Optimizing supply chain network for perishable products using improved bacteria foraging algorithm

Journal Pre-proof Optimizing supply chain network for perishable products using improved bacteria foraging algorithm Amit Kumar Sinha, Ankush Anand

PII: DOI: Reference:

S1568-4946(19)30702-1 https://doi.org/10.1016/j.asoc.2019.105921 ASOC 105921

To appear in:

Applied Soft Computing Journal

Received date : 26 December 2018 Revised date : 25 October 2019 Accepted date : 3 November 2019 Please cite this article as: A.K. Sinha and A. Anand, Optimizing supply chain network for perishable products using improved bacteria foraging algorithm, Applied Soft Computing Journal (2019), doi: https://doi.org/10.1016/j.asoc.2019.105921. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

© 2019 Elsevier B.V. All rights reserved.

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Optimizing Supply Chain Network for Perishable Products using Improved Bacteria Foraging Algorithm Amit Kumar Sinha1*, Ankush Anand1#

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1: School of Mechanical Engineering, Shri Mata Vaishno Devi University, Katra, Jammu and Kashmir, India-182320 *: Corresponding Author: [email protected]; (Mob: +91-8493023061; Fax: +91-01991-285687) #: [email protected]; (Mob: +91-9797598684; Fax: +91-01991-285687)

Biography Amit Kumar Sinha

Amit Kumar Sinha is an Assistant Professor in the School of Mechanical Engineering at Shri

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Mata Vaishno Devi University, Katra, India-182320. He is also a Ph.D. scholar from the same University. He received MS in Human and Systems Engineering from Ulsan National Institute of Science & Technology (UNIST), South Korea. He has B.Tech. Degree in Manufacturing Engineering from National Institute of Foundry & Forge Technology

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(NIFFT), Ranchi, India. He has more than 5 years of teaching and research experience at different levels. He works in the area of Evolutionary Computing, applications, modelling and Simulation of Manufacturing System, Supply Chain Management, Planning and Scheduling of Automated Manufacturing System etc. He has published around 10 articles in leading International Journals and is serving as a Reviewer of International Journals including

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Applied Soft Computing, European Journal of Operation Research (EJOR), IJPR, IJPE, Computers and Industrial Engineering, etc. Ankush Anand

Dr. Ankush Anand is an Associate Professor in the Department of Mechanical Engineering at Shri Mata Vaishno Devi University (SMVDU), Katra, Jammu & Kashmir-182320, India. He

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obtained his Ph.D in Mechanical Engineering in the area of Life Cycle Engineering. He has more than 12 years of teaching and research experience at different levels. His areas of research interest include Sustainable Design, Design Optimization, Life Cycle Engineering, Tribology, etc. He has guided a number of U.G and P.G projects and is also Supervising Ph.D students in the areas of Risk Mitigation in Product Design; New Product Development and, Tribology. He has attended several overseas Conferences and has published Research Papers

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in various International Journals of high repute including American Society of Mechanical Engineering (ASME). He is also a member of ASME and is serving as a Reviewer in reputed International Journals.

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Regarding Contact details are as follows:

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Amit Kumar Sinha School of Mechanical Engineering Shri Mata Vaishno Devi University, Katra; Jammu and Kashmir, India-182320 Phone: 91-1991-285-524-2249 Fax: +91-01991-285687 E-mail: [email protected]

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Optimizing Supply Chain Network for Perishable Products using Improved Bacteria Foraging Algorithm Abstract

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In a supply chain environment, time delay has a significant impact on the success of perishable products. A major concern is therefore aimed at development of a holistic optimized approach in a supply chain environment for perishable products. Thus, integration of production, inventory and, distribution of perishable products in a supply chain environment are the challenging tasks for practitioners and researchers. In general, the standard optimal supply chain model cannot work for perishable products. There is therefore, a need for a holistic model that focuses on the consolidation of the processes. Shorter product

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shelf-life, temperature control, requirement of strict tractability, large number of product variants, and a large volume of goods handled are the major challenges in a supply chain environment for perishable products. The present work focuses on the development of a holistic model which uses improved bacteria forging algorithm (IBFA) for solving the

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formulated model. We have proposed and analyzed some general properties of the model and, finally applied it to a three-stage supply chain problem using an IBFA. Two case studies have been considered for support and demonstration of the integrated perishable supply chain network problem. Results obtained from IBFA reveal that the proposed model is more useful

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for decision makers while considering optimal supply chain network for perishable products. Finally, validation of results has been carried out using bacteria forging algorithm (BFA). The computational performance of the proposed algorithm proves that IBFA is instrumental in effectively handling the proposed approach.

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Keywords: Perishable product, Supply chain management, Meta-heuristic algorithm.

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1. Introduction In today’s globally competitive market environment, suppliers of perishable products are facing tremendous pressure in maintaining inventory and supply of optimal order quantity, so as to fulfill customer demands, thus minimizing the cost [1, 2].The period between manufacturing date and trading of perishable products, up to end customer is of major interest

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for both producers and traders. In general, two major concerns for inventory depletion which occur during the replenishment period are fluctuations of market demands and, deterioration of the product [3]. Deterioration is however, a dominating cause for depletion of perishable product in inventory control systems. Decay, damage, spoilage, obsolescence and, decreasing usefulness are the major reasons for deterioration of perishable products [3]. About 15% of the perishable products in a supermarket are lost by retailers due to spoilage and damage [4]. The deterioration of perishable product is therefore, a challenging task for maintaining the economic inventory level, which is confronting many industries such as, pharmaceutical,

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chemical, blood banks and, agro-based industries [5-6]. In this way, it can be observed that effective and efficient inventory control for perishable product is necessary for achieving and competing within the global market [1-6]. This is possible only, if an appropriate supply

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chain is designed for perishable product.

Unlike conventional supply chain network (SCN), SCN of perishable product changes over time , as it depends significantly upon temperature and, weather conditions. A SCN model of melon and sweet corn has been developed by Blackburn and Scudder [7]. Blackburn and Scudder [7] have separately optimized two segments (efficient and

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responsive) of SCN model of perishable product. Therefore, two distinct SCN will be evolved for perishable product, if we consider the SCN developed by Blackburn and Scudder [7]. A SCN model of blood and food items has been proposed by Diabat [8], which will be applicable during disaster situations. In real life scenarios, the conventional SCN cannot be applied for perishable products. Therefore, a robust SCN is necessary for perishable products. An efficient integrated supply chain network model for perishable products is inevitable due to the following reasons:

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 The shorter shelf life of perishable product causes a significant decline in their quality, thus, leading to a drop in sales efficiency during a longer period [9].

 Storage cost will significantly increase since larger volumes of products which do not fail to fulfill the above-mentioned constraints need to be kept in producers’ warehouses.  Order volume will decrease, since customers will prefer other competitor’s products. Page 4 of 40

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 Generally, most of the perishable products are deteriorated due to micro-organism, growth which is a time-dependent function.  The deterioration rate and inventory carrying cost in each period depends on the age of inventory.  Deterioration of the perishable product begins with an initial point of operation cycle

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but customer demands and sales price remain constant throughout the cycle.  Seasonal variations of the product lifetime of the fast moving consumer goods and the pharmaceutical industries.

Therefore, there is a great need to design a specific integrated optimal supply chain network model for a perishable product. Cost effectiveness is another major challenge for handling the shorter shelf-life product /goods [10].

From the flight transportation point of view, transportation of the perishable product is a challenging task because it needs time sensitivity analysis [11]. Some new products also

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show the characteristics of perishability due to rapid changing environmental conditions. During the inventory stock, if products deteriorate or tend to be ruined or destroyed, these products also come under the category of a perishable product. The shorter shelf-life and

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complex distribution network of perishable product demand a specific supply chain network which can handle some of the crucial supply chain issues like inventory, transportation, and distribution, etc.

In general, pharmaceutical industries very often change the chemical composition of the product, therefore, they require some mechanism which can handle the variable product

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decay rate problem [5]. Food industries are also struggling with the issues of perishability during inventory, transportation and, distribution of milk, meat, vegetables, etc [6]. The health sector also uses the concept of perishability for transplanting kidney, heart, lung, etc [8]. During the inventory, after some time, perishable products particularly pharmaceutical products may become partially or fully unfit for utilization. In a similar way, a trade-off between time and amount of resources play an important role during military relief operations in natural calamities like earthquake, flood, tsunami, cyclone, tornadoes, hurricane, etc [8].

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In this paper, we have developed a cost-efficient integrated supply chain network

model for perishable products by considering the selection of the best ordering policy from the pool of ordering policy at the retailer’s end. The proposed model will act as a decisionmaking tool for retailers for making an effective decision during placing of an order for perishable product, on the basis of stock inventory and allocation of perishable products with respect to shelf-life of the product. Supply chain network model for the perishable product Page 5 of 40

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(SCNMPP) is NP-hard in nature, therefore, for solving this proposed model, we need to develop an efficient meta-heuristic based algorithm. There are a large number of metaheuristic algorithms that solve the NP-hard problem. Several evolutionary based algorithms are available and utilized for solving NP-hard nature of problems like Genetic Algorithm (GA) [12], Vector Evaluated GA (VGEA) [13], Weight-based GA (WGA) [14], Non-

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dominated Sorting Genetic Algorithm (NSGA) [15], Strength Pareto Evolutionary Algorithm (SPEA) [16], and Pareto Archive Evolutionary Strategy (PAES) [17].

Although, GA has very good search ability for finding the global optimal solutions, however its speed for convergences is too slow and it takes more time for generating new offspring. On the other hand, Particle Swarm Optimization (PSO) [18] has faster convergence rate as compare to GA but PSO may be trapped in local optima while solving the NP-hard nature of problems. Lots of effort has been made by researchers for developing hybrid version of GA and PSO. However, a majority of the hybrid version of GA and PSO failed to

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depart from what they were, as they kept the original characteristics of GA and PSO remain intake [12, 18]. These algorithms are thus, not the best suitable algorithms for solving NP hard nature of problem, as they have many issues related to complexity and space problem

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[19].

A new concept of optimization based on foraging behaviour of bacteria was formulated by Passino, which is coined as Bacteria Forging Algorithm (BFA) [20]. BFA contains the characteristics of both different types of evolutionary algorithms (GA, PSO) and neural network. Owing to this fact, BFA gives not only global optimal solutions, but also its

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rate of convergence is very fast as compared to GA or PSO. In case of BFA, apart from chemotactic strategy, other bacterial foraging behaviour like swarming, reproduction, elimination, and dispersal is also incorporated for final global optimal solutions. Again the major problem of BFA is (a) attraction effect reduces the speed of convergence and, (b) repellent effect reduces the precision of the algorithm [20]. These problems decrease the efficiency and effectiveness of BFA.

However, in this paper an Improved Bacteria Forging Algorithm (IBFA) based

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solution methodology has been used for solving the SCNMPP problem. The developed methodology has been coded in MATLAB and a case study has been considered for demonstrating the proposed methodology. Analysis of results reveal that IBFA based methodology gives better results as compared to BFA. Section 2 deals with the literature review of supply chain network for perishable product. Section 3 presents the optimization model of the perishable product. Section 4 Page 6 of 40

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discusses the solution methodology based on IBFA. Section 5 presents a case study. A sensitivity analysis is carried out in section 6. Lastly, concluding remarks have been discussed in section 7.

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2. Literature Review

In today’s market scenario, the increasing demand for fast moving life, which increases the living standard of people converges towards a variety of products, which are easily available, natural and, ready to eat. Often these demandable products having shorter shelf-life, and due to complex supply chain network, it is hard to consume the product by the end customer before the expiry date. SCNMPP becomes more complex in terms of inventory, transportation, and distribution due to a large number of perishable product varieties and lastly, the efficiency of SCNMPP model decreases [21-23]. Therefore, the forecasting of

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consumption of all perishable products becomes difficult and time-consuming [24]. The decisions about the level of safety stock in inventory for a perishable product in a supply chain network cannot ensure to full-fill the demand of the end customer [25]. Excess stock

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and complex nature of flaw stock rotation during an inventory of perishable products create another problem namely spoilage of products. Spoilage problem of product is a very sensitive issue and it also badly affects the sales price of the perishable product. For example, the retailers of Nordic countries charge 10% spoilage charges of the total sale price from the end customers [26]. A yearly loss of millions of dollars occurred in the European grocery store

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from the products that are not consumed by end customer before their sell-by date [26]. Therefore, an effective mechanism of stock rotation is necessary for handling the inventory of perishable product for saving and proper utilization of product before the due date of shelflife [27]. Extensive research should be carried out for handling the problem of supply chain network for perishable product and challenges in SCNMPP provide a good research gap under this domain.

Initial research for the deteriorating inventory model based on constant decay rate for

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perishable product has been carried out by Ghare and Schrader [5]. A review on the perishable inventory was conducted by Nahmias [28], where differentiation between the fixed and variable lifetime of inventory have been highlighted. Fixed lifetime goods mean a perishable product can be consumed used up to a given period of time after that it will expire but variable lifetime goods deteriorate during inventory by fixed or variable decay rate. A single supplier with multiple retailer supply chain inventory model for the perishable product Page 7 of 40

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is discussed by Zhao and Yang [29]. A variable decay rate based on stock’s age and production period inventory model of supply chain network for a perishable product has been analyzed by Hsu et al. [30]. Therefore, the concept of distribution centers and warehouses can be incorporated in the supply chain network of a perishable product under the banner of a multi-stage supply chain network.

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A great success of Wal-Mart was possible only due to the optimal utilization of crossdocking concept. Some notable researches on cross-docks include Shaffer et al [31], who analyses the implementation of cross-docking operations. Napolitano [32] investigated the use of cross-docking in SCN. Information sharing during supply chain management of the perishable product is also a critical issue in front of practitioners and researchers [33].Yan [34] proposed a mathematical model by incorporating the internet of things (IoT) and he suggested the optimal opportunity when suppliers should use the IoT for making an optimal profit during supply chain of a perishable product.

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Evaluation of a product service system based on key performance indicators (KPIs) has been asserted by Mourtzis et al. [35]. Performance evaluation of the supply chain network based on rough set theory and extended rough non-cooperative Stackelberg data envelopment

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analysis under uncertain condition has been discussed by Shafiee [36].Yavari and Zaker [37] designed a resilience green supply chain network for perishable product. Soysal et al. [38] proposed an inventory routine problem based stochastic decision-making model of perishable product which is helpful for the logistic industry which is handling the perishable products. A key performance index has been developed by Soysal et al. [38] which inculcate logistic cost,

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waste cost, inventory cost, and environmental pollution cost. Golden et al. [39] have developed a large number of heuristics by using the concept of saving algorithm [40], sweep algorithm [41], and the generalized assignment of Fisher and Jaikumar [42]. Some of the traditional heuristics like Tabu search are outperformed by a new generation of heuristics, including mathematical programming based heuristics and metaheuristics. A column generation methods for solving linear relaxation is developed by Choi and Tcha[43]which also gives an idea of boundary conditions. Column generation method

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developed by Choi and Tcha [43] gives better solution as compared to the heuristic developed by Golden et al. [39]. The evolutionary algorithm has been utilized for solving fleet vehicle routing problem by Ochi et al. [44], and Lima et al. [45] but, the analysis of their results reveals that some better results can be obtained using some advance meta-heuristic algorithm. Some variants of Tabu search based algorithm which is developed by Wassan and Osman [46], and Brandao [47] give better results as compared to Tabu search for vehicle routing Page 8 of 40

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problem. Ronald [48] improved the efficiency of a genetic algorithm using the concept of a hash table for reducing the number of comparisons. Still, the concept of the hash table does not guarantee non-revisits or duplicacy of the solution in the entire search space and this concept only compares a child with the current population. Therefore, Povinelli and Feng et al. [49]; and Kratica [50] used the concept of a small hash table which store the information

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of all visited individual and restrict the duplicity of the solution.

Rafie-Majd et al. [51] have presented the integrated optimization approach of strategic, tactical and an operational decision in an SCM of a perishable product under uncertain demand scenario. The optimization approach proposed by Rafie-Majd et al. [51] is based on Lagrangian relaxation method. Shrivastava et al. [52] have developed an SC network for a perishable product under the unavoidable circumstance of transportation disturbance. Disturbance in transportation gives tremendous impact on the quality of a perishable product. Therefore, the proposed model of Shrivastava et al. [52] provides a

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realistic solution in today's competitive world. In this model, these researchers have incorporated not only the uncertain demand of market but also uncertainty in manufacturer’s logistics decisions. Kara and Dogan [53] have suggested reinforcement learning based

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modeling for the inventory of perishable products under random demand and deterministic lead time for getting minimum total supply chain cost. The results obtained by Kara and Dogan [53] prove that the model is performing well as compared to other existing metaheuristic based models for the inventory of perishable product. In reality, disruption during disaster directly affects the supply chain network for a

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perishable product. In order to handle real-life supply chain network problem of perishable product, Diabat et al. [8] have suggested a resilient optimization model of perishable product which will be a great thing for decision makers. A periodic review inventory model for perishable product has been developed by Fu et al [54].Lu and Ni [55] developed high dimensional multi-objective bacteria forging algorithm for solving supply chain network model. The global optimum solution obtained by Lu and Ni [55] demonstrate better results as compared to results obtained from other meta-heuristic algorithms. A comparative literature

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survey based on supply chain network for perishable products is carried out in Table 1.

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Table 1: Comparative literature survey based on supply chain network for perishable products S. No.

Issues

Concerns

References

1.

Perishable

-54% ($200billion) of total store sales -[56]

products

and 57% of total store shrink is

perishable products in supermarket of

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US.

-Approximately 10% of perishable - [57] products have been wasting before purchase of customers. 2006,

due

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-In

approximately

to

$300

spoilage, -[10]

million

loss

occurs in apple industry of US whose turnover was $1.7 billion roughly -15%

of

perishable

products

in -[4]

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supermarket are lost by retailers due to spoilage and damage.

-The retailers of Nordic countries -[26] charge 10% spoilage charges of the total

sale

price

from

the

end

customers.

Deterioration

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2.

-Exponential

decay

of

perishable -[5]

inventory model of product for zero shortage of inventory perishable

model while considering constant

products

demand. -Continuous deteriorate occurs in real life perishable products like blood,

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medicine, volatile liquids etc. -Order level of inventory model for -[58] perishable product while considering constant rate of deterioration. -Improved inventory model of Shah -[59] -An

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and Jaiswal [58]. inventory

considering

while -[60]

model

salvage

value

while

deteriorating the perishable products at -[61] constant demand.

-Deterministic order level inventory

models for perishable products while

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considering dynamic demands. -Integrated

production

inventory -[62]

model for perishable product for assessing economic order quantity

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(EOQ).

-perishable inventory model for food -[63] industry

applied

for

Kimchi

processing facility in South Korea. 3.

Supply

chain -Supply chain management model of -[8] for perishable product during disasters.

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network

perishable products

-SCN model based on mixed integer -[64] linear

programming

model

for

perishable products.

Modelling

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4.

perishable

of -Milk distribution problem in Greece -[6] is

formulated

in

the

form

of

products based on Heterogeneous Fixed Fleet Vehicle meta

heuristics Routing Problem (HFFVRP)

algorithms

-Modeling of perishable product based -[65]

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on

vehicle

routine

problem

is

formulated, and solved by GA -Modeling of perishable product based -[66] on green routine vehicle problem is formulated and solved by gradient

evolutionary

pro of

objective

many

algorithm

On the basis of the above literature survey, it has been observed that an integrated supply chain network model for the perishable product has not developed in a systematic way which can be utilized for the real-life situation. Normal supply chain network model cannot be implemented for a perishable product. In nutshell, on the basis of existing meta-heuristic

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algorithms, it is observed that a systematic integrated approach is necessary to develop a supply chain network model for perishable product using IBFA. Therefore, in this paper, we have developed an improved bacteria forging optimization-based supply chain model for a perishable product which will be very useful for the decision makers. Some of the major 

A holistic optimized model of production, inventory and, distribution for perishable products is proposed.

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Solution methodology based on Improved Bacteria Forging Algorithm (IBFA) has been proposed.

A decision-making tool has been developed which helps retailers for making an

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

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contributions of this research paper are as follows:

effective decision during placing of an order of perishable product. Inventory of perishable product is a challenging task. Therefore, in the proposed model we have tried to eliminate the inventory issues by using the concept of cross-dock. We have considered the case in which, cross-docks are used in the supply chain instead of warehouses. This serves the following two purposes: (i) Inventory costs are eliminated and, (ii) the time

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taken by perishable product to reach the retailer reduces. 3. Modeling of SCN for Perishable product The problem of optimization in a multistage supply chain environment with perishable products is considered as a three-echelon supply chain consisting of multi vendors (plants), multi cross-docks and multi retailers (markets). A single perishable item is being produced

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and distributed in supply chain and, a coordination policy is assumed to be used to minimize the various costs of production, which includes: setup cost, ordering cost, transportation cost, deterioration cost and, inventory holding costs throughout the supply chain. To achieve this objective, the problem seeks to determine the flow quantities of the single item from vendors to cross-docks and from cross-docks to retailers, along with the transport selection of the

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rented vehicles that are required to conduct that flow. The work proposes a mathematical model for the considered problem that takes into account the capacity limits at each echelon of the supply chain.

The basic concept of deterioration of perishable product has been modeled in the form of inventory, and transportation cost. In most of the cases, the inventory model assumes that the indefinite amount of storage fulfills the requirement of the market. But, in the proposed model, we have tried to develop an integrating approach of inventory model with optimal transportation approach for a perishable product by considering deterioration. The model is

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solved using an evolutionary algorithm named Improved Bacteria Foraging Algorithm. 3.1 Model Formulation

Supply chain network model for perishable product has been adopted from Dolgui et al. [67].

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In this model, there are Np number of plants, Nc number of cross-docks and, Nm number of markets. The product produced at the plant is going to the market through the cross-dock. The required inventory is held at the market itself. Just for clear understanding, we have shown two plant P1, and P2, three cross-docks C1, C2, and C3, and two markets M1, and M2 in

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Figure.1.

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Market 1

Cross-Dock 1

Plant 1

T1

T3

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Cross-Dock 2

T2

Plant 2

T1

Cross-Dock 3

T2

T4

T3

T4

T5

Market 2

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Figure 1: Multi-Stage Supply Chain

In the proposed model, the following assumptions/limitations have been considered: (1) The capacity of plants, cross-docks and, markets is not going to change with respect to time.

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(2) Zero inventory cost has been considered for the product/unit during the production period. (3) Deterioration of the product/unit has been considered only after reaching in the inventory. (4) Back ordering is not allowed.

(5) Deterioration of the product/unit during the transportation is not considered.

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The total cost of supply chain can be divided into five different components; we can express total cost by an expression given below:

Total Cost of Supply Chain = Set-up Cost at plants + Variable Transportation Cost in Transition

+ Fixed Transportation Cost in Transition + Ordering Cost at Markets

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+ Holding Cost at Markets

Parameters and symbol used:

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Number of Plants

Nc

Number of Cross-docks

Nm

Number of Markets

Nt

Number of Time Periods

t t

A fraction of units produced in t1period that deteriorate in period t2

C pc

Cost of transporting a unit from plant p to cross-dock c

Ccm

Cost of transporting a unit from cross-dock c to market m

D pc

Distance from plant p to cross-dock c

Dcm

Distance from cross-dock c to market m

mt

The demand of market min period t

CF

Fixed cost for hiring a single truck with full truckload (FTL)

Cf

Fixed cost for hiring a single truck with half truckload (HTL)

Chm

Holding cost per unit at market m

p

The capacity of the plant p

c

The capacity of the cross-dock c

A Bp Bm Sl N Fpct

N fpct N Fcmt

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m

The capacity of the market m Lost cost per unit.

Setup cost per production period at plant p

Setup or order cost per order at the market m

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pro of

Np

Service level

Number of FTL trucks shipped from plant i to cross-dock c in period t Number of HTL trucks shipped from plant i to cross-dock c in period t

Number of FTL trucks shipped from cross-dock c to market min period t Page 15 of 40

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Number of HTL trucks shipped from cross-dock c to market min period t

Omt

1, if order placed by market min period t; 0 otherwise.

Pct

1, if production occurs at plant pin period t; 0 otherwise

X cmt

Units shipped from cross-dock c to market min period t

X pct

Units shipped from plant m to cross-dock c in period t

X mt1t2

Units produced in period t1held as inventory at market min period t

' X mt 1t 2

Units produced in period t1used to satisfy the demand of market min period t

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

t  t2  ( t1 1)

t  t 2  t1

Det (t )

…

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t t

amount of unit deterioratied in period t 2  amount of unit produced in period t1

pro of

N cmt f

(1)

The deterioration of perishable product increases with time and, is expressed in equation (2).

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 t t   ( t 1) t 12

1

2

… (2)

For perishable product, due to the high rate of deterioration, the amount of unit produced will decrease with respect to time and, is expressed in equation (3).

X mt1t2  X mt t * (t2*  t1 )

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... (3)

The total cost incurred due to deterioration (which produced at time t1) and holding in the inventory (in period t) is expressed in equation (4). ... (4)

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H ( t1t2 , Ymt1t2 )  Chm (1   t1t2 )Ymt1t2  A t1t2 Ymt1t2

Objective Function

The total cost of supply chain is given below. It consists of set up cost, fixed transportation cost, variable transportation cost, holding cost and, ordering cost. The objective here is to minimize the total cost function which is given by:

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



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

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 Nt Np    B P Setup Cost  p pt     t 1 p1   Nt N p Nc  Nc Nm     Cpc Dpc X pct  Cpc Dpc Xcmt  Variable Transportation Cost    t 1  c1 m1   p1 c1   N p Nc Nc Nm Nc Nm  Nt  N p Nc     min CF NFpct  CF NFcmt  C f N fpct  C f N cmt Fixed Transportation Cost   f c1 m1 p1 c1 c1 m1   t 1  p1 c1   Nt Nm Nt    A  C 1  X Holding Cost  mt1t2 hm mt1t2 mt1t2   t 1 m1 t 1    N     p Nc    BmOmt   Ordering Cost      p1 c1

Constraints Capacity constraint at plants Nc

  p Ppt

p, t

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X

lP

…(5)

c 1

pct

... (6)

Capacity constraint at cross-dock Np

X p 1

pct

 c

c, t

... (7)

Units shipped from plants should be equal to the total unit supplied through the cross-docks Np

Nm

p 1

c, t

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 X pct   X cmt m 1

... (8)

Capacity constraint at market Nc

X c 1

cmt

  mOmt

m, t

... (9)

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Np

X p 1

cmt

 X mt' 1t2  X mt1t2

m, t

... (10)

Units Consumed = Units Demanded

t 1

' mt1t2

  mt

m, t

... (11)

Inventory Balance across the periods

1  

mt1 ( t2 1)

Y

mt1 ( t2 1)

 X mt' 1t2  X mt1t2

m, t1 , t2

4. Bacteria Foraging Algorithm

pro of

Nt

X

... (12)

Bacteria Foraging Algorithm has been developed by Passino [20]. The optimization in BFA comprises the followingprocesses:

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Chemotaxis Swarming Reproduction Eliminationand Dispersal

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The activity of bacteria for gathering of nutrient rich is known as Chemotaxis. A cell-tocellmechanism is established to simulate the biological behavior of bacteria swarming. Reproduction is based on the concept of natural selection. During reproduction, only the best bacteria adapted to their environment will survive and is able to transmit their genetic

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character towards the next generation and, those bacteria which are less adapted tend to be eliminated. During the reproduction, diversity of the species remains intact. Diversity of the species is achieved because during elimination and dispersal, event selects parts of the bacteria to diminish and disperse into random positions in the environment. 4.1 Chemotaxis

Tumble and run are the two alternative ways for the movement of E-coil bacterium. In one way unit walk in a random direction is known as tumble whereas, in another way, a unit walk

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with the same direction as the previous step is known as the run. In the chemotaxis process, the movement of bacterium occurs one step of tumble and followed by an uncertain step of a run where, the step of the run depends upon the environmental conditions. During tumble, the position of the bacterium is updated by equation (13).

 i ( j  1, k , l )   i ( j , k , l )  C (i )

 (i )  (i )  (i ) T

… (13)

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Where,

 i ( j , k , l ) : Position of ith bacterium at the jth chemotactic step of the kth reproduction loop in the lth elimination-dispersal event,

C : The size of a step taken in the random direction specified by the tumble

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𝛥: Indicates a vector in the random direction whose elements lie in [-1, 1]

J i ( j, k , l ) : The fitness value of the ith bacterium at  i ( j , k , l )

J min : The minimum fitness value and it is defined as the global optimum Tumble is taken by satisfying the following condition: (i) J ri 1 ( j, k , l ) is better (lower) than J ri ( j, k , l ) (ii) r< Ns Where,

Ns : Maximum number of steps in a run 4.2 Swarming

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J ri ( j, k , l ) : Fitness value of the ith bacterium in the rth step of the run

It is observed that E-coil bacterium has a specific characteristic like sensing and actuation

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which facilitates in the decision making the process. Therefore, during the movement of a bacterium, it releases a signal of attraction to the other bacteria for swarming towards it. Meanwhile, each bacterium also releases a signal of repellent, which facilitates to warn other bacteria for keeping a safe distance from it to avoid the collision of bacteria themselves. In

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the BFA, the above discussed social behavior of bacteria i.e., attraction and repelling effect is modeled in below equation (14). S

J cc ( i ( j , k , l ),  ( j , k , l ))   J cci ( i ,  ) i 1

2 2 P P     S         d attrac tan t exp   wattrac tan t  ( mi  mt ) 2      hrepallant exp   wrepallant  ( mi  mt ) 2   i 1  m 1 m 1    i 1      S

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… (14) Where,

 i  1 ,  2 ,...,  P  : Location of the ith bacterium on the P-dimensional optimization domain T

   i | i  1, 2,..., S  : Position of each member in the population of the S bacteria

 mi : mth Component of  i

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 mt : mth component of position  t for the tth bacterium d attrac tan t : Quantification of attractant wattrac tan t : Rate of diffusion of the chemical signal

wrepallant : Width of the repelling effect J cct  i ,   : Signals released by the tth bacterium

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hrepallant : Magnitude of the repelling effect

J cc  i ,   : Combined attraction and repelling effects received by the ith bacterium J cc  i ,   is time-varying

i

The swarming effect is thus produced by updating the fitness value J ( j  1, k , l ) after each step of tumble and run. The ith bacteria will hill-climb on the updated J i ( j  1, k , l ) :

re-

i

J ( j  1, k , l )  J i ( j  1, k , l )  J cc ( i ( j , k , l ),  ( j , k , l ))

… (15) i

… (16)

4.3 Reproduction

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J i ( j  1, k , l )  J ( j  1, k , l )

The fitness value for the ith bacterium in the chemotactic loop iscalculated by using equation (17).

… (17) Where,

Nc 1

 J  j, k , l  j 1

i

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i J health 

i J health : Health of the ith bacterium

i The smaller value of J health represents the healthier bacterium. For simulating the

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reproduction model all the bacteria are arranged on the ascending order of their health i ) and each of the first Sr  Sr  S / 2  bacteria splits into two bacteria. (smaller value of J health

The characteristics of mother bacteria particularly location and step length are reproduced to the children bacteria. During this reproduction process, all the remaining S r unhealthier

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bacteria are eliminated. In this way, the total number of bacteria will remain constant throughout the steps of the algorithm.

4.4 Elimination-dispersal In this algorithm, global search ability is improved by the elimination-dispersal event. Elimination-dispersal is occurring after the completion of N re steps of reproduction. The

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elimination and dispersion of bacteria to a random position in the search space is defined by probability Ped . Due to elimination-dispersal step, bacteria able to search the global optimum and do not trapped into local optima. N ed has the number of events occurred in this step.

4.5 Improved Bacteria Foraging Algorithm The major problem of BFA is (a) Attraction effect reduces the speed of convergence (b) Repellent effect reduces the precision of the algorithm. These two problems decrease the efficiency and effectiveness of BFA. Therefore, if some mechanism is necessary for BFA

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which can either eliminate or reduces the attraction as well as repellent effects. Without cellto-cell attraction and repellent communication BFA will definitely give a better result and this mechanism is known as improved bacteria forging algorithm (IBFA). Flow chart of

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IBFA is illustrated in Figure 2.

In IBFA,the position of each bacterium after every move (tumble or run) is updated by equation (18).

 i ( j  1, k , l )   i ( j , k , l )  Ccc ( b ( j , k , l )   i ( j , k , l ))

... (18)

where,

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if J i ( j  1, k , l )  J min ( j  1, k , l )

 i ( j , k , l ) : Position of the ith bacteria

J min ( j  1, k , l ) : Fitness value of the best bacterium in the previous chemotactic process Ccc : Attraction factor

In BFA,we assume the step length C as a constant.

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If C is too large: Bacteria may miss the global optimum. If C is too small: Bacteria will take more time to search the global optimum Therefore, the optimum value of ‘C’ is important for finding the global optimum. In other words, we can say that the value of ‘C’ influences the accuracy and efficiency (search speed) of the algorithm.

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For ensuring the bacteria movement, the initial point should be in such a manner that local optimum reached quickly and lastly the movement of bacteria should converge towards the global optimum and it is possible only by adjusting appropriate step length ‘C’ which is modeled in equation (19).

C ( k , l )  L in / n k  l  1

pro of

... (19)

Where Lin is the initial size of the chemotactic step length, n is a constant controlling the decreasing rate of the step length at the kth reproduction loop in the lth elimination dispersal

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re-

event.

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Start Initialize all variables. Set all the loop counters and bacterium indexi to zero.

St op

Increase elimination-dispers ion loo p counter l = l + 1

l
P erform Elim inati on Dispersal

Increase rep ro ducti on lo op count er k = k +1

No

K
pro of

Yes

No

Increase Chemot act ic lo op co un t er j = j + 1

j
No

Increase bact erium i ndex i = i + 1

Let t he it h bact erium t ak e a st ep of h ei ght C (i ) alo ng a random ly generat ed t umble vect or

1. Com put e t he object iv e funct ion val ue for t he it h bact eri um as J(i,j ,k,l), addi ng th e cell t o cell at tracti on effect t o n ut rient concen t rat ion and set Ji +1 = J (i,j,k ,l)

No

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i
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Com pute t he object iv e funct ion value J(i,j+ 1,k ,l) t ak ing int o acco un t t he cell-t o-cell at tracti on affect

Set swim count er m = 0

m
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Yes

m =m +1

Set m= Ns

No

J(i,j+ 1,k,l )
Set J (last) = J(i,j+1 ,k,l ). Swim (let t he it h bact erium t ake ast ep o f heigh t C(k) along t he direct ion of t he sam e t um ble vect or

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Figure 2: Improved Bacteria Foraging Algorithm Flow-Chart

The pseudo code of IBFA is illustrated in Table 2. Table 2: Pseudo code of IBFA

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Calculate J  i, j , k , l   J  i, j , k , l   J cc  i  j , k , l   ,   j , k , l  Set J last  J  i, j , k , l 

 i  j  1, k , l     m' , n' , i, j  1, k , l  Compute J  i, j  1, k , l  m0 While (m  N s ) m  m 1 If

 J  i, j  1, k , l   J  last

J last  J  i, j  1, k , l  Update   i, j  1, k , l 

re-

Recalculate J  i, j  1, k , l 

N c1

i   J  i, j , k , l  J health j 1

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Else m  Ns End IF End While REPRODUCTION Sum

pro of

Tumble Move

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Sort Arrange in ascending cost J health

5. Case study-I

The data sets have been taken for validation of the proposed methodology. Mean value of the

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demand matrix for the different market at a particular time period is illustrated in Table 3. The variance of the demand matrix for the different market at a particular time period is illustrated in Table 4. The distance between different cross-docks and markets are shown in Table 5. In the same way, the distance between plant and cross-docks are shown in Table 6. The required cost parameters and a number of different algorithm parameters are illustrated as follows: Page 24 of 40

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Table 3: Mean Demand Matrix (no of units) 1

Period

2

3

4

5

Market 5500 6500 4500 5500 4000

2

5000 5800 7500 8500 3000

3

6500 7000 5500 7000 4500

4

7500 8500 5000 6500 7000

pro of

1

Table 4: Demand Variance Matrix (no of units) 1

Period

2

3

5

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Market

4

500

500

600

500

400

2

500

800

500

850

300

3

500

600

500

7000 350

4

500

450

400

600

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1

700

Table 5: Cross-dockMarket Distance Matrix (Kilometer) 1

2

3

4

1

48

72

60

50

2

67

91

49

74

3

42

55

89

93

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Market

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Cross-dock

Table 6: PlantCross-dock Distance Matrix Cross-dock

1

2

3

4

3

6

Plant 1

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2

5

4

9

Cost Parameters

FTL = 250 Units HTL = 125 Units

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A = 0.025 / Unit time B = 0.72 Period ff= Rs. 5000 fh= Rs. 3000 Set up cost for plants = Rs. 10000 Ordering Cost for all markets = Rs. 1000 Production Capacity = [50000, 50000] Capacity of each market = 10000 Algorithm Parameters

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Capacity of each cross-dock = 100000

Dimension of search space

s=25;

Number of bacteria

Nc=50;

Number of chemotactic steps

Ns=4;

Limits the length of a swim

Nre=4;

Number of reproduction steps

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p=2;

Number of elimination-dispersal events

Sr=s/2;

Number of bacteria reproductions (splits) per generation

Ped=0.25;

Probability that each bacteria will be eliminated/dispersed

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Ned=2;

5.1 Sensitivity analyses Results using Bacteria Foraging Algorithm are listed in below Table 7:

Table 7: Results Obtained from Bacteria Foraging Algorithm

2 3 4

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No of Inventory Holding Transportation Instances and Lost Cost ($) ($) 1 70660 976200

Cost

Total Cost ($)

1725976

74188

961500

1693534

67996

957000

1741279

79896

980700

1806456

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77376

999800

1701400

6

44868

1094000

1732935

7

78236

1059000

1655988

8

44836

983600

1801959

9

58080

1031600

1697992

10

78200

1062500

1799802

pro of

5

From the results, it can be observed that the contribution of transportation cost to total cost is more than half in almost all cases. One can judge that transportation cost is very crucial in terms of cost minimization. The effort made to reduce the transportation cost will lead to a low total cost (see Table7).

re-

Results using Improved Bacteria Foraging Algorithm are listed in below Table 8: Table 8: Results Obtained from Improved Bacteria Foraging Algorithm No of Inventory Holding Transportation Instances and Lost Cost ($) ($)

45460

2

84020

3

54280

4

66410

5

96185

7 8 9

1641957 1669781

1002400

1688767

946000

1740842

988200

1647544

47210

976600

1720374

49775

995100

1676514

57570

1010000

1794951

63095

1070700

1668004

49160

1054100

1716699

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10

Total Cost ($)

1021400

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6

953400

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1

Cost

The Table 8 above shown represents results obtained using Improved Bacteria Foraging Algorithm, further we have compared the different costs obtained by Bacteria Foraging Algorithm and Improved Bacteria Foraging Algorithm. Comparison of Inventory Holding cost for both algorithms is shown in Figure 3.

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4

10

Combine Results of BFA & IBFA

x 10

BFA IBFA

8

pro of

Inventory Holding & Lost Cost ($)

9

7

6

4 1

2

3

4

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5

5 6 7 Number of Instances

8

9

10

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Figure 3: Comparison of Holding Cost If we analyze the graph of inventory holding cost for Bacteria Foraging Algorithm and Improved Bacteria Foraging Algorithm, it can be observed that the Improved Bacteria Foraging Algorithm has performed better except for very few points (see Figure 3).

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Minimum inventory holding cost is almost the same for both the algorithm. Now let us compare the transportation costs and total costs obtained by Bacteria Foraging Algorithm and Improved Bacteria Foraging Algorithm. Comparison of Transportation cost for both

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algorithm is shown in Figure. 4.

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6

1.1

Combined Results of BFA & IBFA

x 10

BFA IBFA

1.08

pro of

Transportation Cost ($)

1.06 1.04 1.02 1 0.98

0.94

1

2

3

4

re-

0.96

5 6 7 Number of Instances

8

9

10

Figure 4: Comparison of transportation cost

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When we compared the transportation costs for Bacteria Foraging Algorithm and Improved Bacteria Foraging Algorithm, we found that for few times Bacteria Foraging Algorithm performed better than Improved Bacteria Foraging Algorithm but if we look at the minimum transportation cost it almost same even smaller for Improved Bacteria Foraging Algorithm.

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For a few points, it can be observed that the difference in transportation cost is large and, Improved Bacteria Foraging Algorithm has performed better for these points (see Figure 4). Now finally, we will compare the total cost of the supply chain resulted from Bacteria Foraging Algorithm and Improved Bacteria Foraging Algorithm. Comparison of Total cost

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for both algorithms is shown in Figure 5.

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6

1.82

Combined Results of BFA & IBFA

x 10

BFA IBFA

1.8 1.78

pro of

Total Cost ($)

1.76 1.74 1.72 1.7 1.68

1.64

1

2

3

4

re-

1.66

5 6 7 Number of Instances

8

9

10

Figure 5: Comparison of the total cost

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Now finally, we compared the total cost of the supply chain resulted by Bacteria Foraging Algorithm and Improved Bacteria Foraging Algorithm. If we look closely at the total cost graph almost all the points for Improved Bacteria Foraging Algorithm are below the graph obtained by Bacteria Foraging Algorithm. The minimum total cost has also resulted in an

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Improved Bacteria Foraging Algorithm. For most of the points, Improved Bacteria Foraging Algorithm has performed better (see Figure 5) but the difference is not very large in the minimum cost obtained by both the algorithms.

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6. Case study-II (Hajiaghaei-Keshteli and Fard[68]) A real life industrial example based on manufacturing centre has been considered for

validating our proposed model. This example is taken from Hajiaghaei-Keshteli and Fard [68]. A detailed information about the case study can been found at Hajiaghaei-Keshteli and Fard [68]. Some of the raw date has been considered in Table 9, Table 10, Table 11, and Table 12, and Table 13. In this case study, there are four manufacturing centres (A, B, C, and Page 30 of 40

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D); Seven Distribution centers (E, F, G, H, I, J, and K); three collection centers (L, M, and N); two recycling centers (O, and P); and two recovering centers (R, and S).In this case study,

t t

12

(Which is a fraction of units produced in t1period that deteriorate in period t2) is

considered 0.82. Manufacturing Fixed Centers opening cost ($) A B C D

134 168 295 165

pro of

Table 9: Details information about manufacturing centers Manufacturing Capacity of Cost/unit manufacturing ($/unit) centers (Units) 128 1300 160 1100 147 1560 155 1700

Table 10: Detail information about distribution centers Fixed opening cost ($)

Handling cost /unit ($/unit)

E F G H I J K

68 56 48 42 38 72 64

78 92 102 88 89 73 76

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Capacity of distribution centers (Units) 1500 1700 1860 1700 1750 1150 1150

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Distribution Centers

Table 11: Detail information about collection centers Fixed opening cost ($)

Handling cost /unit ($/unit)

L M N

48 46 34

18 12 16

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Collection Centers

Capacity collection centers (Units) 100 170 110

of

Table 12: Detail information about recycling centers

Distribution

Fixed

Handling cost

Capacity

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Centers

opening cost ($)

/unit ($/unit)

O P

68 74

14 12

collection centers (Units) 300 310

pro of

Table 13: Detail information about recycling centers Recycling Centers

Fixed opening cost ($)

Handling cost /unit ($/unit)

R S

58 43

11 10

Capacity collection centers (Units) 400 510

of

Now the combined results of BFA & IBFA is illustrated in Figure 6. The results reveals that IBFA is performing better results as compare to BFA. The results obtained in Case study-II

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confirms the robustness of the proposed model in terms of IBFA. Combined Results of BFA & IBFA 6

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4

3

2

1

1

2

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0

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Total Cost in X1000$

5

BFA IBFA

3

4

5 6 7 Number of Instances

8

9

10

Figure 6: Comparison of the total cost

7. Conclusions

In this work, a model for the optimization of total cost for perishable items in a multistage supply chain has been developed. The total cost consists of set up cost, variable transportation

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cost, fixed transportation cost, inventory holding cost and, ordering cost has to be minimized. For minimization of total cost techniques of evolutionary algorithms have been used. We have used Bacteria Foraging Algorithm (BFA) and Improved Bacteria Foraging Algorithm (IBFA) to solve our model. Improved Bacteria Foraging Algorithm differs from Bacteria Foraging Algorithm in terms of cell-to-cell communication; Bacteria Foraging Algorithm

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considers cell-to-cell communication whereas, Improved Bacteria Foraging Algorithm does not.

Results obtained from both the algorithms indicate that out of total cost more than half is contributed by transportation cost. This means that we should try to minimize the transportation cost by other means, if possible, reduce transportation cost will lead to a minimum total cost.

When we compared the results obtained by Bacteria Foraging

Algorithm with Improved Bacteria Foraging Algorithm, we observed that Improved Bacteria Foraging Algorithm gives better performance. In this research, not only development of a

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holistic optimized model of production, inventory and, distribution for perishable products is formulated but also a solution methodology based on Improved Bacteria Forging Algorithm (IBFA) has been analyzed. Lastly, a decision-making tool has been developed which helps

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retailers in effective decision making while placing an order for perishable product. The objective of this research is to analyse the degree of variation by comparing the IBFA with existing BFA. However, this may be future scope of work, where algorithms like RCGA, NSGA-II etc. may be analysed in the similar type of work.

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Acknowledgement

The authors would like to thank the Editor and the anonymous reviewers for their constructive and valuable comments. References

1. Mallidis, I., Vlachos, D., Yakavenka, V. and Eleni, Z., 2018,“Development of a single period inventory planning model for perishable product redistribution,” Annals of Operations Research, pp.1-17.

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2. Vaughan, T.S., 1994,“A model of the perishable inventory system with reference to consumer-realized product expiration,” Journal of the Operational Research Society, 45(5), pp.519-528.

3. Ali, S. S., Madaan, J., Chan, F. T., & Kannan, S. 2013, “Inventory management of perishable products: a time decay linked logistic approach”, International Journal of Production Research, 51(13), pp. 3864-3879. Page 33 of 40

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4. Ferguson, M. E., & Ketzenberg, M. E. 2005, “Information Sharing to Improve Retail Product Freshness of Perishables” (ed. 3). 5. Ghare, P. N., and Schrader, G. F., 1963, “A model for exponentially decaying inventories,” Journal of Industrial Engineering, 15, pp. 238–243. 6. Tarantilis, C. D., & Kiranoudis, C. T. 2001, “A meta-heuristic algorithm for the 9.

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efficient distribution of perishable foods”, Journal of food Engineering, 50(1), pp. 17. Blackburn, J., & Scudder, G. 2009, “Supply chain strategies for perishable products: the case of fresh produce”, Production and Operations Management, 18(2), pp. 129137.

8. Diabat, A., Jabbarzadeh, A., & Khosrojerdi, A. 2019. “A perishable product supply chain

network

design

problem

with

reliability

and

disruption

considerations”, International Journal of Production Economics, 212, pp.125-138.

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Highlights



A holistic optimized model of production, inventory and, distribution for perishable products is proposed



Solution methodology based on Improved Bacteria Forging Algorithm (IBFA) has

A decision making tool has been developed which helps for retailers for taking

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effective decision during placing of order of perishable product

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

pro of

been proposed

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