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IIM Indore

We present a location–allocation problem for wastewater treatment in a cluster of processing units. The problem involves installing wastewater treatment plants of appropriate capacities at suitable locations and allocating processing units to these plants. The operational cost of treatment plants is a nonconvex function of wastewater volume and pollutant concentration. This problem is formulated as a non-convex mixed integer <a href="https://www.sciencedirect.com/topics/computer-science/nonlinear-programming">nonlinear programming</a> (MINLP) problem. For the general class of this problem, solution strategies based on adapted <a href="https://www.sciencedirect.com/topics/mathematics/mccormick">McCormick</a> envelopes and also piece-wise linear approximation are presented. For a special case of this problem, an exact convexification strategy is proposed and an outer approximation based branch-and-cut approach is presented as an exact solution method. For solving larger instances, we present a matheuristic solution method based on multi-start neighborhood search incorporated into a MINLP solution approach. A real-world case study is presented to illustrate the results of the model along with a multi-objective analysis.

Operation Management and Quantitative Techniques

World University root node

Computers & Operations Research

Capacitated facility location–allocation problem for wastewater treatment in an industrial cluster

We study the problem of assigning doctors to existing, non-operational Primary Health Centers (PHCs). We do this in the presence of clear guidelines on the maximum population that can be served by any PHC, and uncertainties in the availability of the doctors over the planning horizon. We model the problem as a robust capacitated multi-period maximal covering location problem with server uncertainty. Such supply-side uncertainties have not been accounted for in the context of multi-period facility location in the extant literature. We present an MIP formulation of this problem, which turns out to be too difficult for an off-the-shelf solver like CPLEX. We, therefore, present several dominance rules to reduce the size of the model. We further propose a Benders decomposition based solution method with several refinements that exploit the underlying structure of the problem to solve it extremely efficiently. Our computational experiments show one of the variants of our Benders decomposition based method to be on average almost 1000 times faster, compared to the CPLEX MIP solver, for problem instances containing 300 demand nodes and 10 facilities. Further, while the CPLEX MIP solver could not solve most of the instances beyond 300 demand nodes and 10 facilities even after 20 hours, two of our variants of Benders decomposition could solve instances upto the size of 500 demand nodes and 15 facilities in less than 0.5 hour, on average.

European Journal of Operational Research

Capacitated multi-period maximal covering location problem with server uncertainty

Primary Health Centers (PHCs), which are single doctor clinics and vital to health care in rural areas of developing countries, often remain inoperative due to shortage of doctors. When doctors become available, the health administrator needs to decide that which of the inoperative PHCs should the doctor be assigned. For transparency and operational efficiency, the sequence in which the inoperative PHCs will be assigned doctors needs to be decided in advance at the start of the planning horizon. Further, the number of doctors that will become available in each period of the planning horizon is uncertain. Moreover, the health guidelines set the capacity target, i.e. the maximum population a PHC can provide service to. We introduce and study the capacitated multi period maximal covering facility location problem under server uncertainty. We provide a formulation for the problem based on the minimax regret approach. Further, we solve the problem using CPLEX MIP solver and observe that it can only solve very small instances. Hence, we provide Benders decomposition based solution methods and refinements thereof, which is 100 − 5000 times faster and could solve practical size instances in reasonable time.

IIMA Working Papers

Benders decomposition for capacitated multi-period maximal covering location problem with server uncertainty

Benders decomposition for capacitated multi-period maximal covering location problem with server uncertainty

Facility location problems reported in the literature generally assume the problem parameter values (like cost, budget, etc.) to be known with complete certainty, even if they change over time (as in multi-period versions). However, in reality, there may be some uncertainty about the exact values of these parameters. Specifically, in the context of locating primary health centers (PHCs) in developing countries, there is generally a high level of uncertainty in the availability of servers (doctors) joining the facilities in different time periods. For transparency and efficient assignment of the doctors to PHCs, it is desirable to decide the facility opening sequence (assigning doctors to unmanned PHCs) at the start of the planning horizon. We present a new formulation for a multi-period maximal covering location problem with server uncertainty. We further demonstrate the superiority of our proposed formulation over the only other formulation reported in the literature. For instances of practical size, we provide a Benders decomposition based solution method, along with several refinements. For instances that the CPLEX MIP solver could solve within a time limit of 20 hours, our proposed solution method turns out to be of the order of 150–250 times faster for the problems with complete coverage, and around 1000 times faster for gradual coverage.

A new formulation and Benders decomposition for the multi-period maximal covering facility location problem with server uncertainty

Facility location problems reported in the literature generally assume the problem parameter values (like cost, budget, etc.) to be known with complete certainty, even if they change over time (as in multi-period versions). However, in reality, there may be some uncertainty about the exact values of these parameters. Specifically, in the context of locating primary health centers (PHCs) in developing countries, there is generally a high level of uncertainty in the availability of servers (doctors) joining the facilities in different time periods. For transparency and efficient assignment of the doctors to PHCs, it is desirable to decide the facility opening sequence (assigning doctors to unmanned PHCs) at the start of the planning horizon. For, this we present a new formulation for a multi-period maximal coverage location problem with server uncertainty (MMCLPSU). We further demonstrate the superiority of our proposed formulation over the only other formulation reported in the literature. For instances of practical size, we provide Benders’ decomposition based solution method, along with several refinements. For instances that CPLEX MIP solver could solve within a time limit of 20 hours, our proposed solution method turns out to be of the order of 150 - 250 times faster for the problems with complete coverage, and around 1000 times faster for gradual coverage

A New Formulation and Benders' Decomposition for Multi-period facility Location Problem with Server Uncertainty

We study the problem of allocating doctors to primary health centers. We model the problem as a multi-period uncapacitated facility location problem under uncertainty. The problem is unconventional in that the uncertainty is in the number and period of availability of doctors. We use a minmax regret approach to solve the problem. We present solution techniques using local search and tabu search and compare our solutions with optimal solutions obtained using commercial solvers. We see that one of our tabu search algorithms is faster and yields optimal solutions in the problems we tested on.