A New MILP Approach for the Facility Layout Design Problem with Rectangular and L/T Shaped Departments

In this paper we propose a new approach for the facility layout problem (FLP) and suggest new mixed-integer linear programming (MILP) formulations. The proposed approach considers simultaneously the location of the departments within the facility and the internal arrangement of the machines. Two models are suggested, where the first addresses the rectangular department case and the second allows nonrectangular departments defined by an L/T shape. New regularity constraints are developed to avoid irregular department shapes

Combinatorial Contracts Beyond Gross Substitutes

We study the combinatorial contracting problem of D\"utting et al. [FOCS '21], in which a principal seeks to incentivize an agent to take a set of costly actions. In their model, there is a binary outcome (the agent can succeed or fail), and the success probability and the costs depend on the set of actions taken. The optimal contract is linear, paying the agent an $\alpha$ fraction of the reward. For gross substitutes (GS) rewards and additive costs, they give a poly-time algorithm for finding the optimal contract. They use the properties of GS functions to argue that there are poly-many "critical values" of $\alpha$, and that one can iterate through all of them efficiently in order to find the optimal contract. In this work we study to which extent GS rewards and additive costs constitute a tractability frontier for combinatorial contracts. We present an algorithm that for any rewards and costs, enumerates all critical values, with poly-many demand queries (in the number of critical values). This implies the tractability of the optimal contract for any setting with poly-many critical values and efficient demand oracle. A direct corollary is a poly-time algorithm for the optimal contract in settings with supermodular rewards and submodular costs. We also study a natural class of matching-based instances with XOS rewards and additive costs. While the demand problem for this setting is tractable, we show that it admits an exponential number of critical values. On the positive side, we present (pseudo-) polynomial-time algorithms for two natural special cases of this setting. Our work unveils a profound connection to sensitivity analysis, and designates matching-based instances as a crucial focal point for gaining a deeper understanding of combinatorial contract settings.Comment: 22 pages, 3 figure

Flexibility and Complexity in Periodic Distribution Problems

In this paper, we explore trade-offs between operational flexibility and operational complexity in periodic distribution problems. We consider the gains from operational flexibility in terms of vehicle routing costs and customer service benefits, and the costs of operational complexity in terms of implementation difficulty. Periodic distribution problems arise in a number of industries, including food distribution, waste management and mail services. The period vehicle routing problem (PVRP) is a variation of the classic vehicle routing problem in which driver routes are constructed for a period of time; the PVRP with service choice (PVRP-SC) extends the PVRP to allow service (visit) frequency to become a decision of the model. While introducing operational flexibility in periodic distribution systems can increase efficiency, it poses three challenges: the difficulty of modeling this flexibility accurately; the computational effort required to solve the problem as modeled with such flexibility; and the complexity of operationally implementing the resulting solution. This paper considers these trade-offs between the system performance improvements due to operational flexibility and the resulting increases in operational and computational complexity as they relate to periodic vehicle routing problems. In particular, increasing the operational complexity of driver routes can be problematic in industries where some level of system regularity is required. As discussed in the paper, recent work in the literature suggests that dispatching drivers consistently to the same geographic areas results in driver familiarity and improved driver performance. Additionally, having the same driver visit a customer on a continual basis can foster critical relationships. According to UPS, such driver-customer relationships are a key competitive advantage in its package delivery operations, attributing 60 million packages a year to sales leads generated by drivers. In this paper, we develop a set of quantitative measures to evaluate the trade-offs between flexibility and complexity

Learning in Setups: Analysis, Minimal Forecast Horizons, and Algorithms

We analyze the dynamic lot-sizing model in which the cost of a setup depends on the number of setups that have occurred prior to it. This arises, for example, when there exist learning effects in setups. Our model is more general than most learning models in the literature since it allows the total setup cost to be a general nondecreasing (but not necessarily concave) function of the number of setups. We explore tight relationships between our model and special cases of the classical dynamic lot-sizing model. On the basis of these we find minimal forecast and planning horizons for our model, which determine the first decision when the model is solved on a rolling horizon basis. When a forecast horizon cannot be found, we provide guidelines regarding the optimal first decision. We also provide an algorithm to solve the finite horizon problem, which uses as sub-problems variations of the classical dynamic lot-sizing problem. The advantage of this approach is the ability to use the extensive literature available on the latter, to generalize the results of this paper. As many of our results are qualitative in nature, they provide insights which can be useful for other models with a similar setup cost behavior.learning in setups, general setup cost function, minimal forecast horizon

A Simple Forward Algorithm to Solve General Dynamic Lot Sizing Models with n Periods in 0(n log n) or 0(n) Time

This paper is concerned with the general dynamic lot size model, or (generalized) Wagner-Whitin model. Let n denote the number of periods into which the planning horizon is divided. We describe a simple forward algorithm which solves the general model in 0(n log n) time and 0(n) space, as opposed to the well-known shortest path algorithm advocated over the last 30 years with 0(n 2) time. A linear, i.e., 0(n)-time and space algorithm is obtained for two important special cases: (a) models without speculative motives for carrying stock, i.e., where in each interval of time the per unit order cost increases by less than the cost of carrying a unit in stock; (b) models with nondecreasing setup costs. We also derive conditions for the existence of monotone optimal policies and relate these to known (planning horizon and other) results from the literature.dynamic lot sizing models, dynamic programming, complexity