Optimal Selling Policies Research Articles

Optimal Selling Policies for Farmer Cooperatives

Sustainability and long‐term prosperity are chronic challenges in the agriculture sector of many countries. To address such challenges, farmer cooperatives are formed as an innovative approach to improve the livelihoods of millions of farmers around the world. Inspired by real‐life practice in the Kenya coffee industry, we study a class of stochastic and dynamic inventory models for storable agricultural products with random exogenous supply and price. For a variety of cost functions relevant in practice, we characterize the optimal selling policies to maximize the farmer cooperatives’ expected profit. We show that for concave inventory holding cost, the sell‐all‐or‐retain‐all ( r, R) (or sell‐all‐or‐retain‐all R) policies are optimal with (without) the fixed selling cost; for convex holding cost, the sell‐down‐to ( S, s) (or sell‐down‐to s) policies are optimal with (without) the fixed selling cost. For the special case of linear holding cost, the optimal policy is a cut‐off price policy and we derive closed‐form expressions for the optimal policy and the optimal total discounted profit. We discuss the model extensions to include general stochastic harvest and price processes, selling/storage capacity limits, price‐dependent random demand with a spot market, and the flexibility of procurement from other producers, and then perform a numerical study to quantify the impact of the optimal solutions. Reconciling the theory with practice, useful insights and guidelines are provided to help farmer cooperatives make strategic selling decisions.

An optimal stopping policy for car rental businesses with purchasing customers

We analyze decisions for a car rental firm using a common pool of cars to serve both rental, and purchasing customers. The rental customers arrive successively, and rent out cars for random durations while effectuating random incremental mileages on them. This stochastic rental behavior makes the decision of when to sell a rental car quite a crucial one for the firm because it involves a certain amount of risk. On one hand, selling a car when its mileage is low proactively avoids a huge decline in the car’s residual market value (even though it could also cause the firm to forfeit income from future rental customers who intend to rent that car for long durations while driving it sparingly). On the other hand, delaying selling is equally risky because the sample of early rental customers whom that car serves may only successively keep the car for short durations while significantly adding to its mileage. Such opportunistic customers would therefore have the effect of drastically diminishing the car’s market value without providing sufficient rental income to compensate. We use optimal stopping theory to derive the firm’s optimal selling policy algorithm which unfortunately comes with a practical implementation challenge. To address this issue, we propose three heuristic selling policies, one of which is based on the restart-in formulation introduced by Katehakis and Veinott (Math Oper Res 12(2):262–268, 1987). Numerical experiments using real and artificial parameter settings (1) reveal conditions under which the proposed heuristic policies outperform the firm’s current (suboptimal) policy, and (2) demonstrate how all four suboptimal policies compare to the optimal policy.

Managing Inventories for Agricultural Products: The Optimal Selling Policies

The inventory control theory in the operations research and management literature is primarily concerned about manufactured products, for which, the typical assumptions are ample external supplies and predictable prices, and the decision is on how much to order or produce. Agricultural products, on the other hand, are rarely studied in this literature but deserve an attention long overdue because of their unique features, such as the limited and unpredictable supply, inelastic demand and thus highly unpredictable prices, and a decision of how much to sell. Such features are considered in the agricultural economics literature which provides insights and strategic guidelines but with limited structural results on the theory. Inspired by real-life practice in the Kenya coffee industry, we consider a class of stochastic and dynamic inventory models for storable agricultural products with random exogenous supply and price. We characterize the optimal inventory (selling) policies for a variety of cost functions relevant in practice. In the cases of linear cost functions, we derive closed-form expressions for the optimal policies and the optimal discounted profits. Our theoretical advancement reveals additional insights and deepens the understanding of inventory management for agricultural products. Applying the theory to practice, we show that making selling decisions judiciously can significantly outperform the prevailing practice of selling-all regardless.

Dynamic selling of quality-graded products under demand uncertainties

This paper examines dynamic selling (DS) problems under demand uncertainties. Quality-graded products with fully downward substitutable demands are considered. Downward demand substitution indicates that demands for lower quality grade products can be fulfilled by either designated or higher quality grade products. In this dynamic selling problem, decision makers need to choose an optimal selling policy in each decision epoch. The objective is to identify an optimal policy for the dynamic selling of quality-graded inventory. DS problems are formulated as a discrete-time Markov decision process (MDP) model. In the MDP model, demand type and inventory levels are state variables. The objective is to maximize expected profits. In such a multi-dimensional dynamic decision problem, computational complexity is a chief concern. This study proves the structure of optimal policies that significantly reduce computational complexity. Performance of optimal dynamic selling policies is evaluated in detailed numerical studies.

Optimal Selling Policy for a Trading Network System with Return Handling in an e-commerce Environment

Optimal Selling Policy for a Trading Network System with Return Handling in an e-commerce Environment

Selling to Heterogeneous Customers with Uncertain Valuations under Returns Policies

This paper considers a firm that sells a product with a finite inventory to a market with a random population size. Each customer's valuation of the product consists of two components: the customer's intrinsic value of the product which is privately known by him before purchase, and the product's fitness which is ex ante uncertain and revealed to him only after purchase. The customers are heterogeneous in the sense that they differ in their intrinsic values. Because of its lack of knowledge about the customer's intrinsic value before purchase and about the fitness value after purchase, the firm essentially faces a sequential screening problem. It has been established in the literature that when the firm has an unlimited inventory, the firm's optimal selling policy takes the form of a menu of returns policies that serves the entire population. However, we show that this result no longer holds when the firm has a finite inventory or has to decide the inventory level before selling to the market. In particular, we study three commonly-used selling policies: 1) the single returns policy that serves only a fraction of the population; 2) the menu of returns policies that serves the entire population; 3) the inventory rationing policy. We show that the firm may be better off under 1) or 3) relative to 2). We derive the results as to how the firm's preference over these selling policies depends on the customer heterogeneity, the customer's ex ante valuation uncertainty, the firm's inventory level, and the firm's procurement cost.

A theory of clearance sales

Clearance sales are widely used by firms as an intertemporal selling policy, in particular in markets where firms face demand uncertainty and need to choose capacity in advance. Clearance sales consist in charging a high price initially but then lowering the price in the sales period. High-valuation consumers purchase the good at the high initial price so as to avoid rationing at the low price, while low-valuation consumers wait for the price to drop. We develop a simple model of intertemporal monopoly pricing under demand uncertainty, and show that clearance sales may be the optimal intertemporal selling policy. In this article, we derive the optimal selling policy of a monopolist who faces uncertainty about the level of demand. In contrast to much of the literature, we show that uniform pricing is not necessarily optimal. The optimal selling policy may instead involve clearance sales: the monopolist commits to supply only a limited quantity, and then charge a high price in the first period (palatable to high-valuation consumers) and a low price (affordable to low-valuation consumers) in the last period. The premium associated with the rent offered to consumers that buy at the high price and the probability of being rationed at the low price induce all high-valuation consumers to separate themselves and purchase the good in the first period. In contrast, lowvaluation consumers purchase the good in the last period but are rationed with an optimally determined probability.

Monopoly Pricing Under Demand Uncertainty: Final Sales Versus Introductory Offers, Second Version

We study rationing as a tool of the monopolist's selling policy when demand is uncertain. Three selling policies are potentially optimal in our environment: uniform pricing, final sales, and introductory offers. Final sales consist in charging a high price initially, but then lowering the price while committing to a total capacity. Consumers with a high valuation may decide to buy at the high price since the endogenous probability of rationing is higher at the lower price. Introductory offers consist in selling a limited quantity at a low price initially, and then raising price. Those consumers with high valuations who were rationed initially at the lower price may find it optimal to buy the good at the higher price. We show that the optimal selling policy involves either uniform pricing or final sales. Introductory offers may dominate uniform pricing, but can never be optimal if the monopolist can also use final sales.

Recursive Algorithms for Stock Liquidation: A Stochastic Optimization Approach

This work develops stochastic optimization algorithms for a class of stock liquidation problems. The stock liquidation rules are based on hybrid geometric Brownian motion models allowing regime changes that are modulated by a continuous-time finite-state Markov chain. The optimal selling policy is of threshold type and can be obtained by solving a set of two-point boundary value problems. The total number of equations to be solved is the same as the number of states of the underlying Markov chain. To reduce the computational burden, using a stochastic optimization approach, recursive algorithms are constructed to approximate the optimal threshold values. Convergence and rates of convergence of the algorithm are studied. Simulation examples are presented, and the computation results are compared with the analytic solutions. Finally, the algorithms are tested using real market data.

Optimal Strategies for Selling an Asset

This paper considers the problem of selling an asset on the open market. The seller receives a random sequence of price offers, which may arrive either periodically or randomly over time. After each offer is received, the seller must decide whether or not to sell, weighing the possibility of obtaining a better offer against the cost of waiting. A number of authors have established the properties of optimal selling policies when the distribution of offers is known and offers are received periodically. This paper investigates the conditions under which these same properties hold for an unknown offer distribution which is updated as successive offers are received.

Optimal Stopping in the Stock Market When the Future is Discounted

In the random walk stock market model, a stock is purchased at price $x$ and is sold at time $t$ for the price $x + S_t$ where $S_t = \sum^t_0 X_i, X_i$ is the price change during the $i$th epoch, and $X_1, X_2, \cdots$ are i.i.d. random variables with $\mu = E(X_1) > 0$ and finite $\sigma^2 = E(X^2_1) - \mu^2 > 0$. Discounting the future by a factor of $\gamma$ per epoch, $0 < \gamma < 1$, a selling or stopping policy $t$ has expected payoff or utility $u(t) = E\{\gamma^t(x + S_t)\}$. This article determines second order asymptotic properties of the optimal selling policy $s$, the first passage time of $S_n$ across a straight line boundary $c$, whose utility is equal to the value $V(x) = \sup_tu(t)$ of the stock purchased at price $x$. Specifically, as $\gamma \rightarrow 1$, renewal theory is utilized to evaluate the limiting distribution of $s, E(s), V(x)$, and the first passage boundary $c$ up to second order terms.