, at the same time, energy demand is subject to seasonal variations and peak-hour factors that cause it to be extremely volatile and unpredictable, thereby complicating the decision-making process. We investigate how power companies can optimise their capacity-expansion decisions while facing uncertainty and examine how expansion and forward contracts can be used as suitable tools for hedging against risk under market power. The problem is solved through a mixed-complementarity approach. Scenario-specific numerical results are analysed, and conclusions are drawn on how risk aversion, competition, and uncertainty interact in hedging, generation, and expansion decisions of a power company. We find that forward markets not only provide an effective means of risk hedging but also improve market efficiency with higher power output and lower prices. Power producers with higher levels of risk aversion tend to engage less in capacity expansion with the result that together with the option to sell in forward markets, very risk-averse producers generate at a level that hardly varies with scenarios.     

基于现货市场的供应链博弈模型

构建了现货市场价格及市场需求均不确定的供应链博弈模型,分析了风险中性供应商的批发价决策、风险厌恶制造商的采购决策,证明博弈均衡存在且唯一。通过解析分析及数值实验探讨了风险态度、现货价格和需求波动及其相关性对供应链博弈的影响。结果表明:1)制造商因规避风险会减小采购,这降低了供应链总效用;2)现货市场存在时制造商会减小向供应商采购的数量,这降低了供应商的利润,但提高了供应链总效用;3)现货价格与市场需求的相关关系让供应商掌握主动,但会降低制造商及整个供应链的绩效。     

A study on the effect of yield uncertainty in price-setting newsvendor models with additive-multiplicative demand

Random yield and uncertain demand usually both exist in many industries, such as the semiconductor industry. In this paper, the price-setting newsvendor model is studied which involves a single manufacturer and a single retailer with random yield and uncertain demand respectively. Under the condition of additive-multiplicative demand, we investigate the varying effects of random yield on the optimal price, order quantity, and expected profit in two situations with different cost structures. The first case is an in-house production case where the firm pays for the raw material quantity it has ordered, and the second one is a procurement case where the firm pays for the real product quantity it receives only. By using the theory of stochastic comparisons, we find that a less variable and a stochastically larger yield rate both lead to a lower optimal price and a higher expected profit for the in-house production case. Moreover, a less variable yield rate also results in a lower optimal price and a higher profit for the procurement case, but this is not true for a stochastically larger yield rate. Numerical examples illustrate that the effect of yield randomness on the optimal order quantity is not general.     

总量控制和交易环境下考虑随机需求的企业决策分析

总量控制和交易(Cap-and-Trade, C&T)给排放企业运营决策带来了新的挑战。本文提出一个非线性优化模型分析C&T环境下的企业最优产量,并在绿色改进和碳权交易之间有效权衡。模型不仅考虑了随机需求和碳价波动,还考虑了绿色改进的边际递减效果和实施绿色生产的风险。理论分析证明了最优解的存在性,并给出了排放企业的最优决策及C&T环境下企业新的生产条件。解析分析表明:与非C&T环境相比,新的最优产量更低,实际排放下降;碳配额虽然影响企业利润和碳权交易量,但不影响最优产量和最优改进投资;碳价和绿色改进系数越大,越有利于促进企业实施绿色改进减少排放;企业利润随绿色改进系数和碳配额的增加而上升,随单位产品碳排放的增加而下降。数值分析验证了理论模型及其分析结果;蒙特卡洛模拟揭示利润波动受需求风险、绿色改进风险和碳价波动的影响,但需求风险对利润波动的影响更为显著。     

The natural hedge of a gas-fired power plant

Electricity industries worldwide have been restructured in order to introduce competition. As a result, decision makers are exposed to volatile electricity prices, which are positively correlated with those of natural gas in markets with price-setting gas-fired power plants. Consequently, gas-fired plants are said to enjoy a “natural hedge.” We explore the properties of such a built-in hedge for a gas-fired power plant via a stochastic programming approach, which enables characterisation of uncertainty in both electricity and gas prices in deriving optimal hedging and generation decisions. The producer engages in financial hedging by signing forward contracts at the beginning of the month while anticipating uncertainty in spot prices. Using UK energy price data from 2006 to 2011 and daily aggregated dispatch decisions of a typical gas-fired power plant, we find that such a producer does, in fact, enjoy a natural hedge, i.e., it is better off facing uncertain spot prices rather than locking in its generation cost. However, the natural hedge is not a perfect hedge, i.e., even modest risk aversion makes it optimal to use gas forwards partially. Furthermore, greater operational flexibility enhances this natural hedge as generation decisions provide a countervailing response to uncertainty. Conversely, higher energy-conversion efficiency reduces the natural hedge by decreasing the importance of natural gas price volatility and, thus, its correlation with the electricity price.     

Two-settlement electricity markets with price caps and Cournot generation firms

We compare two alternative mechanisms for capping prices in two-settlement electricity markets. With sufficient lead time, forward market prices are implicitly capped by competitive pressure of potential entry that will occur when forward prices rise above some backstop price. Another more direct approach is to cap spot prices through a regulatory intervention. In this paper we explore the implications of these two alternative mechanisms in a two-settlement Cournot equilibrium framework. We formulate the market equilibrium as a stochastic equilibrium problem with equilibrium constraints (EPEC) capturing congestion effects, probabilistic contingencies and horizontal market power. As an illustrative test case, we use the 53-bus Belgian electricity network with representative generator costs but hypothetical demand and ownership structure. Compared to a price-uncapped two-settlement system, a forward cap increases firms’ incentives for forward contracting, whereas a spot cap reduces such incentives. Moreover, in both cases, more forward contracts are committed as the generation resource ownership structure becomes more diversified.     

Robust strategies for natural gas procurement

In order to serve their customers, natural gas local distribution companies (LDCs) can select from a variety of financial and non-financial contracts. The present paper is concerned with the choice of an appropriate portfolio of natural gas purchases that would allow a LDC to satisfy its demand with a minimum tradeoff between cost and risk, while taking into account risk associated with modeling error. We propose two types of strategies for natural gas procurement. Dynamic strategies model the procurement problem as a mean-risk stochastic program with various risk measures. Naive strategies hedge a fixed fraction of winter demand. The hedge is allocated equally between storage, futures and options. We propose a simulation framework to evaluate the proposed strategies and show that: (i) when the appropriate model for spot prices and its derivatives is used, dynamic strategies provide cheaper gas with low risk compared to naive strategies. (ii) In the presence of a modeling error, dynamic strategies are unable to control the variance of the procurement cost though they provide cheaper cost on average. Based on these results, we define robust strategies as convex combinations of dynamic and naive strategies. The weight of each strategy represents the fraction of demand to be satisfied following this strategy. A mean–variance problem is then solved to obtain optimal weights and construct an efficient frontier of robust strategies that take advantage of the diversification effect.     

The valuation of options on capacity with cost and demand uncertainty

Options contracts can provide trading partners with enhanced flexibility to respond to uncertain market conditions and allow for superior capacity planning thanks to early information on future demand. We develop an analytical framework to value options on capacity for production of non-storable goods or dated services. The market consists of a sequence of contract and spot market. Reservations are made during the contract market session in period 0, where the buyer’s future demand, the seller’s future marginal costs as well as the future spot price are uncertain, the latter being impacted neither by the buyer nor the seller. During the spot market session in period 1, the buyer may execute his options or satisfy his entire or additional demand from a competing seller in the spot market. The seller allocates reserved capacity now being called and attempts to sell remaining capacity into the spot market. Analytical expressions for the buyer’s optimal reservation quantity and the seller’s tariff are derived, making explicit the risk-sharing benefits of options contracts. The combination of an options contract and a spot market is demonstrated to be Pareto improving as compared to alternative market schemes. An analysis of the determinants of the efficiency gain characterizes industries particularly suitable to the options approach.     

Electricity price modeling and asset valuation: a multi-fuel structural approach

We introduce a new and highly tractable structural model for spot and derivative prices in electricity markets. Using a stochastic model of the bid stack, we translate the demand for power and the prices of generating fuels into electricity spot prices. The stack structure allows for a range of generator efficiencies per fuel type and for the possibility of future changes in the merit order of the fuels. The derived spot price process captures important stylized facts of historical electricity prices, including both spikes and the complex dependence upon its underlying supply and demand drivers. Furthermore, under mild and commonly used assumptions on the distributions of the input factors, we obtain closed-form formulae for electricity forward contracts and for spark and dark spread options. As merit order dynamics and fuel forward prices are embedded into the model, we capture a much richer and more realistic dependence structure than can be achieved by classical reduced-form models. We illustrate these advantages by comparing with Margrabe’s formula and a simple cointegration model, and highlight important implications for the valuation of power plants.