Michal Tzur

Static repositioning in a bike-sharing system: models and solution approaches

Bike-sharing systems allow people to rent a bicycle at one of many automatic rental stations scattered around the city, use them for a short journey and return them at any station in the city. A crucial factor for the success of a bike-sharing system is its ability to meet the fluctuating demand for bicycles and for vacant lockers at each station. This is achieved by means of a repositioning operation, which consists of removing bicycles from some stations and transferring them to other stations, using a dedicated fleet of trucks. Operating such a fleet in a large bike-sharing system is an intricate problem consisting of decisions regarding the routes that the vehicles should follow and the number of bicycles that should be removed or placed at each station on each visit of the vehicles. In this paper, we present our modeling approach to the problem that generalizes existing routing models in the literature. This is done by introducing a unique convex objective function as well as time-related considerations. We present two mixed integer linear program formulations, discuss the assumptions associated with each, strengthen them by several valid inequalities and dominance rules, and compare their performances through an extensive numerical study. The results indicate that one of the formulations is very effective in obtaining high quality solutions to real life instances of the problem consisting of up to 104 stations and two vehicles. Finally, we draw insights on the characteristics of good solutions.

Design of flexible assembly line to minimize equipment cost

Abstract In this paper we develop an optimal and a heuristic algorithm for the problem of designing a flexible assembly line when several equipment alternatives are available. The design problem addresses the questions of selecting the equipment and assigning tasks to workstations, when precedence constraints exist among tasks. The objective is to minimize total equipment costs, given a predetermined cycle time (derived from the required production rate). We develop an exact branch and bound algorithm which is capable of solving practical problems of moderate size. The algorithms efficiency is enhanced due to the development of good lower bounds, as well as the use of some dominance rules to reduce the size of the branch and bound tree. We also suggest the use of a branch-and-bound-based heuristic procedure for large problems, and analyze the design and performance of this heuristic.

The Period Vehicle Routing Problem and its Extensions

This chapter presents an overview of the Period Vehicle Routing Problem, a generalization of the classic vehicle routing problem in which driver routes are constructed over a period of time. We survey the evolution of the PVRP and present a synopsis of modeling and solution methods, including classical heuristics, metaheuristics, and mathematical programming based methods. We review three important variants of the problem: the PVRP with Time Windows, the Multi-Depot PVRP, and the PVRP with Service Choice. We present case studies and highlight related implementation issues, including metrics that quantify the operational complexity of implementing periodic delivery routes. Finally, we discuss potential directions for future work in the area.

The Period Vehicle Routing Problem with Service Choice

The period vehicle routing problem (PVRP) is a variation of the classic vehicle routing problem in which delivery routes are constructed for a period of time (for example, multiple days). In this paper, we consider a variation of the PVRP in which service frequency is a decision of the model. We refer to this problem as the PVRP with service choice (PVRP-SC). We explore modeling issues that arise when service choice is introduced, and suggest efficient solution methods. Contributions are made both in modeling this new variation of the PVRP and in introducing an exact solution method for the PVRP-SC. In addition, we propose a heuristic variation of the exact method to be used for larger problem instances. Computational tests show that adding service choice can improve system efficiency and customer service. We also present general insights on the impact of node distribution on the value of service choice.

Transshipments: An emerging inventory recourse to achieve supply chain leagility

Abstract Supply chain designs are constrained by the cost-service trade-off. Cost minimization typically leads to physically efficient or lean supply chains at the expense of customer responsiveness or agility. Recently, the concept of leagility has been introduced. Research on leagility, defined as the capability of concurrently deploying the lean and agile paradigms, hinges heavily on the identification of the decoupling point, which, in turn, is enabled by postponement. Postponement strategies, however, present a cross-functional challenge for implementation. As a tactical solution to achieve leagility without postponement, we introduce transshipments, which represent a common practice in multi-location inventory systems involving monitored movement of stock between locations at the same echelon level of the supply chain. Through a series of models, we establish how transshipments can be used to enhance both agility and leanness.

Progressive Interval Heuristics for Multi-Item Capacitated Lot-Sizing Problems

We consider a family of N items that are produced in, or obtained from, the same production facility. Demands are deterministic for each item and each period within a given horizon of T periods. If in a given period an order is placed, setup costs are incurred. The aggregate order size is constrained by a capacity limit. The objective is to find a lot-sizing strategy that satisfies the demands for all items over the entire horizon without backlogging, and that minimizes the sum of inventory-carrying costs, fixed-order costs, and variable-order costs. All demands, cost parameters, and capacity limits may be time dependent. In the basic joint setup cost (JS) model, the setup cost of an order does not depend on the composition of the order. The joint and item-dependent setup cost (JIS) model allows for item-dependent setup costs in addition to the joint setup costs. We develop and analyze a class of so-called progressive interval heuristics. A progessive interval heuristic solves a JS or JIS problem over a progressively larger time interval, always starting with period 1, but fixing the setup variables of a progressively larger number of periods at their optimal values in earlier iterations. Different variants in this class of heuristics allow for different degrees of flexibility in adjusting continuous variables determined in earlier iterations of the algorithm. For the JS-model and the two basic implementations of the progressive interval heuristics, we show under some mild parameter conditions that the heuristics can be designed to be e-optimal for any desired value of e > 0 with a running time that is polynomially bounded in the size of the problem. They can also be designed to be simultaneously asymptotically optimal and polynomially bounded. A numerical study covering both the JS and JIS models shows that a progressive interval heuristic generates close-to-optimal solutions with modest computational effort and that it can be effectively used to solve large-scale problems.

Shipping Multiple Items by Capacitated Vehicles: An Optimal Dynamic Programming Approach

We consider a system in which multiple items are transferred from a warehouse or a plant to a retailer through identical capacitated vehicles, or by identical freight wagons. Any mixture of the items may be loaded onto a vehicle. The retailer is facing dynamic deterministic demand for several items, over a finite planning horizon. A vehicle incurs a fixed cost for each trip made from the warehouse to the retailer. In addition, there exist item-dependent variable shipping costs and inventory holding costs at the retailer, which are both constant over time. The objective is to find a shipment schedule that minimizes the total cost, while satisfying demand on time.We address and partially resolve the question regarding the problems complexity by introducing a dynamic programming algorithm whose complexity is polynomial for a fixed number of items, but exponential otherwise. Our dynamic programming formulation is based on properties satisfied by the optimal solution, and uses an innovative way for partitioning the problem into subproblems.

Time-partitioning heuristics: Application to one warehouse, multiitem, multiretailer lot-sizing problems

We describe effective time partitioning heuristics for dynamic lot-sizing problems in multiitem and multilocation production/distribution systems. In a time-partitioning heuristic, the complete horizon of (say) N periods, is partitioned into smaller intervals. An instance of the problem is solved, to optimality, on each of these intervals, and the resulting solution coalesced into a solution for the complete horizon. The intervals are selected to be of a size which permits the use of exact and effective solution methods (e.g., branch-and-bound methods). Each inter- vals problem is specified to include options for starting conditions which adequately comple- ment the solutions obtained for prior intervals. The heuristics can usually be designed to be of low polynomial complexity as well as to guarantee e-optimality for any desired precision e . 0, and asymptotic optimality as N goes to infinity. We first give a general description of the design of time-partitioning heuristics for dynamic lot-sizing problems. We subsequently develop such a heuristic in detail, for the one warehouse multiretailer model representing a two-echelon distribution network with m retailers, selling J distinct items. A comprehensive numerical study exhibits that the partitioning heuristics are very efficient and close-to-optimal. Even problems with a planning horizon of up to 150 periods can be solved within 1.5% of optimality, employing intervals of 5-10 periods only and in a matter of CPU seconds, or up to a few minutes, using longer intervals and when the number of items and retailers is large. These CPU times refer to a SUN 4M (SPARC) workstation.

The dynamic lot‐sizing model with backlogging: A simple o(n log n) algorithm and minimal forecast horizon procedure

We develop a simple O(n log n) solution method for the standard lot-sizing model with backlogging and a study horizon of n periods. Production costs are fixed plus linear and holding and backlogging costs are linear with general time-dependent parameters. The algorithm has linear [O(n)] time complexity for several important subclasses of the general model. We show how a slight adaptation of the algorithm can be used for the detection of a minimal forecast horizon and associated planning horizon. The adapted algorithm continues to have complexity O(n log n) or O(n) for the above-mentioned subclasses of the general model.

Allocation of bandwidth and storage

We consider two allocation problems in this paper, namely, allocation of bandwidth and storage. In these problems, we face a number of independent requests, respectively, for reservation of bandwidth of a communication channel of fixed capacity and for storage of items into a space of fixed size. In both problems, a request is characterized by: (i) its required period of allocation; (ii) its required bandwidth (item width, respectively); and (iii) the profit of accepting the request. The problem is to decide which requests to accept so as to maximize the total profit. These problems in general are NP-hard. In this paper we provide polynomial-time algorithms for solving various special cases, and develop polynomial-time approximation algorithms for very general NP-hard cases with good performance guarantees.