Samir Elhedhli

Hub-and-spoke network design with congestion

We consider a hub-and-spoke network design problem with congestion. The model we propose extends current models by taking congestion effects into account. This is achieved through a non-linear cost term in the objective function. We first linearize the model, and then provide a Lagrangean heuristic that finds high-quality solutions within reasonable computational time. The results of the model provide new and realistic insights into the hub-and-spoke network design problem.

Integrated Production-Inventory-Distribution System Design with Risk Pooling: Model Formulation and Heuristic Solution

In this paper, we consider a multiproduct two-echelon production-inventory-distribution system design model that captures risk-pooling effects by consolidating the safety-stock inventory of the retailers at distribution centers (DCs). We propose a model that determines plant and DC locations, shipment levels from plants to the DCs, safety-stock levels at DCs, and the assignment of retailers to DCs by minimizing the sum of fixed facility location costs, transportation costs, and safety-stock costs. The model is formulated as a nonlinear mixed-integer programming problem and linearized using piecewise-linear functions. The formulation is strengthened using redundant constraints. Lagrangean relaxation is applied to decompose the problem by echelon. A lower bound is provided by the Lagrangean relaxation, while a heuristic is proposed that uses the solution of the subproblems to construct an overall feasible solution. Computational results reveal that the Lagrangean relaxation provides a sharp lower bound and a heuristic solution that is within 5% of the optimal solution.

The integration of an interior-point cutting plane method within a branch-and-price algorithm

Abstract.This paper presents a novel integration of interior point cutting plane methods within branch-and-price algorithms. Unlike the classical method, columns are generated at a ‘‘central’’ dual solution by applying the analytic centre cutting plane method (ACCPM) on the dual of the full master problem. First, we introduce some modifications to ACCPM. We propose a new procedure to recover primal feasibility after adding cuts and use, for the first time, a dual Newton’s method to calculate the new analytic centre after branching. Second, we discuss the integration of ACCPM within the branch-and-price algorithm. We detail the use of ACCPM as the search goes deep in the branch and bound tree, making full utilization of past information as a warm start. We exploit dual information from ACCPM to generate incumbent feasible solutions and to guide branching. Finally, the overall approach is implemented and tested for the bin-packing problem and the capacitated facility location problem with single sourcing. We compare against Cplex-MIP 7.5 as well as a classical branch-and-price algorithm.

A Lagrangean Heuristic for Hub-and-Spoke System Design with Capacity Selection and Congestion

Hub-and-spoke networks are widely applied in a variety of industries such as transportation, postal delivery, and telecommunications. Although they are designed to exploit economies of scale, hub-and-spoke networks are known to favour congestion, jeopardizing the performance of the entire system. This paper looks at incorporating congestion and capacity decisions in the design stage of such networks. The problem is formulated as a nonlinear mixed-integer program (NMIP) that explicitly minimizes congestion, capacity acquisition, and transportation costs. Congestion at hubs is modeled as the ratio of total flow to surplus capacity by viewing the hub-and-spoke system as a network of M/M/1 queues. To solve the NMIP, we propose a Lagrangean heuristic where the problem is decomposed into an easy subproblem and a more difficult nonlinear subproblem. The nonlinear subproblem is first linearized using piecewise functions and then solved to optimality using a cutting plane method. The Lagrangean lower bound is found using subgradient optimization. The solution from the subproblems is used to find a heuristic solution. Computational results indicate the efficiency of the methodology in providing a sharp bound and in generating high-quality feasible solutions in most cases.

Service System Design with Immobile Servers, Stochastic Demand, and Congestion

The service system design problem seeks to locate a set of service facilities, allocate enough capacity, and assign stochastic customer demand to each of them, so as to minimize the fixed costs of opening facilities and acquiring service capacity, as well as the variable access and waiting costs. This problem is commonly known in the location literature as the facility location problem with immobile servers, stochastic demand, and congestion. It is often set up as a network of M/M/1 queues and modeled as a nonlinear mixed-integer program (MIP). Because of the complexity of the resulting model, the current literature focuses on approximate and/or heuristic solution methods. This paper proposes a linearization based on a simple transformation and piecewise linear approximations and an exact solution method based on cutting planes. This leads to the exact solution of models with up to 100 customers, 20 potential service facilities, and 3 capacity levels.

A Stochastic Programming Model for a Day-Ahead Electricity Market With Real-Time Reserve Shortage Pricing

We present a multi-period stochastic mixed integer programming model for power generation scheduling in a day-ahead electricity market. The model considers various scenarios and integrates the idea of reserve shortage pricing in real time. Instead of including all the possible scenarios, we parsimoniously select a certain number of scenarios to limit the size of the model. As realistic size models are still intractable for exact methods, we propose a heuristic solution methodology based on scenario-rolling that is capable of finding good quality feasible solutions within reasonable computation time.

Efficient Production-Distribution System Design

The production-distribution system design is an integral part of the general supply chain design. This paper proposes a novel solution methodology for this problem that is based on Lagrangean relaxation, interior-point methods, and branch and bound. Unlike classical approaches, Lagrangean relaxation is applied in a two-level hierarchy, branch and bound is based on a Lagrangean lower bound and column generation (branch and price), while interior-point methods are used within a cutting-plane context (analytic centre cutting-plane method---ACCPM). Numerical results demonstrate that the two-level approach outperforms the classical approach and provides a very sharp lower bound that is the (proven) optimal in most cases.

Energy and Content Aware Multi-Homing Video Transmission in Heterogeneous Networks

This paper studies video transmission using a multi-homing service in a heterogeneous wireless access medium. We propose an energy and content aware video transmission framework that incorporates the energy limitation of mobile terminals (MTs) and the quality-of-service (QoS) requirements of video streaming applications, and employs the available opportunities in a heterogeneous wireless access medium. In the proposed framework, the MT determines the transmission power for the utilized radio interfaces, selectively drops some packets under the battery energy limitation, and assigns the most valuable packets to different radio interfaces in order to minimize the video quality distortion. First, the problem is formulated as MINLP which is known to be NP-hard. Then we employ a piecewise linearization approach and solve the problem using a cutting plane method which reduces the associated complexity from MINLP to a series of MIPs. Finally, for practical implementation in MTs, we approximate the video transmission framework using a two-stage optimization problem. Numerical results demonstrate that the proposed framework exhibits very close performance to the exact problem solution. In addition, the proposed framework, unlike the existing solutions in literature, offers a choice for desirable trade-off between the achieved video quality and the MT operational period per battery charging.

Integrated design of supply chain networks with three echelons, multiple commodities and technology selection

We consider a strategic supply chain design problem with three echelons, multiple commodities and technology selection. We model the problem as a tri-echelon, capacitated facility location problem that decides on the location of plants and warehouses, their capacity and technology planning, the assignment of commodities to plants and the flow of commodities to warehouses and customer zones. We use a mixed-integer programming formulation strengthened by valid but redundant constraints and apply Lagrangean relaxation to decompose the problem by echelon. Lagrangean relaxation provides a lower bound that is calculated using an interior-point cutting plane method. Feasible solutions are generated using a primal heuristic that uses the solution of the subproblems. Unlike common practice in the literature, the decomposition does not aim at getting easy subproblems, but rather at getting subproblems that preserve most of the characteristics of the original problem. Not only does this provide a sharp lower bound but also leads to a simple and efficient primal heuristic. We can afford to have relatively difficult subproblems because the interior-point cutting plane method used to solve the Lagrangean dual makes clever and selective choices of the Lagrangean multipliers leading to fewer calls to the subproblems. Computational results indicate the efficiency of the approach in providing a sharp bound and in generating feasible solutions that are of high quality.

A stochastic optimization model for real-time ambulance redeployment

When ambulances are engaged in responding to emergency calls, the ability to respond quickly to future calls is considerably compromised. The available ambulances are typically relocated to reestablish maximal coverage. We present a two-stage stochastic optimization model for the ambulance redeployment problem that minimizes the number of relocations over a planning horizon while maintaining an acceptable service level. We conduct computational testing based on the real historical data from the Region of Waterloo Emergency Medical Services. The results show that the optimal relocation strategies can be computed within 40s of computational time for a desired service level of 90%.