Mathematics

Research

27 pages, 2790 KiB

Open AccessArticle

A Robust Mixed-Integer Linear Programming Model for Sustainable Collaborative Distribution

by Islem Snoussi, Nadia Hamani, Nassim Mrabti and Lyes Kermad

Mathematics 2021, 9(18), 2318; https://doi.org/10.3390/math9182318 - 19 Sep 2021

Cited by 10 | Viewed by 2177

In this paper, we propose robust optimisation models for the distribution network design problem (DNDP) to deal with uncertainty cases in a collaborative context. The studied network consists of collaborative suppliers who satisfy their customers’ needs by delivering their products through common platforms. [...] Read more.

In this paper, we propose robust optimisation models for the distribution network design problem (DNDP) to deal with uncertainty cases in a collaborative context. The studied network consists of collaborative suppliers who satisfy their customers’ needs by delivering their products through common platforms. Several parameters—namely, demands, unit transportation costs, the maximum number of vehicles in use, etc.—are subject to interval uncertainty. Mixed-integer linear programming formulations are presented for each of these cases, in which the economic and environmental dimensions of the sustainability are studied and applied to minimise the logistical costs and the CO₂ emissions, respectively. These formulations are solved using CPLEX. In this study, we propose a case study of a distribution network in France to validate our models. The obtained results show the impacts of considering uncertainty by comparing the robust model to the deterministic one. We also address the impacts of the uncertainty level and uncertainty budget on logistical costs and CO₂ emissions. Full article

(This article belongs to the Special Issue Mathematical Models and Exact and Heuristic Algorithms for Solving Complex Optimization Problems)

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20 pages, 1413 KiB

Open AccessArticle

Optimization of the Real-Time Response to Roadside Incidents through Heuristic and Linear Programming

by Roman Buil, Jesica de Armas, Daniel Riera and Sandra Orozco

Mathematics 2021, 9(16), 1982; https://doi.org/10.3390/math9161982 - 19 Aug 2021

Cited by 3 | Viewed by 2113

Abstract

This paper presents a solution for a real-world roadside assistance problem. Roadside incidents can happen at any time. Depending on the type of incident, a specific resource from the roadside assistance company can be sent on site. The problem of allocating resources to [...] Read more.

This paper presents a solution for a real-world roadside assistance problem. Roadside incidents can happen at any time. Depending on the type of incident, a specific resource from the roadside assistance company can be sent on site. The problem of allocating resources to these road-side incidents can be stated as a multi-objective function and a large set of constraints, including priorities and preferences, resource capacities and skills, calendars, and extra hours. The request from the client is to a have real-time response and to attempt to use only open source tools. The optimization objectives to consider are the minimization of the operational costs and the minimization of the time to arrive to each incident. In this work, an innovative approach to near-optimally solving this problem in real-time is proposed, combining a heuristic approach and linear programming. The results show the great potential of this approach: operational costs were reduced by 19%, the use of external providers was reduced to half, and the productivity of the resources owned by the client was significantly increased. Full article

(This article belongs to the Special Issue Mathematical Models and Exact and Heuristic Algorithms for Solving Complex Optimization Problems)

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16 pages, 1544 KiB

Open AccessArticle

Optimal Price and Lot Size for an EOQ Model with Full Backordering under Power Price and Time Dependent Demand

by Luis A. San-José, Joaquín Sicilia, Manuel González-de-la-Rosa and Jaime Febles-Acosta

Mathematics 2021, 9(16), 1848; https://doi.org/10.3390/math9161848 - 5 Aug 2021

Cited by 6 | Viewed by 1684

Abstract

In this paper, we address an inventory system where the demand rate multiplicatively combines the effects of time and selling price. It is assumed that the demand rate is the product of two power functions, one depending on the selling price and the [...] Read more.

In this paper, we address an inventory system where the demand rate multiplicatively combines the effects of time and selling price. It is assumed that the demand rate is the product of two power functions, one depending on the selling price and the other on the time elapsed since the last inventory replenishment. Shortages are allowed and fully backlogged. The aim is to obtain the lot sizing, the inventory cycle and the unit selling price that maximize the profit per unit time. To achieve this, two efficient algorithms are proposed to obtain the optimal solution to the inventory problem for all possible parameter values of the system. We solve several numerical examples to illustrate the theoretical results and the solution methodology. We also develop a numerical sensitivity analysis of the optimal inventory policy and the maximum profit with respect to the parameters of the demand function. Full article

(This article belongs to the Special Issue Mathematical Models and Exact and Heuristic Algorithms for Solving Complex Optimization Problems)

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25 pages, 387 KiB

Open AccessArticle

Production/Inventory Policies for a Two-Echelon System with Credit Period Incentives

by Beatriz Abdul-Jalbar, Roberto Dorta-Guerra, José M. Gutiérrez and Joaquín Sicilia

Mathematics 2021, 9(15), 1725; https://doi.org/10.3390/math9151725 - 22 Jul 2021

Cited by 1 | Viewed by 1452

Abstract

Trade credit is a crucial source of capital particularly for small businesses with limited financing opportunities. Inventory models considering trade credit financing have been widely studied. However, while there is extensive research on the single-vendor single-buyer inventory model allowing delays in payments, the [...] Read more.

Trade credit is a crucial source of capital particularly for small businesses with limited financing opportunities. Inventory models considering trade credit financing have been widely studied. However, while there is extensive research on the single-vendor single-buyer inventory model allowing delays in payments, the systems where the vendor supplies to more than one buyer have received less attention. In this paper, we analyze a two-echelon inventory system where a single vendor supplies an item to two buyers who face a constant deterministic demand. The vendor produces the items at a finite rate and offers the buyers a delay payment period. That is, the buyers can delay the payment for the purchased items until the end of the credit period. Therefore, during such a period, the buyers sell the items and use the sales revenue to earn interest. At the end of the credit period, the buyers should pay the purchasing cost to the vendor for which external funding may be necessary. It is widely accepted that, in general, centralized policies reduce the total cost of the supply chain. Therefore, we first deal with an integrated model assuming that the vendor and the buyers make decisions jointly. However, in some cases, the buyers are not willing to collaborate, and the management of the supply chain has to be carried out in a decentralized manner. Hence, we also address the problem under a non-cooperative setting. Numerical examples are presented to illustrate both models. Additionally, we perform a computational experiment to compare both strategies, and a sensitivity analysis of the parameters is also carried out. From the results, we derived that, in general, it was more profitable to follow the integrated policy excepting when the replenishment costs for the buyers were high. Finally, in order to validate the computational results, a statistical analysis is performed. Full article

(This article belongs to the Special Issue Mathematical Models and Exact and Heuristic Algorithms for Solving Complex Optimization Problems)

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44 pages, 1750 KiB

Open AccessArticle

A Multi-Depot Vehicle Routing Problem with Stochastic Road Capacity and Reduced Two-Stage Stochastic Integer Linear Programming Models for Rollout Algorithm

by Wadi Khalid Anuar, Lai Soon Lee, Hsin-Vonn Seow and Stefan Pickl

Mathematics 2021, 9(13), 1572; https://doi.org/10.3390/math9131572 - 4 Jul 2021

Cited by 6 | Viewed by 4117

Abstract

A matheuristic approach based on a reduced two-stage Stochastic Integer Linear Programming (SILP) model is presented. The proposed approach is suitable for obtaining a policy constructed dynamically on the go during the rollout algorithm. The rollout algorithm is part of the Approximate Dynamic [...] Read more.

A matheuristic approach based on a reduced two-stage Stochastic Integer Linear Programming (SILP) model is presented. The proposed approach is suitable for obtaining a policy constructed dynamically on the go during the rollout algorithm. The rollout algorithm is part of the Approximate Dynamic Programming (ADP) lookahead solution approach for a Markov Decision Processes (MDP) framed Multi-Depot Dynamic Vehicle Routing Problem with Stochastic Road Capacity (MDDVRPSRC). First, a Deterministic Multi-Depot VRP with Road Capacity (D-MDVRPRC) is presented. Then an extension, MDVRPSRC-2S, is presented as an offline two-stage SILP model of the MDDVRPSRC. These models are validated using small simulated instances with CPLEX. Next, two reduced versions of the MDVRPSRC-2S model (MDVRPSRC-2S1 and MDVRPSRC-2S2) are derived. They have a specific task in routing: replenishment and delivering supplies. These reduced models are to be utilised interchangeably depending on the capacity of the vehicle, and repeatedly during the execution of rollout in reinforcement learning. As a result, it is shown that a base policy consisting of an exact optimal decision at each decision epoch can be obtained constructively through these reduced two-stage stochastic integer linear programming models. The results obtained from the resulting rollout policy with CPLEX execution during rollout are also presented to validate the reduced model and the matheuristic algorithm. This approach is proposed as a simple implementation when performing rollout for the lookahead approach in ADP. Full article

(This article belongs to the Special Issue Mathematical Models and Exact and Heuristic Algorithms for Solving Complex Optimization Problems)

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19 pages, 1272 KiB

Open AccessArticle

A Hybrid Whale Optimization Algorithm for Global Optimization

by Chun-Yao Lee and Guang-Lin Zhuo

Mathematics 2021, 9(13), 1477; https://doi.org/10.3390/math9131477 - 24 Jun 2021

Cited by 17 | Viewed by 2560

Abstract

This paper proposes a hybrid whale optimization algorithm (WOA) that is derived from the genetic and thermal exchange optimization-based whale optimization algorithm (GWOA-TEO) to enhance global optimization capability. First, the high-quality initial population is generated to improve the performance of GWOA-TEO. Then, thermal [...] Read more.

This paper proposes a hybrid whale optimization algorithm (WOA) that is derived from the genetic and thermal exchange optimization-based whale optimization algorithm (GWOA-TEO) to enhance global optimization capability. First, the high-quality initial population is generated to improve the performance of GWOA-TEO. Then, thermal exchange optimization (TEO) is applied to improve exploitation performance. Next, a memory is considered that can store historical best-so-far solutions, achieving higher performance without adding additional computational costs. Finally, a crossover operator based on the memory and a position update mechanism of the leading solution based on the memory are proposed to improve the exploration performance. The GWOA-TEO algorithm is then compared with five state-of-the-art optimization algorithms on CEC 2017 benchmark test functions and 8 UCI repository datasets. The statistical results of the CEC 2017 benchmark test functions show that the GWOA-TEO algorithm has good accuracy for global optimization. The classification results of 8 UCI repository datasets also show that the GWOA-TEO algorithm has competitive results with regard to comparison algorithms in recognition rate. Thus, the proposed algorithm is proven to execute excellent performance in solving optimization problems. Full article

(This article belongs to the Special Issue Mathematical Models and Exact and Heuristic Algorithms for Solving Complex Optimization Problems)

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29 pages, 390 KiB

Open AccessArticle

Profitability Index Maximization in an Inventory Model with a Price- and Stock-Dependent Demand Rate in a Power-Form

by Valentín Pando, Luis A. San-José, Joaquín Sicilia and David Alcaide-López-de-Pablo

Mathematics 2021, 9(10), 1157; https://doi.org/10.3390/math9101157 - 20 May 2021

Cited by 6 | Viewed by 2203

Abstract

This paper presents the optimal policy for an inventory model where the demand rate potentially depends on both selling price and stock level. The goal is the maximization of the profitability index, defined as the ratio income/expense. A numerical algorithm is proposed to [...] Read more.

This paper presents the optimal policy for an inventory model where the demand rate potentially depends on both selling price and stock level. The goal is the maximization of the profitability index, defined as the ratio income/expense. A numerical algorithm is proposed to calculate the optimal selling price. The optimal values for the depletion time, the cycle time, the maximum profitability index, and the lot size are evaluated from the selling price. The solution shows that the inventory must be replenished when the stock is depleted, i.e., the depletion time is always equal to the cycle time. The optimal policy is obtained with a suitable balance between ordering cost and holding cost. A condition that ensures the profitability of the financial investment in the inventory is established from the initial parameters. Profitability thresholds for several parameters, including the scale and the non-centrality parameters, keeping all the others fixed, are evaluated. The model with an isoelastic price-dependent demand is solved as a particular case. In this last model, all the optimal values are given in a closed form, and a sensitivity analysis is performed for several parameters, including the scale parameter. The results are illustrated with numerical examples. Full article

(This article belongs to the Special Issue Mathematical Models and Exact and Heuristic Algorithms for Solving Complex Optimization Problems)

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21 pages, 1506 KiB

Open AccessArticle

Effective Algorithms for the Economic Lot-Sizing Problem with Bounded Inventory and Linear Fixed-Charge Cost Structure

by José M. Gutiérrez, Beatriz Abdul-Jalbar, Joaquín Sicilia and Inmaculada Rodríguez-Martín

Mathematics 2021, 9(6), 689; https://doi.org/10.3390/math9060689 - 23 Mar 2021

Cited by 1 | Viewed by 1439

Abstract

Efficient algorithms for the economic lot-sizing problem with storage capacity are proposed. On the one hand, for the cost structure consisting of general linear holding and ordering costs and fixed setup costs, an

O (T^{2})

dynamic programming algorithm is introduced, where

T

[...] Read more.

Efficient algorithms for the economic lot-sizing problem with storage capacity are proposed. On the one hand, for the cost structure consisting of general linear holding and ordering costs and fixed setup costs, an

O (T^{2})

dynamic programming algorithm is introduced, where

T

is the number of time periods. The new approach induces an accurate partition of the planning horizon, discarding most of the infeasible solutions. Moreover, although there are several algorithms based on dynamic programming in the literature also running in quadratic time, even considering more general cost structures and assumptions, the new solution uses a geometric technique to speed up the algorithm for a class of subproblems generated by dynamic programming, which can now be solved in linearithmic time. To be precise, the computational results show that the average occurrence percentage of this class of subproblems ranges between 13% and 45%, depending on both the total number of periods and the percentage of storage capacity availability. Furthermore, this percentage significantly increases from 13% to 35% as the capacity availability decreases. This reveals that the usage of the geometric technique is predominant under restrictive storage capacities. Specifically, when the percentage of capacity availability is below 50%, the average running times are on average 100 times faster than those when this percentage is above 50%. On the other hand, an

O (T)

on-line array searching method in Monge arrays can be used when the costs are non-speculative costs. Full article

(This article belongs to the Special Issue Mathematical Models and Exact and Heuristic Algorithms for Solving Complex Optimization Problems)

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17 pages, 2407 KiB

Open AccessFeature PaperArticle

Close-Enough Facility Location

by Alejandro Moya-Martínez, Mercedes Landete and Juan Francisco Monge

Mathematics 2021, 9(6), 670; https://doi.org/10.3390/math9060670 - 21 Mar 2021

Cited by 2 | Viewed by 2206

Abstract

This paper introduces the concept of close-enough in the context of facility location. It is assumed that customers are willing to move from their homes to close-enough pickup locations. Given that the number of pickup locations is expanding every day, it is assumed [...] Read more.

This paper introduces the concept of close-enough in the context of facility location. It is assumed that customers are willing to move from their homes to close-enough pickup locations. Given that the number of pickup locations is expanding every day, it is assumed that pickup locations can be placed everywhere. Conversely, the set of potential location for opening facilities is discrete as well as the set of customers. Opening facilities and pickup points entails an installation budget and a distribution cost to transport goods from facilities to customers and pickup locations. The

(p, t)

-Close-Enough Facility Location Problem is the problem of deciding where to locate p facilities among the finite set of candidates, where to locate t pickup points in the plane and how to allocate customers to facilities or to pickup points so that all the demand is satisfied and the total cost is minimized. In this paper, it is proved that the set of initial infinite number of pickup locations is finite in practice. Two mixed-integer linear programming models are proposed for the discrete problem. The models are enhanced with valid inequalities and a branch and price algorithm is designed for the most promising model. The findings of a comprehensive computational study reveal the performance of the different models and the branch and price algorithm and illustrate the value of pickup locations. Full article

(This article belongs to the Special Issue Mathematical Models and Exact and Heuristic Algorithms for Solving Complex Optimization Problems)

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