Coordinating Inventory Control and Pricing Strategies with Random Demand and Fixed Ordering Cost: The Finite Horizon Case

Abstract

We analyze a finite horizon, single product, periodic review model in which pricing and production/inventory decisions are made simultaneously. Demands in different periods are random variables that are independent of each other and their distributions depend on the product price. Pricing and ordering decisions are made at the beginning of each period and all shortages are backlogged. Ordering cost includes both a fixed cost and a variable cost proportional to the amount ordered. The objective is to find an inventory policy and a pricing strategy maximizing expected profit over the finite horizon. We show that when the demand model is additive, the profit-to-go functions are k-concave and hence an s S p policy is optimal. In such a policy, the period inventory is managed based on the classical s S policy and price is determined based on the inventory position at the beginning of each period. For more general demand functions, i.e., multiplicative plus additive functions, we demonstrate that the profit-to-go function is not necessarily k-concave and an s S p policy is not necessarily optimal. We introduce a new concept, the symmetric k-concave functions, and apply it to provide a characterization of the optimal policy.

Fully Polynomial Time Approximation Schemes for Stochastic Dynamic Programs

“…Let us consider the well-studied model of "Markov-modulated demand" (or "world-driven demand"); see, for example, [12,42,46,71] ) is independent of all past events. For example, in a generalization of the cash management problem, the world may represent the economy [37].…”

The Stochastic Cash Balance Problem with Fixed Costs: The Risk-averse Case

“…Note that the (K, 0)-convexity is exactly the K-convexity introduced by Scarf [18] for the classical stochastic inventory control problem with fixed ordering costs. Moreover, the (K, K)-convexity is the symmetric K-convexity, a concept introduced and applied in Chen and SimchiLevi [19] to analyze a joint inventory control and pricing problem with fixed ordering costs and a general demand distributions. Similar to the proof of Theorems 3.1 and 3.2 in [7], we have the following main results for the traditional risk-neutral stochastic cash balance problem.…”

Integrated market selection and production planning: complexity and solution approaches

“…Under this assumption, observe that the demand pattern in (7) satisfies a feasible market selection if and only if the exclusive disjunction requirement in (3) is satisfied. The pattern in (8) then ensures that m α i is selected if and only if m l i 1 or m l i 2 or both are selected, which corresponds to the requirement in (4). Similarly, the pattern in (9) guarantees a feasible market selection if and only if (5) is satisfied by either selecting m α i or m l i 3 or both.…”

“…Recent research recognizes the importance of accounting for the levers suppliers have for managing or shaping demand in order to ensure the most profitable match between supply and demand (see, e.g., [21] and [9]). Spurred on by the development of research in the area of revenue management in the past 20 years (see [24] for a thorough discussion of this field), a number of recent papers have considered the pricing lever that suppliers can use to help shape demand within procurement planning contexts (see, e.g., [2][3][4][5]7,13,15,16,29]). We note that these more recent works built on the foundations provided by [11,26,27,31], and [20].…”