Optimal Purchasing Policy Research Articles

On maximizing a loss-averse buyer’s expected utility in a multi-sourcing problem

The paper studies optimal purchasing policies for buyers in a multi-sourcing problem. For a multi-souring problem, the study first obtains the optimal purchasing policy for a loss-neutral buyer to maximize expected profit. In a multi-sourcing, we find that a loss-neutral buyer could purchase no commodities and can rely only on spot buying to meet the whole market demand if the wholesale price is larger than the spot price. Further, the research finds that a loss-neutral buyer’s optimal purchasing quantity is independent of the retail price. To reveal the loss aversion effect, the study introduces the well-known loss aversion function and obtains the optimal purchasing policy to maximize a loss-averse buyer’s expected utility in a multi-sourcing. To maximize expected utility, it is proven that a loss-averse buyer who becomes more loss-averse should purchase fewer items in the long-term contract. Moreover, under the optimal purchasing policy that maximizes expected utility, a loss-averse buyer’s expected profit is decreasing when the loss aversion coefficient is increasing. If a loss-averse buyer intends to get a bigger expected profit, he/she should be less loss-averse and select an optimal purchasing policy with a smaller loss aversion coefficient.

Optimal Purchasing Policy for Fresh Products from Multiple Supply Sources with Considerations of Random Delivery Times, Risk, and Information

ABSTRACTWe consider a problem in which a distributor purchases fresh products from multiple supply sources for subsequent sale at a wholesale market. Due to uncontrollable factors such as traffic congestion or bad weather conditions, the time taken for the product to be delivered from each production origin is uncertain. As the wholesale market is open for trading only in a fixed‐time window, any fresh products arriving earlier face the risk of decay, and any arriving later cannot be sold. The market demand for the product is random and follows a general distribution. We formulate the basic problem as a multi‐source selection model with random yield, taking into account the trade‐off between purchasing cost and other costs such as deterioration loss arising from delivery uncertainty. We derive the optimal order quantities from different origins and propose algorithms to search for them. We also investigate the effects of delivery uncertainty on service level and transport mode selection. Finally, we examine two extensions. One considers a capacity limit in each production origin, and the other incorporates information updating. For the first extension, we propose algorithms to compute the optimal solution based on its structural properties. For the second problem, we derive the optimal purchasing policy according to updated information such as details of the arrivals of previous orders.

On Optimizing Airline Ticket Purchase Timing

Proper timing of the purchase of airline tickets is difficult even when historical ticket prices and some domain knowledge are available. To address this problem, we introduce an algorithm that optimizes purchase timing on behalf of customers and provides performance estimates of its computed action policy. Given a desired flight route and travel date, the algorithm uses machine-learning methods on recent ticket price quotes from many competing airlines to predict the future expected minimum price of all available flights. The main novelty of our algorithm lies in using a systematic feature-selection technique, which captures time dependencies in the data by using time-delayed features, and reduces the number of features by imposing a class hierarchy among the raw features and pruning the features based on in-situ performance. Our algorithm achieves much closer to the optimal purchase policy than other existing decision theoretic approaches for this domain, and meets or exceeds the performance of existing feature-selection methods from the literature. Applications of our feature-selection process to other domains are also discussed.

Obsolescence of durable goods and optimal purchase timing

We study a model with a durable good subject to periodic obsolescence and analytically characterize the optimal purchasing policy. The key result is that consumers optimally synchronize new purchases with the innovation cycle. The model simultaneously explains coordinated adoption without invoking network effects and provides a theoretical underpinning for a diffusion curve that features a temporary adoption slowdown.

Enhancing value in reverse supply chains by sorting before product recovery

Offering a well-designed reverse supply chain programme can significantly improve the ability of any organisation to differentiate itself, and even to take market share away, from its competitors. Reverse supply chain considerations should be a part of an organisation's corporate strategy. From a macro-level perspective, value propositions of reverse supply chains in for an organisation, and/or the industry in which the organisation operates, include considerations for both strategies (to reuse, repair, refurbish, remanufacture, retrieve parts or cannibalise components, recycle, scrap, redesign returned products, etc.) and effective operations (to handle and sort returns by value and ease of remanufacture) to sustain and even enhance organisational competency. In this article, we examine the role of sorting used products before disassembly for parts retrieval and remanufacturing under stochastic variability based on customer demand using a Markov decision process. We address a problem of managing costs in a remanufacturing environment with stochastically variable demand and model it for two types of used parts. Each part type is assumed to have varying quality and acquisition costs. The cost function includes manufacturing, holding and backlog costs components. We compute optimal purchase policy that maximises the expected average profit per period. Using a case example from photocopier industry, numerical analysis is performed to study the implications of various holding cost for two parts on optimal purchase policy in remanufacturing environments.

Optimal purchasing policy in a two-component assembly system with different purchasing contracts for each component

This paper considers an assembly system where a firm produces a single product which is assembled using two types of components (component 1 and component 2). The components are provided by individual suppliers (supplier 1 and supplier 2). We assume that the firm makes different procurement contracts with supplier 1 and supplier 2. To supplier 1, the firm specifies the maximum inventory level of component 1 and makes a commitment to purchase the component as long as its inventory level is below this target level. To supplier 2, the firm has the option of purchasing or rejecting component 2 at each instant supplier 2 provides it. Formulating our model as a Markov decision problem, we identify a component 2 purchasing policy which maximizes the firm’s profits subject to the costs of rejecting component 1, holding component 2, and purchasing component 2. We also investigate how the changes in the sales price and cost parameters affect the optimal purchasing policy. Finally, we present numerical study for the optimal performance evaluation.

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.

N-period contracts with ordering constraints and total minimum commitments: Optimal and heuristic solutions

We consider long-term supply contracts with commitments and purchasing constraints. Under a commitment, the buyer agrees that the quantities purchased over the length of the contract will exceed a pre-specified amount. Purchasing constraints restrict the size of the quantity purchased each period. We present the optimal purchasing policy for an N-period contract, and a more readily implementable heuristic policy. By computing the expected profits of following such a policy, the user can easily choose between contracts with different commitments, constraints, and prices. We present computational results demonstrating this procedure.

The Demand for Food Products: An Analysis of Interpurchase Times and Purchased Quantities

In this paper we study household purchase behaviour of storable food products. An inventory model is developed in which the household chooses an optimal stock level of the product. Storage of the product is costly, there is a fixed cost per purchase occasion, and the market price is sometimes discounted because of price promotions. We show that the optimal purchase policy is an s, S policy. The model is used to derive predictions on the correlations between interpurchase times and purchased quantities on the one hand, and prices on the other. These predictions are empirically verified using consumer panel data.

Rational Shopping Behavior and the Option Value of Variable Pricing

When a product's price fluctuates at a store, how should rational, cost-minimizing shoppers shop for it? Specifically, how frequently should they visit the store, and how much of the product should they buy when they get there? Would this rational shopping behavior differ across Every Day Low Price (EDLP) and Promotional Pricing (HILO) stores? If shoppers are rational, which retail price format is more profitable, EDLP or HILO? To answer these questions, we develop a normative model that shows how rational customers should shop when the price of the product is random. We derive a closed-form expression for the optimal purchasing policy and show that the optimal quantity to purchase under a given price scenario is linearly decreasing in the difference between the price under that scenario and the average price. This purchase flexibility due to price variability has a direct impact on shopping frequency. Indeed, the benefit of this purchase flexibility can be captured via an “option value” that implicitly reduces the fixed cost associated with each shopping trip. Consequently, rational shoppers should shop more often and buy fewer units per trip when they face higher price variability. Our results suggest that if two stores charge the same average price for a product, rational shoppers incur a lower level of expenditure at the store with a higher price variability. Since stores with different price variabilities coexist in practice, we expect stores with higher price variability to charge a higher average price. Thus, given two stores, a higher relative mean price for a given item should be indicative of higher price variability, and vice versa. These model implications are tested using multicategory scanner panel data from 513 households and pricing data for three stores (two EDLP stores and one HILO store) and 33 product categories over a two-year period. We find strong empirical support for the model implications.

Analysis of supply contracts with total minimum commitment

In this paper we analyze a supply contract for a single product that specifies that the cumulative orders placed by a buyer, over a finite horizon, be at least as large as a (contracted upon) given quantity. We assume that the demand for the product is uncertain, and the buyer places orders periodically. We derive the optimal purchase policy for the buyer for a given total minimum quantity commitment and a discounted price. We show that the policy is characterized by the order-up-to levels of the corresponding finite horizon and a single-period standard newsboy problem with no commitment but with discounted price. We show that this policy can be computed easily. We can use this to evaluate any discount schedule characterized by a set of (price discount, minimum commitment) pairs.

The Impact of Dealing Patterns on Purchase Behavior

We explore the effect of dealing patterns on consumer purchase behavior by developing a normative purchase quantity model that can incorporate any dealing pattern. The model adds to the stream of research on optimal purchasing policy by demonstrating how dealing patterns can be incorporated in a simple manner in dynamic programming models. Implications for purchase behavior are derived by employing the model in a numerical simulation in which time between deals is characterized by a Weibull distribution. The flexibility of the Weibull distribution enables us to establish how particular facets of the dealing distribution (e.g., certainty in deal timing, minimum time between deals) affect consumer behavior with respect to optimal purchase quantity, inventory, etc. One of the implications of the model is that the average quantity purchased on deal should be larger when there is greater certainty in deal timing. The model also shows that the average quantity purchased on deal should be larger when deals are spaced further apart. even though the buyer is presented with the same number of deals. We test certain model implications in a laboratory experiment and find actual behavior varying across dealing patterns in a manner consistent with model implications.

Commodity Purchasing

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.