Using tropical optimization to solve constrained minimax single-facility location problems with rectilinear distance

Krivulin, Nikolai

doi:10.1007/s10287-017-0289-2

Cited by 11 publications

(14 citation statements)

References 43 publications

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“…Applications of tropical optimization cover various problems in project scheduling, location analysis, decision making and in other fields. Some related examples can be found, e.g., in [16,18,19,20,21,22,24]. There are multidimensional tropical optimization problems that can be solved directly to describe all solutions in a compact closed vector form, whereas for other problems, only algorithmic solutions are available, which offer numerical iterative procedures to find a solution if one exists.…”

Section: Introductionmentioning

confidence: 99%

Using tropical optimization techniques in bi-criteria decision problems

Krivulin

2018

Comput Manag Sci

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We consider decision problems of rating alternatives based on their pairwise comparisons according to two criteria. Given pairwise comparison matrices for each criterion, the problem is to find the overall scores of the alternatives. We offer a solution that involves the minimax approximation of the comparison matrices by a common consistent matrix of unit rank in terms of the Chebyshev metric in logarithmic scale. The approximation problem reduces to a bi-objective optimization problem to minimize the approximation errors simultaneously for both comparison matrices. We formulate the problem in terms of tropical (idempotent) mathematics, which focuses on the theory and applications of algebraic systems with idempotent addition. To solve the optimization problem obtained, we apply methods and results of tropical optimization to derive a complete Pareto-optimal solution in a direct explicit form ready for further analysis and straightforward computation. We then exploit this result to solve the bi-criteria decision problem of interest. As illustrations, we present examples of the solution of two-dimensional optimization problems in general form, and of a decision problem with four alternatives in numerical form.

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Section: Introductionmentioning

confidence: 99%

Using tropical optimization techniques in bi-criteria decision problems

Krivulin

2018

Comput Manag Sci

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“…There are several options in choosing a location such as deterministic and stochastic. The former uses known and constant input as seen in a research by Boonmee et al (2017), Ahmadi-Javid et al (2017) and Krivulin (2017). The latter uses varied inputs to predict probability which is more likely to the actual situation as seen in a research by Amiri-Aref et al 2018 X ijk : 1, (If the vehicle k is traveling from node i to node j) X ijk : 0, Otherwise Y i : 1, If the central market is opened at the location i Y i : 0, Otherwise Z ik : 1, If the vehicle k is traveling out of the node i), Z ik : 0, Otherwise P jk : The amount of palm that the vehicle k receives is still farmers in each township j (kilogram)…”

Section: Hypothesis and Characteristics Of The Problemmentioning

confidence: 99%

A Mathematical Model for Locating Potential Central Markets and Arranging Transportation Routes for Oil Palm Transportation in the Upper Southern Part of Thailand

Ponatong¹,

Sindhuchao²

2019

J. Eng. Appl. Sci.

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There is no central market for oil palm in Thailand. Knowing that the central market can help reduce logistic costs and provide a suitable price for oil palm transaction, this research studies a problem of selecting locations of oil palm central markets and constructing vehicle routes for oil palm collection under vehicle speed and capacity limitation with the objective of minimizing the total cost that consists of the vehicle routing cost, the fixed operating cost and the depreciation of the central markets and the vehicles. A mathematic model of the problem is formulated and then validated using comparison and cutting methods. The problem instances are generated from real data of 439 townships of 6 provinces in the upper Southern area of Thailand. Forty-two problems are tested and solved by using a computer software. The computational results show that based on all problem instances, the total cost can be reduced by at least 5.76% when compared to the one of the existing process. In addition with a reduction of the total travel distance, the amount of carbon dioxide emission goes down by at least 13.35%.

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“…A similar algebraic approach based on the theory of max-separable functions is implemented in [31,32,15,16,30] to solve constrained minimax location problems. Further examples include the solutions, given in terms of idempotent algebra in [23,18,24,25,21], to unconstrained and constrained minimax single-facility location problems with Chebyshev and rectilinear distances.…”

Section: Introductionmentioning

confidence: 99%

“…A solution for the weighted problem with rectilinear distance is given in [6], which involves decomposition into independent one-dimensional subproblems solved by reducing to equivalent network flow problems. In [23,18,24,25,21], an approach based on idempotent algebra is applied to solve unweighted unconstrained and constrained location problems. Further results on both unweighted and weighted location problems can be found in the survey papers [11,1,28,2,3], as well as in the books [29,17,9,7,26].…”

Section: Introductionmentioning

confidence: 99%

Algebraic Solution of Weighted Minimax Single-Facility Constrained Location Problems

Krivulin

2018

Relational and Algebraic Methods in Computer Science

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We consider location problems to find the optimal sites of placement of a new facility, which minimize the maximum weighted Chebyshev or rectilinear distance to existing facilities under constraints on a feasible location domain. We examine Chebyshev location problems in multidimensional space to represent and solve the problems in the framework of tropical (idempotent) algebra, which deals with the theory and applications of semirings and semifields with idempotent addition. The solution approach involves formulating the problem as a tropical optimization problem, introducing a parameter that represents the minimum value of the objective function in the problem, and reducing the problem to a system of parametrized inequalities. The necessary and sufficient conditions for the existence of a solution to the system serve to evaluate the minimum, whereas all corresponding solutions of the system present a complete solution of the optimization problem. With this approach, we obtain direct, exact solutions represented in a compact closed form, which is appropriate for further analysis and straightforward computations with polynomial time complexity. The solutions of the Chebyshev problems are then used to solve location problems with rectilinear distance in the two-dimensional plane. The obtained solutions extend previous results on the Chebyshev and rectilinear location problems without weights and with less general constraints.Key-Words: tropical mathematics, idempotent semifield, constrained optimization problem, single-facility location problem, Chebyshev and rectilinear distances MSC (2010): 65K10, 15A80, 90B85, 90C47, 90C48

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Using tropical optimization to solve constrained minimax single-facility location problems with rectilinear distance

Cited by 11 publications

References 43 publications

Using tropical optimization techniques in bi-criteria decision problems

Using tropical optimization techniques in bi-criteria decision problems

A Mathematical Model for Locating Potential Central Markets and Arranging Transportation Routes for Oil Palm Transportation in the Upper Southern Part of Thailand

Algebraic Solution of Weighted Minimax Single-Facility Constrained Location Problems

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