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A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Engineering Mathematics".

Deadline for manuscript submissions: 30 June 2024 | Viewed by 14297

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Special Issue Editors

Prof. Dr. Shiv Raj Singh

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Guest Editor

Department of Mathematics, Chaudhary Charan Singh University, Meerut 250004, India
Interests: operations research; mathematical modeling; production planning and control

Dr. Dharmendra Yadav

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Guest Editor

Department of Mathematics, Vardhaman College, Bijnor 246701, India
Interests: supply chain management; production planning; inventory modeling; mathematical modeling

Dr. Himani Dem

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Guest Editor

Department of Mathematics, Ramjas College, University of Delhi, Delhi 110015, India
Interests: applied mathematics; inventory; logistics; operations management; inventory management; production planning; optimization

Special Issue Information

Dear Colleagues,

Operations research usually uses existing scientific and technological knowledge to solve specific problems and provides a basis for decision makers to make the best decisions. In recent years, operations research has been continuously innovated and developed; however, there are still some exciting and challenging problems in technology and methods that are worthy of further exploration. The common element in all of the scientific areas that this Special Issue will address is the need for some optimization methodology for determining viable solutions to problems using computers and the techniques of operations research. This Special Issue will therefore concern itself with these scientific fields of application and will be accordingly broad in scope in regard to subject matter. This will involve theoretical and computational issues, as well as application studies. The multidimensional nature of these problems raises relevant mathematical and algorithmic challenges.

The aim of the Special Issue is to publish original articles dealing with every aspect of mathematical programming, optimization, operations research, and everything of direct or indirect use concerning the related problem.

Contributions are solicited in all subjects related to Mathematical Programming, Optimization and Operations Research.

Potential topics include, but are not limited to, the following:

Operations research and applications;
Optimization theory and its applications;
Stochastic optimization and applications;
Game theory and its application;
Supply chain optimization;
Data envelopment analysis;
Group decision making analysis;
Multi-criteria decision analysis and applications;
Fuzzy programming;
Production planning and inventory control;
Sustainability;
Multi-objective problems;
Stochastic and robust multi-objective optimization;
Approximation and representation algorithms.

Prof. Dr. Shiv Raj Singh
Dr. Dharmendra Yadav
Dr. Himani Dem
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

mathematical programming
optimization methods and algorithms
operations research
scheduling and planning
resource optimization
game theory
supply chain management
stochastic process
decision theory and applications
multi-objective problems
evolutionary computation
computational intelligence

Published Papers (13 papers)

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Research

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21 pages, 4219 KiB

Open AccessArticle

A Hybrid Adaptive Simulated Annealing and Tempering Algorithm for Solving the Half-Open Multi-Depot Vehicle Routing Problem

by Shichang Xiao, Pan Peng, Peng Zheng and Zigao Wu

Mathematics 2024, 12(7), 947; https://doi.org/10.3390/math12070947 - 22 Mar 2024

Viewed by 458

Abstract

The half-open multi-depot vehicle routing problem (HOMDVRP) is a typical decision optimization problem in the field of collaborative logistics that considers resource sharing. This study aims to develop an effective meta-heuristic algorithm for solving the HOMDVRP. Firstly, a mixed-integer programming model of HOMDVRP is established to minimize the total travel distance of the vehicles. After that, a novel hybrid adaptive simulated annealing and tempering algorithm (HASATA) is proposed based on the features of HOMDVRP. The proposed algorithm combines the strengths of the simulated annealing algorithm and the large-neighborhood search algorithm to balance the algorithm’s searching capabilities in both breadth and depth. Meanwhile, an adaptive Markov chain length mechanism and a tempering mechanism are designed to improve the algorithm’s computational efficiency and convergence ability. Finally, simulation experiments are conducted to verify the effectiveness of the proposed model and the computational performance of the proposed algorithm. Four comparison algorithms are selected and analyzed using 24 groups of problem instances. The comparison results show that the proposed HASATA can solve the HOMDVRP more efficiently and obtain a solution with better optimization performance and satisfactory stability. Full article

(This article belongs to the Special Issue Mathematical Programming, Optimization and Operations Research)

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45 pages, 5204 KiB

Open AccessArticle

An Inventory Model for Growing Items When the Demand Is Price Sensitive with Imperfect Quality, Inspection Errors, Carbon Emissions, and Planned Backorders

by Cynthia Griselle De-la-Cruz-Márquez, Leopoldo Eduardo Cárdenas-Barrón, J. David Porter, Imelda de Jesús Loera-Hernández, Neale R. Smith, Armando Céspedes-Mota, Gerardo Treviño-Garza and Rafael Ernesto Bourguet-Díaz

Mathematics 2023, 11(21), 4421; https://doi.org/10.3390/math11214421 - 25 Oct 2023

Cited by 1 | Viewed by 859

Abstract

Inventory models that consider environmental and quality concerns have received some attention in the literature, yet no model developed to date has investigated these features in combination with growing items. Therefore, there is a need to incorporate these three relevant aspects together in a single inventory model to support decisions, compare results, and obtain new knowledge for the complexities of the real world. Moreover, current sustainable inventory management practices aim at mitigating the ecological consequences of an industry while preserving its profitability. The present study aligns with this perspective and introduces an economic order quantity (EOQ) model that considers imperfect quality while also accounting for sustainability principles. More specifically, the model addresses growing items, which have a demand dependent on selling price and the unique ability to grow while being stored in inventory. Additionally, the analysis acknowledges the possibility of classification errors during the inspection process, encompassing both Type-I and Type-II inspection errors. Furthermore, the model permits shortages and ensures that any shortage is completely fulfilled through backorders. The optimization model produces an optimal solution for the proposed model that is derived by optimizing three decision variables: order quantity of newborn items, backordering quantity, and the selling price of perfect items. A numerical example is presented, and the results are discussed. Finally, a sensitivity analysis on variations of parameters such as Type-I and Type-II errors shows that it is advantageous to reduce the percentage of good items that are misclassified as defective (i.e., Type-I error). As there is a direct impact of such errors on sales, it is imperative to address and mitigate this issue. When defective items are mistakenly classified as good Type-II errors, adverse consequences ensue, including a heightened rate of product returns. This, in turn, results in additional costs for the company, such as penalties and diminished customer confidence. Hence, the findings clearly suggest that the presence of Type-I and Type-II errors has a negative effect on the ordering policy and on the total expected profit. Moreover, this work provides a model that can be used with any growing item (including plants), so the decision-maker has the opportunity to analyze a wide variety of scenarios. Full article

(This article belongs to the Special Issue Mathematical Programming, Optimization and Operations Research)

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13 pages, 323 KiB

Open AccessArticle

The Sufficiency of Solutions for Non-smooth Minimax Fractional Semi-Infinite Programming with (B_K,ρ)−Invexity

by Hong Yang and Angang Cui

Mathematics 2023, 11(20), 4240; https://doi.org/10.3390/math11204240 - 10 Oct 2023

Viewed by 593

Abstract

Minimax fractional semi-infinite programming is an important research direction for semi-infinite programming, and has a wide range of applications, such as military allocation problems, economic theory, cooperative games, and other fields. Convexity theory plays a key role in many aspects of mathematical programming and is the foundation of mathematical programming research. The relevant theories of semi-infinite programming based on different types of convex functions have their own applicable scope and limitations. It is of great value to study semi-infinite programming on the basis of more generalized convex functions and obtain more general results. In this paper, we defined a new type of generalized convex function, based on the concept of the K−directional derivative, that is, uniform

(B_{K}, ρ) -

invex, strictly uniform

(B_{K}, ρ) -

invex, uniform

(B_{K}, ρ) -

pseudoinvex, strictly uniform

(B_{K}, ρ) -

pseudoinvex, uniform

(B_{K}, ρ) -

quasiinvex and weakly uniform

(B_{K}, ρ) -

quasiinvex function. Then, we studied a class of non-smooth minimax fractional semi-infinite programming problems involving this generalized convexity and obtained sufficient optimality conditions. Full article

(This article belongs to the Special Issue Mathematical Programming, Optimization and Operations Research)

22 pages, 456 KiB

Open AccessArticle

Young Duality for Variational Inequalities and Nonparametric Method of Demand Analysis in Input–Output Models with Inputs Substitution: Application for Kazakhstan Economy

by Seyit Kerimkhulle, Nataliia Obrosova, Alexander Shananin and Akylbek Tokhmetov

Mathematics 2023, 11(19), 4216; https://doi.org/10.3390/math11194216 - 09 Oct 2023

Viewed by 807

Abstract

The global macroeconomic shocks of the last decade entail the restructuring of national production networks and induce processes of input substitution. We suggest mathematical tools of Young duality for variational inequalities for studying these processes. Based on the tools we provide, a new mathematical model of a production network with several final consumers is created. The model is formulated as a pair of conjugated problems: a complementarity problem for optimal resource allocation with neoclassical production functions and the Young dual problem for equilibrium price indices on network products. The solution of these problems gives an equilibrium point in the space of network inter-industry flows and price indices on goods. Based on our previous results, we suggest an algorithm for model identification with an official economic statistic in the case of constant elasticity of substitution production functions. We give an explicit solution to the complementarity problems in this case and develop the algorithm of the inter-industry flows scenario projection. Since the algorithm needs the scenario projection of final sales structure as its input, we suggest a modified methodology that allows the calculation of scenario shifts in final consumer spending. To do this, we employ the generalized nonparametric method of demand analysis. As a result, we develop new technology for scenario calculation of a national input–output table, including shifts in final consumer spending. The technology takes into account a substitution of inputs in the network and is based on officially published national statistics data. The application of the methodology to study tax collection scenarios for Kazakhstan’s production network is demonstrated. Full article

(This article belongs to the Special Issue Mathematical Programming, Optimization and Operations Research)

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21 pages, 2154 KiB

Open AccessArticle

A Two-Machine Learning Date Flow-Shop Scheduling Problem with Heuristics and Population-Based GA to Minimize the Makespan

by Jian-You Xu, Win-Chin Lin, Yu-Wei Chang, Yu-Hsiang Chung, Juin-Han Chen and Chin-Chia Wu

Mathematics 2023, 11(19), 4060; https://doi.org/10.3390/math11194060 - 25 Sep 2023

Cited by 1 | Viewed by 721

Abstract

This paper delves into the scheduling of the two-machine flow-shop problem with step-learning, a scenario in which job processing times decrease if they commence after their learning dates. The objective is to optimize resource allocation and task sequencing to ensure efficient time utilization and timely completion of all jobs, also known as the makespan. The identified problem is established as NP-hard due to its reduction to a single machine for a common learning date. To address this complexity, this paper introduces an initial integer programming model, followed by the development of a branch-and-bound algorithm augmented with two lemmas and a lower bound to attain an exact optimal solution. Additionally, this paper proposes four straightforward heuristics inspired by the Johnson rule, along with their enhanced counterparts. Furthermore, a population-based genetic algorithm is formulated to offer approximate solutions. The performance of all proposed methods is rigorously evaluated through numerical experimental studies. Full article

(This article belongs to the Special Issue Mathematical Programming, Optimization and Operations Research)

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21 pages, 4604 KiB

Open AccessArticle

A Heuristic Model for Spare Parts Stocking Based on Markov Chains

by Ernesto Armando Pacheco-Velázquez, Manuel Robles-Cárdenas, Saúl Juárez Ordóñez, Abelardo Ernesto Damy Solís and Leopoldo Eduardo Cárdenas-Barrón

Mathematics 2023, 11(16), 3550; https://doi.org/10.3390/math11163550 - 17 Aug 2023

Viewed by 1078

Abstract

Spare parts management has gained significant attention in recent years due to the considerable costs associated with backorders or excess inventory. This article addresses the challenge of determining the optimal number of spare parts to stock, assuming that the parts can be repaired. When an item fails, it is promptly sent for repair in a workshop. The time between failures and the repair time are assumed to follow an exponential distribution, although it should be noted that the results could be adapted to other distributions as well. This study introduces a heuristic method to find the optimal inventory level that minimizes the total cost, considering holding inventory, backorder, and repair costs. The research offers a valuable decision-making framework for determining the number of spare parts needed to minimize inventory costs, based on just two parameters: (1) the ratio of time to repair and time to failure, and (2) the ratio of the inventory holding cost of a spare part per day to the daily cost of an idle machine. To the best of our knowledge, there are no similar methodologies in the existing literature. The proposed method is straightforward to implement, employing graphs and simple computations. Therefore, it is anticipated to be highly beneficial for practitioners seeking a quick and reliable estimator of the optimal number of spare parts to stock for critical components. Full article

(This article belongs to the Special Issue Mathematical Programming, Optimization and Operations Research)

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11 pages, 1375 KiB

Open AccessArticle

Discontinuous Economic Growing Quantity Inventory Model

by Amir Hossein Nobil, Erfan Nobil, Leopoldo Eduardo Cárdenas-Barrón, Dagoberto Garza-Núñez, Gerardo Treviño-Garza, Armando Céspedes-Mota, Imelda de Jesús Loera-Hernández and Neale R. Smith

Mathematics 2023, 11(15), 3258; https://doi.org/10.3390/math11153258 - 25 Jul 2023

Cited by 1 | Viewed by 798

Abstract

The classical economic growing quantity (EGQ) model is a key concept in the inventory control problems research literature. The EGQ model is commonly employed for the purpose of inventory control in the management of growing items, such as fish and farm animals, within industries such as livestock, seafood, and aviculture. The economic order quantity (EOQ) model assumes that customer demand is satisfied without interruption in each cycle; however, this assumption is not always true for some companies as they do not have continuous operations, except for item storage, during non-working times such as weekends, natural idle periods, or spare time. In this study, we extend the traditional EGQ model by incorporating the concept of working and non-working periods, resulting in the development of a new model called discontinuous economic growing quantity (DEGQ). Unlike the conventional EGQ model, the DEGQ model considers the presence of intermittent operational periods, in which the firm is actively engaged in its activities, and non-working periods, during which only storage-related operations occur. By incorporating this discontinuity, the DEGQ model provides a more accurate representation of real-world scenarios where businesses operate in a non-continuous manner, thus enhancing the effectiveness of inventory control and management strategies. The study aims to obtain the optimal number of periods in each cycle and the optimal slaughter age for the breeding items, and, subsequently, to find the optimal order size to minimize the total cost. Finally, we propose an optimal analytical procedure to determine the optimal solutions. This procedure entails finding the optimal number of periods using a closed-form equation and determining the optimal slaughter age by exhaustively searching the entire range of possible growth times. Full article

(This article belongs to the Special Issue Mathematical Programming, Optimization and Operations Research)

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25 pages, 1181 KiB

Open AccessArticle

Compact Integer Programs for Depot-Free Multiple Traveling Salesperson Problems

by José Alejandro Cornejo-Acosta, Jesús García-Díaz, Julio César Pérez-Sansalvador and Carlos Segura

Mathematics 2023, 11(13), 3014; https://doi.org/10.3390/math11133014 - 06 Jul 2023

Cited by 2 | Viewed by 1438

Abstract

Multiple traveling salesperson problems (mTSP) are a collection of problems that generalize the classical traveling salesperson problem (TSP). In a nutshell, an mTSP variant seeks a minimum cost collection of m paths that visit all vertices of a given weighted complete graph. This paper introduces novel compact integer programs for the depot-free mTSP (DFmTSP). This fundamental variant models real scenarios where depots are unknown or unnecessary. The proposed integer programs are adapted to the main variants of the DFmTSP, such as closed paths, open paths, bounding constraints (also known as load balance), and the minsum and minmax objective functions. Some of these integer programs have

O (n^{2} m)

binary variables and

O (n^{2})

constraints, where m is the number of salespersons and

n = | V (G) |

. Furthermore, we introduce more compact integer programs with

O (n^{2})

binary variables and

O (n^{2})

constraints for the same problem and most of its main variants. Without losing their compactness, all the proposed programs are adapted to fixed-destination multiple-depots mTSP (FD-MmTSP) and a combination of FD-MmTSP and DFmTSP, where fewer than m depots are part of the input, but the solution still consists of m paths. We used off-the-shelf optimization software to empirically test the proposed integer programs over a classical benchmark dataset; these tests show that the proposed programs meet desirable theoretical properties and have practical advantages over the state of the art. Full article

(This article belongs to the Special Issue Mathematical Programming, Optimization and Operations Research)

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16 pages, 18361 KiB

Open AccessArticle

Space Splitting and Merging Technique for Online 3-D Bin Packing

by Thanh-Hung Nguyen and Xuan-Thuan Nguyen

Mathematics 2023, 11(8), 1912; https://doi.org/10.3390/math11081912 - 18 Apr 2023

Cited by 3 | Viewed by 1275

Abstract

This paper introduces a novel method for online 3-D bin packing, which is a strongly NP-hard problem, based on a space splitting and merging technique. In this scenario, the incoming box is unknown and must be immediately packed. The problem has many applications in industries that use manipulators to automate the packing process. The main idea of the approach is to divide the bin into spaces. These spaces are then categorized into one of two types of data structures: main and secondary data structures. Each node in the main data structure holds the information of a space that can be used to fit a new box. Each node in the secondary data structure holds the information of a space that cannot be used to place a box. The search algorithm based on these two data structures reduces the required search effort and simplifies the organizing and editing of the data structure. The experimental results demonstrate that the proposed method can achieve a packed volume ratio of up to 83% in the case of multiple bins being used. The position of a placed box can be found within milliseconds. Full article

(This article belongs to the Special Issue Mathematical Programming, Optimization and Operations Research)

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23 pages, 4086 KiB

Open AccessFeature PaperArticle

Machine Downtime Effect on the Warm-Up Period in an Economic Production Quantity Problem

by Erfan Nobil, Leopoldo Eduardo Cárdenas-Barrón, Dagoberto Garza-Núñez, Gerardo Treviño-Garza, Armando Céspedes-Mota, Imelda de Jesús Loera-Hernández, Neale R. Smith and Amir Hossein Nobil

Mathematics 2023, 11(7), 1740; https://doi.org/10.3390/math11071740 - 05 Apr 2023

Cited by 3 | Viewed by 1887

Abstract

Success in the industrial sector is compromised by diverse conditions such as imperfect product production, manufacturing line interruptions, and unscheduled maintenance. The precise use of common practices in production environments is an available solution to eliminate some of these issues. Applying a warm-up period in a manufacturing process is adequate and cost-effective for almost all companies. It improves the equipment’s productivity and helps the manufacturing line generate fewer defective products. Even though several inventory management studies have included a warm-up phase in their models, its use in economic production quantity (EPQ) models remains largely unexplored. Adding a warm-up phase to the production cycle minimizes maintenance expenses and defective products and increases the machine’s performance. In this study, the dependency between the machine downtime and the warm-up length is examined for the first time. The warm-up time depends on the machine’s off-state period: if the machine has a longer operation timeout, then a longer warm-up period is needed. The model includes a function to model the warm-up time relative to the machine downtime and two types of defective products: scrapping and reworking items. The study is concluded with some numerical examples, a sensitivity analysis, and some management insights related to the EPQ. Full article

(This article belongs to the Special Issue Mathematical Programming, Optimization and Operations Research)

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21 pages, 2594 KiB

Open AccessArticle

Economic Order Quantity for Growing Items with Mortality Function under Sustainable Green Breeding Policy

Mathematics 2023, 11(4), 1039; https://doi.org/10.3390/math11041039 - 18 Feb 2023

Cited by 6 | Viewed by 1323

Abstract

Determining the optimal slaughter age of fast-growing animals regarding the mortality rates and breeding costs plays an important and major role for companies that benefit from their meat. Additionally, the effects of carbon dioxide (CO₂) emissions during the growth cycle of animals are a significant concern for governments. This study proposes an economic order quantity (EOQ) for growing items with a mortality function under a sustainable green breeding policy. It assumes that CO₂ production is a practical polynomial function that depends on the age of the animals as well as the mortality function. The aim of the model is to determine the optimal slaughter age and the optimal number of newborn chicks, purchased from the supplier, to minimize the total costs. We propose an analytical approach, with five simple steps, to find the optimal solutions. Finally, we provide a numerical example and some model management insights to help practitioners in this area. Full article

(This article belongs to the Special Issue Mathematical Programming, Optimization and Operations Research)

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Review

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18 pages, 2542 KiB

Open AccessReview

Optimizing Inventory Management: A Comprehensive Analysis of Models Integrating Diverse Fuzzy Demand Functions

by Mandeep Mittal, Vibhor Jain, Jayanti Tripathi Pandey, Muskan Jain and Himani Dem

Mathematics 2024, 12(1), 70; https://doi.org/10.3390/math12010070 - 25 Dec 2023

Cited by 1 | Viewed by 1107

Abstract

This review study provides a comprehensive analysis of the classification of inventory models, with a focus on incorporating various fuzzy demand functions. The incorporation of fuzzy sets theory within inventory models is highlighted as a significant advancement in the field. The study emphasizes the importance of efficiently locating pertinent publications on this topic, rendering it a valuable resource for individuals interested in exploring inventory models that incorporate fuzzy demand functions. There was a need for a systematic and complete examination of recent breakthroughs in fuzzy inventory management. Our objective was to provide an illuminating overview of the significant developments in this field and offer insights into the probable future directions of research. Our evaluation of various model components has unveiled new and underexplored territories that may warrant further exploration. Perhaps it would be prudent to consider the possibility of establishing simpler models or incorporating qualitative methods into existing models and initiating a discourse on this topic. Full article

(This article belongs to the Special Issue Mathematical Programming, Optimization and Operations Research)

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24 pages, 975 KiB

Open AccessReview

Multi-Criteria Analysis for the Evaluation of Urban Freight Logistics Solutions: A Systematic Literature Review

by Sandra Alvarez Gallo and Julien Maheut

Mathematics 2023, 11(19), 4089; https://doi.org/10.3390/math11194089 - 27 Sep 2023

Cited by 1 | Viewed by 922

Abstract

The tension between city logistics and its impact on sustainable urban development is evident. Often, local environmental decisions overlook the effects on urban freight logistics, lacking consideration for stakeholders. To address this, utilizing multi-criteria analysis becomes relevant for informed urban planning and management decision making. In this context, this paper conducts a systematic literature review from 2012 to 2022, focused on implementing the multi-criteria analysis methodology to evaluate alternatives for solutions in urban freight logistics. The PRISMA tool was used in the review to select publications and categorize the information obtained to address the research questions. Results display the most prominent authors and publications, authors’ country affiliations, annual publication frequency, research objectives, used frameworks, involved actors, defined evaluation criteria, types of alternatives for solutions considered, and MCDM methods applied. The main finding is that the most commonly used MCDM methods were AHP hybrid followed by MAMCA. In addition, no clear correlation between the pursued objectives and the MCDM methods employed by the researchers is identified. It is important to note that all publications with the highest number of citations use fuzzy methods in their analyses. Full article

(This article belongs to the Special Issue Mathematical Programming, Optimization and Operations Research)

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Planned Papers

The below list represents only planned manuscripts. Some of these manuscripts have not been received by the Editorial Office yet. Papers submitted to MDPI journals are subject to peer-review.

Title: Department of Mathematics, Gujarat University, Ahmedabad-380009, Gujarat, India
Authors: Nita H. Shah; Kavita Rabari; Ekta Patel
Affiliation: Department of Mathematics, Gujarat University, Ahmedabad-380009, Gujarat, India
Abstract: TBA

Title: Effects of human learning in efficient production and management of an environmentally sensitive supply chain under trade credit and inflation
Authors: Richi Singha; S.R. Singhb; Ashok Kumarc; Dharmendra Yadavd
Affiliation: a Research Scholar, Meerut College Meerut, U.P, India b Department of Mathematics, CCS University, Meerut (UP) India c Meerut College, Meerut, U.P, India d Vardhman College, Bijnore, U.P, India
Abstract: TBA

Title: Effect of Renewable energy on an inventory model with Reverse logistics
Authors: S.R. Singh; Dipti Singh
Affiliation: Department of Mathematics, Chaudhary Charan Singh University Meerut (UP), India
Abstract: TBA

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