Using decomposition to improve greedy solutions of the optimal diversity management problem

Agra, Agostinho; Cerdeira, J. Orestes; Requejo, Cristina

doi:10.1111/itor.12004

Cited by 5 publications

(3 citation statements)

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“…In [2] it is observed that the values q U k obtained with the greedy algorithm are close to the ones from optimal solutions, i.e., the greedy solution gives a good estimate of the optimal number of facilities to choose from each subgraph G k . Based on this observation, matrix U is further refined in a neighborhood of values q U k using a relax-and-fix heuristic as follows.…”

Section: Initial Upper Boundsmentioning

confidence: 86%

“…We address the p-median problem when graph G has several different connected components. This case arises in the automotive industry, namely in the production of electric wiring configurations for vehicles, where the problem is known as the optimal diversity management problem [1,2,4,5,6]. This situation also occurs when location takes place over split administrative regions or regions geographically apart (e.g.…”

Section: Introductionmentioning

confidence: 99%

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A decomposition approach for the p -median problem on disconnected graphs

Agra

Cerdeira

Requejo

2017

Computers & Operations Research

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The p-median problem seeks for the location of p facilities on the vertices (customers) of a graph to minimize the sum of transportation costs for satisfying the demands of the customers from the facilities. In many real applications of the p-median problem the underlying graph is disconnected. That is the case of p-median problem defined over split administrative regions or regions geographically apart (e.g. archipelagos), and the case of problems coming from industry such as the optimal diversity management problem. In such cases the problem can be decomposed into smaller p-median problems which are solved in each component k for different feasible values of p k , and the global solution is obtained by finding the best combination of p k medians. This approach has the advantage that it permits to solve larger instances since only the sizes of the connected components are important and not the size of the whole graph. However, since the optimal number of facilities to select from each component is not known, it is necessary to solve p-median problems for every feasible number of facilities on each component. In this paper we give a decomposition algorithm that uses a procedure to reduce the number of subproblems to solve. Computational tests on real instances of the optimal diversity management problem and on simulated instances are reported showing that the reduction of subproblems is significant, and that optimal solutions were found within reasonable time.

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Section: Initial Upper Boundsmentioning

confidence: 86%

Section: Introductionmentioning

confidence: 99%

A decomposition approach for the p -median problem on disconnected graphs

Agra

Cerdeira

Requejo

2017

Computers & Operations Research

Self Cite

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“…A decomposition approach was also proposed in where

p

‐median heuristics and a MCKP dynamic programming algorithm were developed, taking into account the particularity of ODMP instances. Recently, the same authors proposed an exact approach using LP relaxation bounds for sub‐PMP s and variable fixing in MCKP to reduce the number of sub‐PMP s to be solved .…”

Section: Literature Reviewmentioning

confidence: 99%

A three‐stage p‐median based exact method for the optimal diversity management problem

et al. 2018

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The optimal diversity management problem (ODMP) arises in many application fields when a company, producing a good and/or a service customizable with options, has to satisfy many different client demands with various subset of options, but only a limited number of option combinations can be produced. ODMP can be represented by a disconnected network and formulated as a large-scale p-median problem (PMP). In this article we improve a known decomposition approach where smaller PMPs, related to the network components, can be solved instead of the initial large problem. The proposed method is structured in three stages and it combines Lagrangian relaxation-based techniques, variable fixing and reduction tests, and a dynamic programming algorithm. It drastically reduces the number and the dimensions of the p-median subproblems to be solved to optimality by a MIP solver and to be combined to determine the optimal solution of the original PMP by a multiple choice knapsack problem. A sequential and a parallel implementation of the method are provided and tested. Obtained results on known and new test instances show that our approach considerably outperforms state-of-the-art algorithms for large-scale ODMPs.decomposition, Lagrangian relaxation, multiple choice knapsack, optimal diversity management, p-median, parallel computing I N T R O D U C T I O NThe optimal diversity management problem (ODMP) is a well-known optimization problem arising in many application fields, every time a company produces a good and/or a service which can be provided with options. In this case the product can be personalized by the customer who can choose different option combinations (configurations) depending on her/his needs or preferences. In this context, satisfying all the possible demands with exactly the required options would impose a company to produce all the possible configurations in advance and manage all of them at the assembly lines. Moreover, the production operations could start only after a demand is received, so providing huge delays in satisfying a request. To overcome these drawbacks, a company usually produces a limited number of opportunely chosen configurations to cover all the possible ones. In this way a demand, if no available configuration contains exactly the required options, can be satisfied by a compatible configuration, that is, by a configuration containing all the required options plus some others undemanded by the customer. This implies that a client could receive some unrequired options, so generating an over-cost for the company.

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