29 research outputs found
Lp-Based Artificial Dependency for Probabilistic Etail Order Fulfillment
We consider an online multi-item retailer with multiple fulfillment facilities and finite inventory, with the objective of minimizing the expected shipping cost of fulfilling customer orders over a finite horizon. We approximate the stochastic dynamic programming formulation of the problem with an equivalent deterministic linear program, which we use to develop a probabilistic fulfillment heuristic that is provably optimal in the asymptotic sense. This first heuristic, however, relies on solving an LP that is exponential in the size of the input. Therefore, we subsequently provide another heuristic which solves an LP that is polynomial in the size of the input, and prove an upper bound on its asymptotic competitive ratio. This heuristic works by modifying the LP solution with artificial dependencies, with the resulting fractional variables used to probabilistically fulfill orders. A hardness result shows that asymptotically optimal policies that are computationally efficient cannot exist. Finally, we conduct numerical experiments that show that our heuristic's performance is very close to optimal for a range of parameters.http://deepblue.lib.umich.edu/bitstream/2027.42/108712/1/1250_ASinha.pd
Certainty Equivalent Planning for Multi-Product Batch Differentiation: Analysis and Bounds
We consider a multi-period planning problem faced by a firm that must coordinate the production and allocations of batches to end products for multiple markets. Motivated by a problem faced by a biopharmaceutical firm, we model this as a discrete-time inventory planning problem where in each period the firm must decide how many batches to produce and how to differentiate batches to meet demands for different end products. This is a challenging problem to solve optimally, so we derive a theoretical bound on the performance of a Certainty Equivalent (CE) control for this model, in which all random variables are replaced by their expected values and the corresponding deterministic optimization problem is solved. This is a variant of an approach that is widely used in practice. We show that while a CE control can perform very poorly in certain instances, a simple re-optimization of the CE control in each period can substantially improve both the theoretical and computational performance of the heuristic, and we bound the performance of this re-optimization. To address the limitations of CE control and provide guidance for heuristic design, we also derive performance bounds for two additional heuristic controls -- (1) Re-optimized Stochastic Programming (RSP), which utilizes full demand distribution but limits the adaptive nature of decision dynamics, and (2) Multi-Point Approximation (MPA), which uses limited demand information to model uncertainty but fully capture the adaptive nature of decision dynamics. We show that although RSP in general outperforms the re-optimized CE control, the improvement is limited. On the other hand, with a carefully chosen demand approximation in each period, MPA can significantly outperform RSP. This suggests that, in our setting, explicitly capturing decision dynamics adds more value than simply capturing full demand information.http://deepblue.lib.umich.edu/bitstream/2027.42/116386/1/1296_Ahn.pd
Near-Optimal Bisection Search for Nonparametric Dynamic Pricing with Inventory Constraint
We consider a single-product revenue management problem with an inventory constraint and unknown, noisy, demand function. The objective of the fi rm is to dynamically adjust the prices to maximize total expected revenue. We restrict our scope to the nonparametric approach where we only assume some common regularity conditions on the demand function instead of a speci fic functional form. We propose a family of pricing heuristics that successfully balance the tradeo ff between exploration and exploitation. The idea is to generalize the classic bisection search method to a problem that is a ffected both by stochastic noise and an inventory constraint. Our algorithm extends the bisection method to produce a sequence of pricing intervals that converge to the optimal static price with high probability. Using regret (the revenue loss compared to the deterministic pricing problem for a clairvoyant) as the performance metric, we show that one of our heuristics exactly matches the theoretical asymptotic lower bound that has been previously shown to hold for any feasible pricing heuristic. Although the results are presented in the context of revenue management problems, our analysis of the bisection technique for stochastic optimization with learning can be potentially applied to other application areas.http://deepblue.lib.umich.edu/bitstream/2027.42/108717/1/1252_Sinha.pd
Dynamic Joint Pricing and Order Fulfillment for E-Commerce Retailers
We consider an e-commerce retailer (e-tailer) who sells a catalog of products to customers from different regions during a finite selling season and fulfills orders through multiple fulfillment centers. The e-tailer faces a Joint Pricing and Fulfillment (JPF) problem: At the beginning of each period, she needs to jointly decide the price for each product and how to fulfill an incoming order. The objective is to maximize the total expected profits defined as total expected revenues minus total expected shipping costs (all other costs are fixed in this problem). The exact optimal policy for JPF is difficult to solve; so, we propose two heuristics that have provably good performance compared to reasonable benchmarks. Our first heuristic directly uses the solution of a deterministic approximation of JPF as its control parameters whereas our second heuristic improves the first heuristic by adaptively adjusting the original control parameters at the beginning of every period. An important feature of the second heuristic is that it decouples the pricing and fulfillment decisions, making it easy to implement. We show theoretically and numerically that the second heuristic significantly outperforms the first heuristic and is very close to a benchmark that jointly re-optimizes the full deterministic problem at every period.http://deepblue.lib.umich.edu/bitstream/2027.42/117573/1/1310_Jasin.pd
Shipping Consolidation with Delivery Deadline and Expedited Shipment Options
Problem definition: Shipment consolidation is commonly used to take advantage of the economies of scale by avoiding some of the shipping costs. However, when pending current orders are consolidated with future orders it may require more expensive expedited shipment in order to meet shorter deadlines. In this paper, we study the optimal consolidation policy focusing on the trade-off between economies of scale and expedited shipping costs. Academic/Practical Relevance: Our work is motivated by the prevalence of consolidation in the supply chain industry and also by its potential application for online and omni-channel retailing, especially with the rise of, so-called, on-demand logistic services. In such situations, sellers, have the flexibility to take advantage of consolidation, by deciding from which warehouse to fulfill the orders and also when to ship the orders, as long as the orders deadlines are met. Methodology: We use Dynamic Programming to study the optimal policy and its structure. We also conduct intensive simulation tests to show the good performance of heuristics which we proposed based on structures of the optimal policy. Results: The optimal policies and their structures are characterized in settings with up to two warehouses, where the impact of expedited shipment on both shipping policy and order fulfillment policy are explored. Utilizing the insights of these structural properties, two easily implementable heuristics are proposed, which perform within 1-2% of the optimal in intensive numerical tests. Managerial Implications: Despite the complexity of the actual optimal consolidation policy, sellers can apply the two simple heuristic policies we proposed to get near-optimal performance in various cases.https://deepblue.lib.umich.edu/bitstream/2027.42/138942/1/1375_Jasin.pd
An Asymptotically Optimal Heuristic for General Non-Stationary Finite-Horizon Restless Multi-Armed Multi-Action Bandits
We propose an asymptotically optimal heuristic, which we termed the Randomized Assignment Control (RAC) for restless multi-armed bandit problems with discrete-time and fi nite states. It is based on a linear programming relaxation to the original stochastic control formulation. In contrast to most of the existing literature, we consider a fi nite horizon with multiple actions and time-dependent (i.e. non-stationary) upper bound on the total number of bandits that can be activated each time period. The asymptotic setting is obtained by letting the number of bandits and other related parameters grow to in finity. Our main contribution is that the asymptotic optimality of RAC in this general setting does not require indexability properties or the usual stability conditions of the underlying Markov chain (e.g. unichain) or fluid approximation (e.g. global stable attractor). Moreover, our multi-action setting is not restricted to the usual dominant action concept. Numerical simulations con firms that our proposed policy indeed performs well in the asymptotic setting. Perhaps more surprisingly, these simulations show that RAC performs well in the non-asymptotic setting as well. Finally, we show that RAC is asymptotically optimal for a dynamic population, where bandits can randomly arrive and depart the system, and discuss how our framework extends to more general costs and constraints.https://deepblue.lib.umich.edu/bitstream/2027.42/138941/1/1374_Jasin.pd
On (Re-Scaled) Multi-Attempt Approximation of Customer Choice Model and its Application to Assortment Optimization
Motivated by the classic exogenous demand model and the recently developed Markov chain model, we propose a new approximation to the general customer choice model based on random utility called multi-attempt model, in which a customer may consider several substitutes before finally deciding to not purchase anything. We show that the approximation error of multi-attempt model decreases exponentially in the number of attempts. However, despite its strong theoretical performance, the empirical performance of multi-attempt model is not satisfactory. This motivates us to construct a modification of multi-attempt model called re-scaled multi-attempt model. We show that re-scaled 2-attempt model is exact when the underlying true choice model is Multinomial Logit (MNL); if, however, the underlying true choice model is not MNL, we show numerically that the approximation quality of re-scaled 2-attempt model is very close to that of Markov chain model. The key feature of our proposed approach is that the resulting approximate choice probability can be explicitly written. From a practical perspective, this allows the decision maker to use off-the-shelf solvers, or borrow existing algorithms from literature, to solve a general assortment optimization problem with a variety of real-world constraints.http://deepblue.lib.umich.edu/bitstream/2027.42/122455/1/1322_Ahn.pd
Real-Time Dynamic Pricing for Revenue Management with Reusable Resources and Deterministic Service Time Requirements
We consider the setting of a ļ¬rm that sells a ļ¬nite amount of resources to price-sensitive customers who arrive randomly over time according to a speciļ¬ed non-stationary rate. Each customer requires a service that consumes one unit of resource for a deterministic amount of time, and the resource is reusable in the sense that it can be immediately used to serve a new customer upon the completion of the previous service. The ļ¬rmās objective is to set the price dynamically to maximize its expected total revenues. This is a fundamental problem faced by many ļ¬rms in many industries. We formulate this as an optimal stochastic control problem and develop two heuristic controls based on the solution of the deterministic relaxation of the original stochastic problem. The ļ¬rst heuristic control is static since the corresponding price sequence is determined before the selling horizon starts; the second heuristic control is dynamic, it uses the ļ¬rst heuristic control as its baseline control and adaptively adjusts the price based on previous demand realizations. We show that both heuristic controls are asymptotically optimal in the regime with large demand and large number of resources. Finally, we consider two important generalizations of the basic model to the setting with multiple service types requiring diļ¬erent service times and the setting with advance service bookings.http://deepblue.lib.umich.edu/bitstream/2027.42/122970/1/1327_Lei.pd