Supply chain collaboration

In the past, research in operations management focused on single-firm analysis. Its goal was to provide managers in practice with suitable tools to improve the performance of their firm by calculating optimal inventory quantities, among others. Nowadays, business decisions are dominated by the globalization of markets and increased competition among firms. Further, more and more products reach the customer through supply chains that are composed of independent firms. Following these trends, research in operations management has shifted its focus from single-firm analysis to multi-firm analysis, in particular to improving the efficiency and performance of supply chains under decentralized control. The main characteristics of such chains are that the firms in the chain are independent actors who try to optimize their individual objectives, and that the decisions taken by a firm do also affect the performance of the other parties in the supply chain. These interactions among firms’ decisions ask for alignment and coordination of actions. Therefore, game theory, the study of situations of cooperation or conflict among heterogenous actors, is very well suited to deal with these interactions. This has been recognized by researchers in the field, since there are an ever increasing number of papers that applies tools, methods and models from game theory to supply chain problems

High frequency trading and asymptotics for small risk aversion in a Markov renewal model

We study a an optimal high frequency trading problem within a market microstructure model designed to be a good compromise between accuracy and tractability. The stock price is driven by a Markov Renewal Process (MRP), while market orders arrive in the limit order book via a point process correlated with the stock price itself. In this framework, we can reproduce the adverse selection risk, appearing in two different forms: the usual one due to big market orders impacting the stock price and penalizing the agent, and the weak one due to small market orders and reducing the probability of a profitable execution. We solve the market making problem by stochastic control techniques in this semi-Markov model. In the no risk-aversion case, we provide explicit formula for the optimal controls and characterize the value function as a simple linear PDE. In the general case, we derive the optimal controls and the value function in terms of the previous result, and illustrate how the risk aversion influences the trader strategy and her expected gain. Finally, by using a perturbation method, approximate optimal controls for small risk aversions are explicitly computed in terms of two simple PDE's, reducing drastically the computational cost and enlightening the financial interpretation of the results.Comment: 30 pages, new asymptotic results, typos corrected, new bibliographical reference

Algorithmic trading in a microstructural limit order book model

We propose a microstructural modeling framework for studying optimal market making policies in a FIFO (first in first out) limit order book (LOB). In this context, the limit orders, market orders, and cancel orders arrivals in the LOB are modeled as Cox point processes with intensities that only depend on the state of the LOB. These are high-dimensional models which are realistic from a micro-structure point of view and have been recently developed in the literature. In this context, we consider a market maker who stands ready to buy and sell stock on a regular and continuous basis at a publicly quoted price, and identifies the strategies that maximize her P\&L penalized by her inventory. We apply the theory of Markov Decision Processes and dynamic programming method to characterize analytically the solutions to our optimal market making problem. The second part of the paper deals with the numerical aspect of the high-dimensional trading problem. We use a control randomization method combined with quantization method to compute the optimal strategies. Several computational tests are performed on simulated data to illustrate the efficiency of the computed optimal strategy. In particular, we simulated an order book with constant/ symmet-ric/ asymmetrical/ state dependent intensities, and compared the computed optimal strategy with naive strategies. Some codes are available on https://github.com/comeh

Optimal Dynamic Procurement Policies for a Storable Commodity with L\'evy Prices and Convex Holding Costs

In this paper we study a continuous time stochastic inventory model for a commodity traded in the spot market and whose supply purchase is affected by price and demand uncertainty. A firm aims at meeting a random demand of the commodity at a random time by maximizing total expected profits. We model the firm's optimal procurement problem as a singular stochastic control problem in which controls are nondecreasing processes and represent the cumulative investment made by the firm in the spot market (a so-called stochastic "monotone follower problem"). We assume a general exponential L\'evy process for the commodity's spot price, rather than the commonly used geometric Brownian motion, and general convex holding costs. We obtain necessary and sufficient first order conditions for optimality and we provide the optimal procurement policy in terms of a "base inventory" process; that is, a minimal time-dependent desirable inventory level that the firm's manager must reach at any time. In particular, in the case of linear holding costs and exponentially distributed demand, we are also able to obtain the explicit analytic form of the optimal policy and a probabilistic representation of the optimal revenue. The paper is completed by some computer drawings of the optimal inventory when spot prices are given by a geometric Brownian motion and by an exponential jump-diffusion process. In the first case we also make a numerical comparison between the value function and the revenue associated to the classical static "newsvendor" strategy.Comment: 28 pages, 3 figures; improved presentation, added new results and section

Strongly Polynomial Primal-Dual Algorithms for Concave Cost Combinatorial Optimization Problems

We introduce an algorithm design technique for a class of combinatorial optimization problems with concave costs. This technique yields a strongly polynomial primal-dual algorithm for a concave cost problem whenever such an algorithm exists for the fixed-charge counterpart of the problem. For many practical concave cost problems, the fixed-charge counterpart is a well-studied combinatorial optimization problem. Our technique preserves constant factor approximation ratios, as well as ratios that depend only on certain problem parameters, and exact algorithms yield exact algorithms. Using our technique, we obtain a new 1.61-approximation algorithm for the concave cost facility location problem. For inventory problems, we obtain a new exact algorithm for the economic lot-sizing problem with general concave ordering costs, and a 4-approximation algorithm for the joint replenishment problem with general concave individual ordering costs

Economic Lot-Sizing Problem with Bounded Inventory and Lost-Sales

In this paper we consider an economic lot-sizing problem with bounded inventory and lost-sales. Different structural properties are characterized based on the system parameters such as production and inventory costs, selling prices, and storage capacities. Using these properties and the results on the lot-sizing problems with bounded inventory, we present improved and new algorithms for the problem. Specifically, we provide algorithms for the general lot-sizing problem with bounded inventory and lost-sales, the lot-sizing problem with nonincreasing selling prices and the problem with only lost-sales

An Integrated Strategy for a Production Planning and Warehouse Layout Problem: Modeling and Solution Approaches

We study a real-world production warehousing case, where the company always faces the challenge to find available space for their products and to manage the items in the warehouse. To resolve the problem, an integrated strategy that combines warehouse layout with the capacitated lot-sizing problem is presented, which have been traditionally treated separately in the existing literature. We develop a mixed integer linear programming model to formulate the integrated optimization problem with the objective of minimizing the total cost of production and warehouse operations. The problem with real data is a large-scale instance that is beyond the capability of optimization solvers. A novel Lagrangian relax-and-fix heuristic approach and its variants are proposed to solve the large-scale problem. The preliminary numerical results from the heuristic approaches are reported