Predicting Accurate Lagrangian Multipliers for Mixed Integer Linear Programs

Lagrangian relaxation stands among the most efficient approaches for solving a Mixed Integer Linear Programs (MILP) with difficult constraints. Given any duals for these constraints, called Lagrangian Multipliers (LMs), it returns a bound on the optimal value of the MILP, and Lagrangian methods seek the LMs giving the best such bound. But these methods generally rely on iterative algorithms resembling gradient descent to maximize the concave piecewise linear dual function: the computational burden grows quickly with the number of relaxed constraints. We introduce a deep learning approach that bypasses the descent, effectively amortizing the local, per instance, optimization. A probabilistic encoder based on a graph convolutional network computes high-dimensional representations of relaxed constraints in MILP instances. A decoder then turns these representations into LMs. We train the encoder and decoder jointly by directly optimizing the bound obtained from the predicted multipliers. Numerical experiments show that our approach closes up to 85~\% of the gap between the continuous relaxation and the best Lagrangian bound, and provides a high quality warm-start for descent based Lagrangian methods

Mathematical Model and Stochastic Multi-Criteria Acceptability Analysis for Facility Location Problem

This paper studies a real-life public sector facility location problem. The problem fundamentally originated from the idea of downsizing the number of service centres. However, opening of new facilities is also considered in case the current facilities fail to fulfil general management demands. Two operation research methodologies are used to solve the problem and the obtained results are compared. First, a mathematical programming model is introduced to determine where the new facilities will be located, and which districts get service from which facilities, as if there were currently no existing facilities. Second, the Stochastic Multi-criteria Acceptability Analysis-TRI (SMAA-TRI) method is used to select the best suitable places for service centres among the existing facilities. It is noted that the application of mathematical programming model and SMAA-TRI integration approach on facility location problem is the first study in literature. Compression of outcomes shows that mixed integer linear programming (MILP) model tries to open facilities in districts which are favoured by SMAA-TRI solution.</span

Mathematical Model and Stochastic Multi-Criteria Acceptability Analysis for Facility Location Problem

This paper studies a real-life public sector facility location problem. The problem fundamentally originated from the idea of downsizing the number of service centres. However, opening of new facilities is also considered in case the current facilities fail to fulfil general management demands. Two operation research methodologies are used to solve the problem and the obtained results are compared. First, a mathematical programming model is introduced to determine where the new facilities will be located, and which districts get service from which facilities, as if there were currently no existing facilities. Second, the Stochastic Multi-criteria Acceptability Analysis-TRI (SMAA-TRI) method is used to select the best suitable places for service centres among the existing facilities. It is noted that the application of mathematical programming model and SMAA-TRI integration approach on facility location problem is the first study in literature. Compression of outcomes shows that mixed integer linear programming (MILP) model tries to open facilities in districts which are favoured by SMAA-TRI solution.</span

Enhanced cell-based algorithm with dynamic radius in solving capacitated multi-source weber problem

Capacitated Multi-source Weber Problem (CMSWP) is a type of Location Allocation Problem (LAP) which have been extensively researched because they can be applied in a variety of contexts. Random selection of facility location in a Cell-based approach may cause infeasible or worse solutions. This is due to the unprofitable cells are not excluded and maybe selected for locating facilities. As a result, the total transportation cost increases, and solution quality is not much improved. This research finds the location of facilities in a continuous space to meet the demand of customers which minimize the total cost using Enhanced Cell-based Algorithm (ECBA). This method was derived from previous study that divides the distribution of customers into smaller cells of promising locations. The methodology consists of three phases. First, the profitable cells were constructed by applying ECBA. Second, initial facility configuration was determined using fixed and dynamic radius. Third, Alternating Transportation Problem (ATL) was applied to find a new location. The algorithm was tested on a dataset of three sizes which are 50, 654 and 1060 customers. The computational results of the algorithm prove that the results are superior in terms of total distance compared to the result of previous studies. This study provides useful knowledge to other researchers to find strategic facilities locations by considering their capacities

Interactive Multicriteria Approach to Facility Location-Allocation Models Under Stochastic Demand

Industrial Engineering and Managemen

The incorporation of fixed cost and multilevel capacities into the discrete and continuous single source capacitated facility location problem

In this study we investigate the single source location problem with the presence of several possible capacities and the opening (fixed) cost of a facility that is depended on the capacity used and the area where the facility is located. Mathematical models of the problem for both the discrete and the continuous cases using the Rectilinear and Euclidean distances are produced. Our aim is to find the optimal number of open facilities, their corresponding locations, and their respective capacities alongside the assignment of the customers to the open facilities in order to minimise the total fixed and transportation costs. For relatively large problems, two solution methods are proposed namely an iterative matheuristic approach and VNS-based matheuristic technique. Dataset from the literature is adapted to assess our proposed methods. To assess the performance of the proposed solution methods, the exact method is first applied to small size instances where optimal solutions can be identified or lower and upper bounds can be recorded. Results obtained by the proposed solution methods are also reported for the larger instances

Optimizing the positioning of medical facilities using linear programming techniques

The Plant Location Problem is one of the most important branch of operations research concerned with the optimal placement of plants to minimize transportation costs. We had to deal with a real problem related to the positioning of medical facilities on the territory of Emilia-Romagna. We started from the SPLP to create a mathematical model for this problem, but we needed to add some more constraints. The described algorithm was designed to give to the user a rapid feedback from the syste

An Evolutionary Model for Spatial Location of Economic Facilities

Locating an economic facility, warehouse, plant, retail store, etc., is one of the most important questions that a business company faces. In this paper we consider a normative model for a certain class of relocation processes. That is, when one location structure is gradually substituted by another one. This happens in response to external factors such as appearance of competitors or change of demand. Thus, we are facing with sequential decisions and the model and algorithm corresponding to them become endogenously dynamic. An evolutionary model for location of economic facilities is presented. Its application to an empirical case, namely changing locations of alcohol distribution stores, is briefly presented

Two-stage network design in humanitarian logistics.

Natural disasters such as floods and earthquakes can cause multiple deaths, injuries, and severe damage to properties. In order to minimize the impact of such disasters, emergency response plans should be developed well in advance of such events. Moreover, because different organizations such as non-governmental organizations (NGOs), governments, and militaries are involved in emergency response, the development of a coordination scheme is necessary to efficiently organize all the activities and minimize the impact of disasters. The logistics network design component of emergency management includes determining where to store emergency relief materials, the corresponding quantities and distribution to the affected areas in a cost effective and timely manner. In a two-echelon humanitarian relief chain, relief materials are pre-positioned first in regional rescue centers (RRCs), supply sources, or they are donated to centers. These materials are then shipped to local rescue centers (LRCs) that distribute these materials locally. Finally, different relief materials will be delivered to demand points (also called affected areas or AAs). Before the occurrence of a disaster, exact data pertaining to the origin of demand, amount of demand at these points, availability of routes, availability of LRCs, percentage of usable pre-positioned material, and others are not available. Hence, in order to make a location-allocation model for pre-positioning relief material, we can estimate data based on prior events and consequently develop a stochastic model. The outputs of this model are the location and the amount of pre-positioned material at each RRC as well as the distribution of relief materials through LRCs to demand points. Once the disaster occurs, actual values of the parameters we seek (e.g., demand) will be available. Also, other supply sources such as donation centers and vendors can be taken into account. Hence, using updated data, a new location-allocation plan should be developed and used. It should be mentioned that in the aftermath of the disaster, new parameters such as reliability of routes, ransack probability of routes and priority of singular demand points will be accessible. Therefore, the related model will have multiple objectives. In this dissertation, we first develop a comprehensive pre-positioning model that minimizes the total cost while considering a time limit for deliveries. The model incorporates shortage, transportation, and holding costs. It also considers limited capacities for each RRC and LRC. Moreover, it has the availability of direct shipments (i.e., shipments can be done from RRCs directly to AAs) and also has service quality. Because this model is in the class of two-stage stochastic facility location problems, it is NP-hard and should be solved heuristically. In order to solve this model, we propose using Lagrangian Heuristic that is based on Lagrangian Relaxation. Results from the first model are amounts and locations of pre-positioned relief materials as well as their allocation plan for each possible scenario. This information is then used as a part of the input for the second model, where the facility location problem will be formulated using real data. In fact, with pre-positioned items in hand, other supplies sources can be considered as necessary. The resulting multi-objective problem is formulated based on a widely used method called lexicography goal programming. The real-time facility location model of this dissertation is multi-product. It also considers the location problem for LRCs using real-time data. Moreover, it considers the minimization of the total cost as one of the objectives in the model and it has the availability of direct shipments. This model is also NP-hard and is solved using the Lagrangian Heuristic. One of the contributions of this dissertation is the development of Lagrangian Heuristic method for solving the pre-positioning and the real- time models. Based on the results of Lagrangian Heuristic for the pre-positioning model, almost all the deviations from optimal values are below 5%, which shows that the Heuristics works acceptably for the problem. Also, the execution times are no more than 780 seconds for the largest test instances. Moreover, for the real-time model, though not directly comparable, the solutions are fairly close to optimal and the execution time for the largest test instance is below 660 seconds. Hence, the efficiency of the heuristic for real-time model is satisfactory