Designing an optimal multi-objective model for a sustainable closed-loop supply chain: a case study of pomegranate in Iran

Abstract

The competitive environment in the global market makes most countries look for better ways to solve their problems. Food waste is the largest concern facing the food security of the world. Not paying attention to process of pomegranate wastes, such as separating the peel from the other parts and ignoring the cost of using artificial intelligence for pest control in gardens and the cost of maintaining the processed products are the gaps of previous researches. To cope with this challenge, recent studies have presented sustainable closed-loop supply chains (SCLSCs) as a strategic approach and a competitive advantage. The present study distinguishes itself from other studies by using the artificial intelligence technology in a supply chain along with the reverse logistics section, i.e., waste recycling. This paper proposes a design for a CLSC pomegranates. The corresponding logistics network is designed for several periods and covers manufacturers, distribution centers, customers, factories, recycling centers (compost centers), and compost end user (compost markets). Using reverse logistics, the wasted pomegranates are also converted into recycled products including ethanol, as an automotive fuel and a renewable energy, and a type of compost processed as an organic fertilizer. The goal of proposed model is to minimize the costs of supply chains, reduce the supply risks involved, and increase the profits for gardeners and investors in the public and non-profit agriculture sectors in Iran. The first pareto solution is 1,869,908.962, 2172.638 and 65.926, and the CPU time is 412 Ms. The results show a rise in the maximum supply risk occurs in the total cost and risk but a reduction in the accountability of the network and also an increase in the disruption period findings in increased total cost and risk of the network, while it first increases and then decreases the accountability.

Introduction

Nowadays, supply chains (SC) serve as structured circulating complexes that undertake both forward and backward flows. A network of members that work together in a supply cycle is called a closed-loop supply chain (CLSC) (Jabbarzadeh et al., 2016). Define of a SC plays a significant role in its management. SC often face a different of uncertainties as in demand and costs (Tan et al., 2019). Since customers have many choices and resources to meet their needs, it is vital for suppliers to offer their final products in the distribution network at the lowest price and with the best accessibility. The need to improve operations, the growing importance of the global trade, transportation costs, and intense competitive pressure are the major reasons for the necessity of supply chains. The absence of a overall management and information system and a practical model in the country's SC of agri-food products has caused this sector to not be efficient despite its potential. One of the biggest problems in the agricultural and food sectors in any country is the gap of knowledge of farmers about the optimal planting of agricultural products based on the demand in that society. Disruption of this equality causes us to face a surplus of a product throughout the year, which causes the price of the product to drop and farmers face financial loss. In addition, by causing a decrease in the amount of other products, the price of the product increases. This requires the study of logistic models in the agri-food sector (Takavakoglou et al., 2022). Defective products and waste have always been an important challenge for agri-food producers. They have found that product recycling and reuse of products, waste and product residues not only reduce harmful effects for the environment, but also improve their competitive position in the market. From this point of view, the efforts in the field of recycling and logistics activities have caused organizations to focus on closing the SC loop and creating a CLSC. This structure can achieve the economic, social and environmental goals of the organization at the same time (Tirkolaee et al., 2022).

With the fast agricultural industrialization, growing world wide food request and raise challenge about food quality and safety, the concept of transparency is gaining increasingly importance in the agricultural sector. The sustainable management of these circumstances is done by business managers who often make small benefits and have to meet the strict requirements imposed by great customers and retailers (Mangla et al., 2018). An agriculture-food supply chain (AFSC) is a system that includes raw material producers, processors, distributors and consumers (Borodin et al., 2016; Hu et al., 2019; Utomo et al., 2018). Increasing customer information about products has made food safety the most important factor for producers.

Based on the Food Agricultural Organization (FAO), industrial pomegranate-processing centers produce estimately 1.5 million tons of waste yearly, with tremendous nutritional value. Different parts of pomegranate other than aryl are considered as agricultural waste; due to their antioxidant effects, they are used as additives in the cosmetics industry (Borodin et al., 2016). Pomegranate skin and juice have been used as anti-inflammatory in the treatment of diseases for a long time in traditional Chinese medicine (Utomo et al., 2018). A recent study (Hu et al., 2019) reinforces the benefits of pomegranate as a fruit that improves metabolic syndromes. Metabolic syndrome is a common term in medical science that refers to the diseases of diabetes, high blood pressure and obesity. This disease causes people suffering from it to have a heart attack or stroke. The extraction of ethanol from pomegranate waste and peel is another benefit of the fruit which is discussed in the present study. Ethanol has applications in car fuel production (Kalaycıoğlu & Erim, 2017; Liu et al., 2017).

Although several studies have addressed horticultural issues in supply chains, this study addresses reverse logistics and the return of products from different sectors; it focuses on rotten and wasted pomegranate fruit as a research gap. To this end, the research uses artificial intelligence to reduce pest damage and the rate of pre-harvest rotting. The rotten and pest-infested pomegranates are identified and processed by images (Silva et al., 2013) and go from the garden to the compost center to become a fertilizer (Mangla et al., 2018). The ability to use computer imaging and combine it with artificial intelligence algorithms in agriculture is significant (Dhande et al., 2021). Computer image processing has flourished owing to reduced equipment costs, promoted computing power, and increased attention to food evaluation methods (Demiray et al., 2019). The technique offers advantages over traditional manual methods (Fashi et al., 2019).

Agriculture sector is one of the most important economic because it provides valuable rates of employment, GDP and non-oil exports, in addition to meeting the food needs of many countries. A major subsector of agricultural is horticulture, where the diversity and nutritional value of products is of importantance. Therefore, it is necessary to conduct scientific research to identify the influencing factors in this field. As it has often been the case, customers who care about healthy food prefer to buy fresh fruits. This makes it significant to guarantee the quality and availability of fruits all the year round. Over the past decade, the agri-food industry in general and the fruit sector in particular have considered supply chains as critical tools for competitiveness. Pomegranate is one of the favored fruits globally, and the demand for it and its derivatives is increasing. The popularity of pomegranate and the willingness of customers to pay for it have made producers turn to this market more than before, which has increased the global competition for pomegranate products. Some countries have established pomegranate-processing plants and generated income and employment through exports.

The aim of proposed modeling is to minimize the cost and risk of the SC and maximize the profits of gardeners and investors in the public and Non-profit agriculture sectors in Iran. The research question are as follows:

• Where are distribution centers, recycling centers and factories established and how much is the flow between established centers?

• How to manage the severity of disruption or natural disasters, risk and uncertainty in the pomegranate SC?

• How to assess the relationship between appearance, color and size of pomegranate with artificial intelligence approaches?

The unique innovation of this paper as main contribution are as follows:

• Defining a sustainable multi-level CLSC including producers, distributors, customers, compost centers, compost customers, and factories, their customers in the field of food and energy production

• Considering image processing to determine the relationships among pomegranate appearance, color and size.

• Processing of pomegranate wastes, such as separating the peel from the other parts for conversion and reprocessing

• Considering the cost of using artificial intelligence for pest control in gardens and the cost of maintaining the processed products

• Using NSGA-II and MOPSO methods to solve the mathematical model and comparing the performance of them

• Considering risk and uncertainty in pomegranate supply chain in Iran

This paper is organized in several sections. The theoretical foundations, literature review and research background are provided in the Sect. 2. In the third section addresses the mathematical model and its whole components designed for a pomegranate CLSC. A new solution to the mathematical model is presented in the Sect. 4. The Sect. 5 solves the model through the new solution method. Finally, the conclusion of the study and the suggestions for further study end up the paper.

Literature review

The literature on SCM is very widely, but designing sustainable closed-loop networks has not received much attention in supply chain models (Patrício & Rieder, 2018). Recently, the fresh fruit sector in the agri-food industry has focused on CLSC only as a important concept for competitiveness (Mahajan et al., 2015). The idea of a CLSC has been around since at least the early industrial age, when merchants realized that old clothing, bedding, etc., could be repurposed into new textile products. The business emerged slowly and came to the fore in the 1980s, when environmental issues became a sensitive and significant issue. This led to the emergence of the "general and hazardous" waste processing and recycling business. A brief account of the most relevant studies in this regard is presented here. Based on the obtained knowledge, no study has been reported on sustainable closed-loop supply chains for pomegranate.

Agricultural SC

A conventional SC for agricultural products consists of three components. The first part: sending the product from the farmer to the intermediate silos. The second component: sending from intermediate silos to conversion factories and the third component: sending from conversion factories to final customers. In each of the mentioned stages, various decisions are needed, and each decision may make the process of optimization and problem solving complicated. One of the first studies on various agricultural foods, both perishable and non-perishable, as well as fresh fruits and vegetables in SC was studied by Barbedo (2016). Integrated planning for the multifunctional manufacture and distribution of perishable products was also done by Hajiaghaei-Keshteli & Aminnayeri (2014). Another study (Ahumada & Villalobos, 2009) aimed at an optimal planning model of transporting fruit to logistics centers according to their demand in non-harvest seasons (Ahumada & Villalobos, 2009). The main aim and innovation of this paper is minimizig the cost of the SC. Numerous studies are still being done in this field. As some examples, one may refer to Amorim et al.(2012), Plà-Aragonés (2015), Carvajal et al. (2019), Cheraghalipour et al. (2019), Gholamian & Taghanzadeh (2017), Paydar et al. (2018), Alaoui et al. (2018), Sharma et al. (2020) and Kamble et al. (2020). To assess research gaps, a summary list of several other articles is provided in Table 1. In addition, for a detailed study of these articles, two charts are shown in Fig. 1. Chart (A) shows that 88% of these selected studies are single- and multi-objective. Based on Chart (B), however, it can be concluded that no study has been done on a pomegranate SC yet.

Table 1.

Comparison of the models reported in the literature for the SC of agricultural products study

Authors/year	Type of product	Model structure	Kind of model	Solution Approach	Goal	Scope	Implication
Bohle et al. (2010)	Grapes	MILP	Single-objective optimization	Exact	Reduce the harvesting schedule	Grape harvesting planning in a supply chain	Numerical
Asgari et al. (2013)	Wheat	ILP	Single-objective optimization	Exact	Minimize transportation costs	Creating the necessary platform for the optimization process of wheat storage and transportation	Numerical
Catalá et al. (2016)	apples and pears	MILP	Multi-objective optimization	Optimization criteria in the lexicographic method	Minimize the demand violation, maximize the economic benefit of the system	Optimizing the model of supply chains in the fruit industry	Case
Gholamian et al. (2017)	Wheat	MILP	Single-objective optimization	Exact	Minimize total cost	Design of Integrated network of a wheat SC	Case
Cheraghalipour et al. (2019)	Citrus fruits	MILP	Bi-level optimization	Metaheuristic, Hybrid metaheuristics	Minimize total cost	Locating allocating modeling to optimize total costs	Case
Cheraghalipour et al. (2019)	Rice	Nonlinear programming	Two-stage optimization	Metaheuristics	Minimize total cost	Defining and solving a two stage optimization model for a rice SC	Case
Abdolazimi et al. (2021)	Citrus fruits	MILP	Multi-objective optimization	Hybrid metaheuristics	Minimize total cost	Define a CLSC for the walnut industry	Case
Ahmadi Choukolaei et al. (2021)	Walnut	MILP	Multi-objective optimization	Exact, Metaheuristic, Hybrid metaheuristics	Minimize total cost	Design a CLSC for the walnut industry	Case
Chouhan et al. (2021)	Sugarcane	MILP	Multi-objective optimization	Metaheuristic, Hybrid metaheuristics	Minimize total cost	Designing a 2E-CLSC for sugarcane	Case
This study	Pomegranate	MIP	Bi-level optimization	NSGA-II, MOPSO, GAMS	Minimize total cost, Minimize the supply risk and maximize responsiveness to customers request	Defining and solving a two stage optimization model for an agriculture SC	Case

Fig. 1.

Charts based on Table 1

Babatunde (2019) proposed an agricultural supply chain including emission reduction considerations and carbon taxes. They also allowed investment cooperation in their chain and designed a game theory (GT) model to compare different decisions and retailers’ benefit. In another study on agri-supply chain, Mukherjee et al. (2021) used the blockchain technology for sustainable development. They benefited from the opinions of managers in the agricultural sector regarding data privacy, decentralisation and smart contracts. Liu et al. (2021) presented a framework for the use of organic manure instead of chemical fertilizer, and concluded that an increase of subsidies would positively affect the use of organic manure. The benefits thus gained are important for the environment. Mukherjee et al. (2021) also applied a game theory in a blockchain to deal with the problem of an agricultural SC. The goal was to do about the sustainability in the operations of the chain. To test their model at a green level, they designed a numerical simulation and assesed the effect of the green policies on the evolution behaviour of the model. As it is clear from Table 1 and Fig. 1, there are few studies that systematically examine the pomegranate bi-level CLSC in a real case study according to sustainability criteria. In addition, few studies used image processing to determine the relationships among pomegranate appearance, color and size.

Farming with artificial intelligence (AI)

Nowadays, the use of artificial intelligence (AL) is very critical for many problems that have a complex solution and require intelligent solutions. (Jiang et al., 2022). For instance, the diagnosis of fruit diseases is possible using modern image-processing technologies and machine learning. Banana production faces many diseases that cause great harm to poor farmers. With a modern image processing technology, these cases may be identified on time, and appropriate precautions may be taken to prevent further damage, thus enhancing healthy production (Song et al., 2022). Chen et al. (2021) designed a model and a framework to distribute fresh agricultural products using IoT technology. Generally, it is essential to make improvements in the production process, marketing and establishment of agricultural quality control systems and auxiliary measures to enter international markets. For this purpose, agricultural production can be mixed by information technology and Internet objective technology services (Bohle et al., 2010).

In this study, image processing was used to determine the relationships among pomegranate appearance, color and size. Generally, the color and size of pomegranate fruit can be measured by removing its skin. Also, it is classified manually. Therefore, image processing and artificial intelligence have been developed to grade pomegranates based on their color and size (Hajiaghaei-Keshteli & Aminnayeri, 2014). In another study, a method was devised to identify pomegranates on a tree and find their total number using the close-up images taken in the corresponding garden. Since pomegranate has a noticeable red color, a color-based method can identify it on a tree (Fig. 2). The model for the pomegranate SC presented in current study considers the costs of using artificial intelligence in agriculture.

Fig. 2.

Identification of pomegranates on the tree using image processing

In Fig. 2, the images are as follows:

• (a) Source image (pomegranate tree).

• (b) Black and white picture (segmentation, color indexing).

• (c) Picture after the applied threshold (figure analysis).

• (d) Location of geometric centers (estimated count of the pomegranates).

Risk and uncertanty in the SC

Risks play a main role in making supply chains vulnerable and lowering their performance and competitive advantage (Asgari et al., 2013). There are some studies that have addressed this issue. Romero (2000) proposed an efficient cultivation model considering the producers' risk. In another study (Vijayakumar & Balakrishnan, 2021), a bi-objective randomized CLSC network was designed, and the negative aspects of risks as a criterion were included in the objective functions. In the same line of research, risk criteria were integrated to a CLSC network design (Athiraja & Vijayakumar, 2021). A critical issue in a supply chain is resource constraints, which has been dealt with through product recycling. This is a process in reverse logistics through which products are collected from customer and carried to production centers to be recycled. Regarding the risks in this context, a study was conducted to define a CLSC for the motor oil sector (Chen et al., 2021). As it has also been shown (Fakhrzad et al., 2020), by considering an improved neural network and a rooted heuristic algorithm (RRA), the future demands of each product have been estimated based on the calculation of the product risk index.

Some issues are affected by uncertainty in the SC. For example, regional political crises, demand changes, financial instability, strategy changes and natural disasters. Therefore, a mathematical model was developed to measure typical risks such as delivery delays, non-standard quality, natural disasters, and supplier’s financial risk (Romero, 2000). Supply uncertainty is due to the uncertainty of maturity, harvest time and performance. Demand uncertainty is the uncertainty in retailers’ weekly request. A method was proposed to determine the farming areas and the cultivation time of the annual plants that survive only for one raising season so as to maximize the total expected benefit (Fard et al., 2017). New SC network models were also designed with different decision criteria under combined uncertainties to reduce the expected total costs (Soleimani et al., 2014). Sriyanto et al. (2021) evaluated the impact of health system SCM in the COVID-19 pandemic times. They used the regression model to estimate the data in the future. The calculation results confirmed the existence of a relationship between the logistics efficiency index of the supply chains of health systems and the cases of COVID-19. Kumar et al. (2022) investigated the potential effect of Internet of Things (IoT) on the performance of the vaccine SC. To show the model validation, the state of Bihar in India has been selected. The results show that there is a positive relationship between the four components of the chain, including product, supply, demand and social behavior in adopting the Internet of Things. Alhawari et al. (2021) presented body of knowledge of circular economy supply chain literature by presenting a better understanding of the circular economy concepts by examining the different definitions. The results show that studies in the field of Industry 4.0 and circular economy, despite the fact that it has been highly regarded by analysts, but there are still study gaps for it. Usman et al. (2022) have studied a review study on the effect of weather change on the sustainability of supply chains of health systems. Based on this study, carbon and profit pricing indicators have been considered. The obtained results show that weather change predicts a positive impact on the decreasing trend of corona virus infection. Awan et al. (2022) investigated the value distribution factor in industrial revolution 4.0 and circular economy. The results show that value chain activities have a positive effect on industrial revolution 4.0 and circular economy. Ahmadi Choukolaei et al. (2021) presented a model for bioethanol consumption experienced due to unexpected worldwide event. Utilizing machine learning approaches to estimate the bioethanol demand was the main contributions of the paper. Designing sustainable bioethanol supply chain by two meta-heuristic algorithms are the main goal of their research. One of the most important practical aspects of this study is providing favorable policies like as determining the favorable location for facilities that are used in the network.

In order to provide more information about the research in this area, some of the reviewed studies are itemized in Table 2.

Table 2.

Supply chain references itemized according to the applied uncertainty approaches and the types of risk

Authors	Uncertainty approach		Type of risk		Type of supply chain
	Probability	Fuzzy numbers	Operational	Disruption
Aqlan and Lam (2016)	✓		✓		–
Lenort et al. (2016)	✓			✓	–
Zahiri et al. (2017)		✓	✓	✓	Sustainable SC
Behzadi et al. (2017)	✓			✓	Agricultural
Ivanov (2018)	✓			✓	Sustainable global supply chain in electronics
Ghomi-Avili et al. (2018)		✓	✓	✓	Green competitive CLSC
Mohammed et al. (2019)		✓	✓		Green SC
Zhao and You (2019)	✓		✓	✓	Biofuel supply chain
Bottani et al. (2019)	✓		✓	✓	Food supply chain
Zare Mehrjerdi and Lotfi (2019)	✓		✓	✓	Sustainable closed-loop
Ahranjani et al. (2020)		✓	✓	✓	Sustainable, Agricultural
Tucker et al. (2020)	✓			✓	Drug SC
Yavari and Zaker (2020)	✓			✓	Green CLSC
Hosseini-Motlagh et al. (2020)				✓	Blood SC management
Nguyen et al. (2021)	✓		✓		–
Momenitabar et al. (2022b)	✓		✓		Sustainable SC

According to the literature review mentioned above, the gaps identified for this study are stated as follows:

• Ignoring sustainability and concept of CLSC network in the field of food and energy production in the past research

• Ignoring image processing to determine the relationships among pomegranate appearance, color and size.

• Not paying attention to process of pomegranate wastes, such as separating the peel from the other parts

• Ignoring the cost of using artificial intelligence for pest control in gardens and the cost of maintaining the processed products

• Ignoring the use of metaheuristic methods to solve the mathematical model and comparing the performance of them

• Not paying attention to risk and uncertainty in pomegranate supply chain in Iran

Problem definition

Current paper proposes a design for a CLSC of supplying pomegranates. The corresponding logistics network is designed for several periods and covers manufacturers, distribution centers, customers, factories, recycling sites (compost sites), and compost sites (compost markets). As depicted in Fig. 3, goods are shipped from the producer to the consumer, distribution centers and factories in the forward flow. In the considered network, fruits are produced as products in a forward flow during three periods, which are the maximum time of mass production periods. In this flow, the customers also receive their goods from the distribution centers, and the demands are met by the manufacturer. Distribution centers ship products within the eight time periods that are assumed to be the maximum storage time. The fruit demand period is eight months too. Besides, the customer locations are considered fixed.

Fig. 3.

Flowchart of the proposed CLSC

The factories in the supply chain obtain the fruit they need from the producers and the distribution centers in the forward part of the chain. A factory consists of a food and pharmaceutical sector that serves to prepare the products and send them to the market. Another feature of this chain is the processing of pomegranate wastes, such as seperating the peel from the other parts for conversion and reprocessing. As Fig. 4 shows, ethanol can be extracted from pomegranate peel and the other components of the fruit waste (Paydar et al., 2017); one of the applications of this extract is in car fuel production (Barbedo, 2016; Mahajan et al., 2015).

Fig. 4.

Process of extracting bioethanol from pomegranate wastes

Rotten fruits on trees transmit their pests to other fruits. To prevent the spread of pests to the whole garden, the spoiled fruit is identified by image processing and artificial intelligence and carried out of the garden to compost centers. The returned fruit is sent to vermicompost centers, converted into organic fertilizers, and transferred to compost customers in the reverse flow. Since the producers (orchards) can be the customers of fertilizers as well, the network is considered as a supply chain of cyclic packages where the producers are the same as the compost customers (Fig. 5).

Fig. 5.

Flow of the returned pomegranate and its conversion to vermicompost

Millions of tons of biowaste are dumped or incinerated every year, which causes many environmental problems and imposes enormous costs of transporting, disposing, and incinerating the wastes. One of the main ways to reuse organic material waste is to transform it into vermicomposting. Therefore, in addition to the production of vermicomposting organic fertilizer, which has many advantages, it is effective in preserving the environment and the health of society. This useful organic bio fertilizer is obtained through the digestive system of a special earthworm. (Goli et al., 2019; Mostafaeipour et al., 2017; Tan & Çömden, 2012).

Problem modelling

This section introduces the indices, parameters and decision variables of the problem and then describes the proposed multi-objective optimization (MOO). MOO is built according to definition of the problem and takes into account certain assumptions. Its objectives are to reduce the SC costs (i.e., costs of transportation, construction of potential locations for distribution centers, inventory maintenance, production in gardens, and processing), to minimize the risks involved and to maximize the demand response.

Indicators

• $i=1,2,\cdots ,I$ Manufacture sites (gardens).

• $j=1,2,\cdots ,J$ Distribution centers.

• $k=1,2,\cdots ,K$ End-user places (fruit markets).

• $m=1,2,\cdots ,M$ Plant locations.

• $f=1,2,\cdots ,F$ Agro-food market places.

• $e=1,2,\cdots ,E$ Ethanol market places.

• $l=1,2,\cdots ,L$ Composting sites.

• $o=1,2,\cdots ,O$ Compost end-user sites.

• $p=1,2,\cdots ,P$ Crop (fruit) production.

• $t=1,2\text{,}\dots t^{\prime },\dots ,T$ Time periods.

Parameters

• ${\tilde{f}}_j$ Fixed costs of building recycling center j.

• ${\tilde{f}}_l$ Fixed costs of building recycling center l.

• ${\tilde{f}}_m$ Fixed costs of establishing plant m.

• ${\tilde{C}x}_{\mathit{ij}}$ Transporting cost each production from manufacture “i” to distributor j.

• ${\tilde{C}s}_{\mathit{ik}}$ Transporting cost each production from manufacture “i” to end-user k.

• ${\tilde{C}y}_{\mathit{im}}$ Transporting cost each production from manufacture “i” to plant m.

• ${\tilde{C}v}_{\mathit{il}}$ Transporting cost each production (after harvest) from manufacture “i” to recycling center l.

• ${\tilde{C}w}_{\mathit{il}}$ Transporting cost each production (before harvest/pest control) from manufacture “i” to recycling center l.

• ${\tilde{C}u}_{\mathit{jk}}$ Transporting cost each production from distributor “j” to cutomer k.

• ${\tilde{C}d}_{\mathit{jm}}$ Transporting cost each production from distributor “j” to plant m.

• ${\tilde{C}q}_{\mathit{jl}}$ Transporting cost each production from distributor “j” to recycling center l.

• ${\tilde{C}e}_{\mathit{kl}}$ Transporting cost each production from end-user “k” to recycling center l.

• ${\tilde{C}n}_{\mathit{km}}$ Transporting cost each production from end-user “k” to plant m.

• ${\tilde{C}b}_{\mathit{ml}}$ Transporting cost each production from plant “m” to recycling site l.

• ${\tilde{C}f}_{\mathit{lo}}$ Transporting cost each production from composting site “l” to compost market o.

• ${\tilde{C}r}_{\mathit{mf}}$ Transporting cost each production from plant “m” to agro-food factory’s market f.

• ${\tilde{C}r}_{\mathit{me}}$ Transpoting cost each production from plant “m” to agro-food factory’s market e.

• ${\tilde{C}h}_t$ Product storage cost by the distribution cite in time t.

• ${\tilde{C}a}_t$ Cost of pesticides at the production site (gardens) with identification through artificial intelligence in time t.

• ${\tilde{C}p}_t$ Cost of product processing and packaging by the distribution site in time t.

• ${\tilde{C}r}_t$ Cost of compost production by the recycling site in time t.

• ${\tilde{C}}_{\mathit{mft}}$ Cost of producing food and pharmaceuticals by the corresponding plants in time t.

• ${\tilde{C}}_{\mathit{met}}$ Cost of producing ethanol products by the corresponding factory in time t.

• $\tilde{C}p{}^{\prime }$ Cost of manufacturing by the gardens.

• ${\widetilde{\mathit{RC}}}_{\mathit{pit}}$ Cost of reducing the production risk of product p from supplier i in time t.

• ${\widetilde{\mathit{SR}}}_{\mathit{pit}}$ Risk of supplying product p from supplier i in time t.

• ${\widetilde{\mathit{MSR}}}_{\mathit{pt}}$ Maximum supply risk allowed to produce product p in time t.

• ${\widetilde{\mathit{MP}}}_{\mathit{pit}}$ Maximum product p from garden i in time t.

• ${\widetilde{\mathit{RV}}}_{\mathit{it}}$ Disruption or natural disasters period from the supplier i in time t.

• ${\widetilde{R\phi }}_{\mathit{pt}}$ Severity of disruption or natural disasters for crop p in time t.

• ${\widetilde{R\theta }}_{\mathit{pit}}$ Frequency rate of the crop failure or the natural disasters of product p from supplier i in time t.

• ${\tilde{d}}_{\mathit{kt}}$ Request of end-user k (fruit market) in time t.

• ${\tilde{d}{}^{\prime }}_{\mathit{ot}}$ Request for the product produced through recycling (composting) by compost customer o in time t.

• ${\tilde{d}{}^{\prime }{}^{\prime }}_{\mathit{ft}}$ Request for plant-produced products (food and pharmaceuticals) by end-user f in time t.

• ${\tilde{d}{}^{\prime }{}^{\prime }{}^{\prime }}_{\mathit{et}}$ Request for plant-produced product (ethanol) by end-user e in time t.

• ${\widetilde{\lambda c}}_{\mathit{it}}$ Maximum production capacity i in time t.

• ${\widetilde{\lambda h}}_{\mathit{jt}}$ Maintenance capacity of distributor j in time t.

• ${\widetilde{\lambda r}}_{\mathit{lt}}$ Production and storage capacity of recycling site l in time t.

• ${\widetilde{\lambda p}}_{\mathit{mft}}$ Factory storage volume for the production and storage of food and pharmaceuticals in time t.

• ${\widetilde{\lambda p}}_{\mathit{met}}$ Production and storage volume of ethanol plant in time t.

• ${\tilde{\alpha }}_t$ Percentage of product waste harvested by the manufacturer in time t.

• ${\tilde{\alpha }{}^{\prime }}_t$ Percentage of crop waste before harvest (pest control) by the producer in time t.

• ${\tilde{\beta }}_t$ Percentage of product waste storaged by the distribution site in time t.

• ${\tilde{\beta }{}^{\prime }}_t$ Percentage of semi-rotted crop waste stored by the distribution site in time t.

• ${\tilde{\theta }}_t$ Percentage of product waste stored by the customer in time t.

• ${\tilde{\theta }{}^{\prime }}_t$ Percentage of semi-rotten products stored by the customer in time t.

• ${\tilde{\gamma }}_t$ Percentage of product waste after production by the factory in time t.

• $\rho$ Weight factor (importance) of the response to forward currents.

• $1-\rho$ Weight factor (importance) of the response to backward flows.

• $\tilde{\omega }$ Coefficient of product conversion into processed crop in the factory.

• $\tilde{\phi }$ Coefficient of waste products conversion into processed products in the recycling site.

• $\mathrm{M}{}^{\prime }$ A large positive value.

Decision variables

$X_{\mathit{ijt}}$ The products transferred from producer i to distributor j in time t.

$Y_{\mathit{imt}}$ The products transported from producer i to plant m in time t.

$S_{\mathit{ikt}}$ The products sent from producer i to end-user k in time t.

$V_{\mathit{ilt}}$ The returned (rotten) products sent from producer i to recycling site l in time t.

$W_{\mathit{ilt}}$ The products (rotten before harvest) sent from producer i to recycling site l in time t.

$U_{\mathit{jkt}}$ The product transported from distributor j to end-user k in time t.

$Q_{\mathit{jlt}}$ returned (semi rotten) product sent from distributor j to recycling site l in time t.

$D_{\mathit{jmt}}$ (semi-rotten) product carried from distributor j to factory m in time t.

$E_{\mathit{klt}}$ returned (rotten) product sent from en-user k to recycling site l in time t.

$N_{\mathit{kmt}}$ returned (semi-rotten) product sent from end-user k to plant m in time t.

$B_{\mathit{mlt}}$ returned (waste) product sent from plant m to recycling site l in time t.

$F_{\mathit{lot}}$ composts produced and sent from recycling site l to compost market o in time t.

$R_{\mathit{mft}}$ The products produced and transported from plant f to factory market e in time t.

$R_{\mathit{met}}$ The products produced and carried from plant m to factory market e in time t.

${\mathit{Ih}}_{\mathit{jt}}$ The products stored in distribution site j over time t.

${\lambda }_{\mathit{it}}$ Crop production by producer “i” in time “t”.

${\vartheta }_j\left\{\begin{array}{c}0\\ 1\end{array}\right)$ If the distribution site is established in nomination location j, a numeric value of one is adopted; otherwise, a value of zero will be assigned.

${\eta }_l\left\{\begin{array}{c}0\\ 1\end{array}\right)$ If the recycling site is established in a candidate place, a value of one is assigned; otherwise, a value of zero will be allotted.

${\zeta }_m\left\{\begin{array}{c}0\\ 1\end{array}\right)$ If the factory is established in candidate location m, a value of one holds; otherwise, a value of zero should be assigned.

${\sigma }_{\mathit{pi}}\left\{\begin{array}{c}0\\ 1\end{array}\right)$ If product p is generated from supplier i, assign a value of one; otherwise, consider a value of zero.

Objective functions

$$\min z=z_1+z_2+z_3+z_4+z_5$$ 1

$$z_1=\sum _{j=1}^J\widetilde{f_j}\times {\vartheta }_j+\sum _{l=1}^L\widetilde{f_l}\times {\eta }_l+\sum _{m=1}^M{\tilde{f}}_m\times {\zeta }_m+\sum _{p=1}^P\sum _{i=1}^I\sum _{t=1}^T{\widetilde{\mathit{SR}}}_{\mathit{pit}}\times \sum _{p=1}^P\sum _{i=1}^I\sum _{t=1}^T{\widetilde{\mathit{RC}}}_{\mathit{pit}}\times \sum _{p=1}^P\sum _{i=1}^I\sum _{t=1}^T{\sigma }_{\mathit{pit}}$$ 2

$$z_2=\sum _{i=1}^I\sum _{j=1}^J\sum _{t=1}^{t^{\prime }}{\widetilde{\mathit{Cx}}}_{\mathit{ij}}\times X_{\mathit{ijt}}+\sum _{i=1}^I\sum _{k=1}^K\sum _{t=1}^{t^{\prime }}{\widetilde{\mathit{Cs}}}_{\mathit{ik}}\times S_{\mathit{ikt}}+\sum _{i=1}^I\sum _{m=1}^M\sum _{t=1}^{t^{\prime }}{\widetilde{\mathit{Cy}}}_{\mathit{im}}\times Y_{\mathit{imt}}+\sum _{i=1}^I\sum _{l=1}^L\sum _{t=1}^{t^{\prime }}{\widetilde{\mathit{Cv}}}_{\mathit{il}}\times V_{\mathit{ilt}}+\sum _{i=1}^I\sum _{l=1}^L\sum _{t=1}^{t^{\prime }}{\widetilde{\mathit{Cw}}}_{\mathit{il}}\times W_{\mathit{ilt}}+\sum _{j=1}^J\sum _{k=1}^K\sum _{t=1}^T{\widetilde{\mathit{Cu}}}_{\mathit{jk}}\times U_{\mathit{jkt}}+\sum _{j=1}^J\sum _{l=1}^L\sum _{t=1}^T{\widetilde{\mathit{Cq}}}_{\mathit{jl}}\times Q_{\mathit{jlt}}+\sum _{j=1}^J\sum _{m=1}^M\sum _{t=1}^T{\widetilde{\mathit{Cd}}}_{\mathit{jm}}\times D_{\mathit{jmt}}+\sum _{k=1}^K\sum _{l=1}^L\sum _{t=1}^T{\widetilde{\mathit{Ce}}}_{\mathit{kl}}\times E_{\mathit{klt}}+\sum _{k=1}^K\sum _{m=1}^M\sum _{t=1}^T{\widetilde{\mathit{Cn}}}_{\mathit{km}}\times N_{\mathit{kmt}}+\sum _{m=1}^M\sum _{l=1}^L\sum _{t=1}^T{\widetilde{\mathit{Cb}}}_{\mathit{ml}}\times B_{\mathit{mlt}}+\sum _{l=1}^L\sum _{o=1}^O\sum _{t=1}^T{\widetilde{\mathit{Cf}}}_{\mathit{lo}}\times F_{\mathit{lot}}+\sum _{m=1}^M\sum _{f=1}^F\sum _{t=1}^T{\widetilde{\mathit{Cr}}}_{\mathit{mf}}\times R_{\mathit{mft}}+\sum _{m=1}^M\sum _{e=1}^E\sum _{t=1}^T{\widetilde{\mathit{Cr}}}_{\mathit{me}}\times R_{\mathit{met}}$$ 3

$$z_3=\sum _{i=1}^I\sum _{t=1}^{t{}^{\prime }}{\lambda }_{\mathit{it}}\times {\widetilde{\mathit{Ca}}}_t$$ 4

$$z_4=\sum _{j=1}^J\sum _{t=1}^T{\mathit{Ih}}_{\mathit{jt}}\times {\widetilde{\mathit{Ch}}}_t$$ 5

$$z_5=\sum _{i=1}^I\sum _{j=1}^J\sum _{t=1}^{t^{\prime }}{X}_{\mathit{ijt}}\times {\widetilde{\mathit{Cp}}}_t+\sum _{l=1}^L\sum _{o=1}^O\sum _{t=1}^T{F}_{\mathit{lot}}\times {\widetilde{\mathit{Cr}}}_t+\sum _{m=1}^M\sum _{f=1}^F\sum _{t=1}^T{R}_{\mathit{mft}}\times {\widetilde{\mathit{Cm}}}_t+\sum _{m=1}^M\sum _{e=1}^E\sum _{t=1}^T{R}_{\mathit{met}}\times {\widetilde{\mathit{Cm}}}_t+\sum _{i=1}^I\sum _{t=1}^T{\lambda }_{\mathit{it}}\times {\widetilde{\mathit{Cp}}{}^{\prime }}_t$$ 6

$$minzz=\sum _{p=1}^P\sum _{i=1}^I\sum _{t=1}^T{\widetilde{\mathit{SR}}}_{\mathit{pit}}\times \sum _{p=1}^P\sum _{i=1}^I\sum _{t=1}^T{\widetilde{\mathit{RC}}}_{\mathit{pit}}\times \sum _{p=1}^P\sum _{i=1}^I\sum _{t=1}^T{\sigma }_{\mathit{pit}}+\sum _{i=1}^I\sum _{t=1}^T{\widetilde{\mathit{RV}}}_{\mathit{it}}\times \sum _{p=1}^P\sum _{t=1}^T{\widetilde{R\phi }}_{\mathit{pt}}\times \sum _{p=1}^P\sum _{i=1}^I\sum _{t=1}^T{\widetilde{R\theta }}_{\mathit{pit}}$$ 7

$$\text{max}{Z}^{\prime }=\rho \times \left(\sum _{i=1}^I\sum _{k=1}^K\sum _{t=1}^{t^{\prime }}{S}_{\mathit{ikt}}+\sum _{j=1}^j\sum _{k=1}^K\sum _{t=1}^T{U}_{\mathit{jkt}}/(\sum _{k=1}^K\sum _{t=1}^T{\tilde{d}}_{\mathit{kt}}\right)+(1-\rho )\times \left(\sum _{l=1}^L\sum _{o=1}^O\sum _{t=1}^T{F}_{\mathit{lot}}+\sum _{m=1}^M\sum _{f=1}^F\sum _{t=1}^T{R}_{\mathit{mft}}+\sum _{m=1}^M\sum _{e=1}^E\sum _{t=1}^T{R}_{\mathit{met}}\right)/\left(\sum _{o=1}^O\sum _{t=1}^T{\tilde{d}}_{\mathit{ot}}^{\prime }+\sum _{f=1}^F\sum _{t=1}^T{\tilde{d}}_{\mathit{ft}}^{″}+\sum _{e=1}^E\sum _{t=1}^T{\tilde{d}}_{\mathit{et}}^{″\prime }\right)$$ 8

This model has three objective functions including cost minimization, risk reduction and accountability maximization. Equation (1) deals with the minimization of the costs, which consist of five types of costs presented in Eqs. (2) to (6). Equation (2) addresses the fixed costs of implementation the distribution, recycling and manufacturing centers. It should be noted that distribution and recycling centers can be both actual and potential points. To make it possible, rather than the addition of an index, the cost of constructing actual points is assumed to be zero in the parameters. Equations (3)–(5) are respectively concerned with the following:

• Transportation costs (comprising forward and backward costs)

• The cost of using artificial intelligence for pest control in gardens

• The cost of maintaining the processed products

Equation (6) regards operating costs, including the costs of energy, processing, packaging, reprocessing as well as food industries. Equation (7) defines other function, which is to decrease the risk in the chain by considering the cost of reducing the risk of supplying the product. It is possible to determine the magnitude of the risk by multiplying the severity, frequency rate and probability of the chain disorder, as three indicators. Equation (8) defines the third objective function, namely accountability maximization. In this regard, the response is divided into two parts. The first Pertains to the customers of the main product, and second concerns the customers of the processed product as well as the food and energy conversion industries.

Constraints

$${\lambda }_{\mathit{it}}\times \left(1-({\alpha }_t+{\alpha }_t^{\prime })\right)=\sum _{j=1}^J{X}_{\mathit{ijt}}+\sum _{k=1}^K{S}_{\mathit{ikt}}+\sum _{m=1}^M{Y}_{\mathit{imt}}\forall i\in I,\forall t\in t^{\prime }$$ 9

$$\sum _{i=1}^I\sum _{t=1}^{T=t^{\prime }}{X}_{\mathit{ijt}}\le M{}^{\prime }\times {\vartheta }_j\forall j\in J$$ 10

$${\lambda }_{\mathit{it}}\le {\widetilde{\lambda c}}_{\mathit{it}}\forall i\in I,\forall t\in t^{\prime }$$ 11

$${\mathit{Ih}}_{j(t-1)}+\sum _{j=1}^J{X}_{\mathit{ijt}}={\mathit{Ih}}_{\mathit{jt}}+\sum _{k=1}^K{U}_{\mathit{jkt}}+\sum _{l=1}^L{Q}_{\mathit{jlt}}+\sum _{m=1}^M{D}_{\mathit{jmt}}\forall i\in I,\forall t\in T$$ 12

$${\mathit{Ih}}_{\mathit{jt}}\le {\widetilde{\lambda h}}_j\forall j\in J,\forall t\in T$$ 13

$$\sum _{j=1}^j{U}_{\mathit{jkt}}+\sum _{i=1}^I{S}_{\mathit{ikt}}\le {\tilde{d}}_{\mathit{kt}}\forall k\in K,\forall t\in T$$ 14

$$\sum _{l=1}^L{V}_{\mathit{ilt}}\le {\tilde{\alpha }}_t\times {\lambda }_{\mathit{it}}\forall i\in I,\forall t\in t^{\prime }$$ 15

$$\sum _{t=1}^{t^{\prime }}\sum _{i=1}^I{V}_{\mathit{ilt}}\le M^{\prime }\times {\eta }_l\forall l\in L$$ 16

$$\sum _{l=1}^L{W}_{\mathit{ilt}}\le {\tilde{\alpha }}_{t^{\prime }}\times {\lambda }_{\mathit{it}}\forall i\in I,\forall t\in t^{\prime }$$ 17

$$\sum _{t=1}^{t^{\prime }}\sum _{i=1}^I{W}_{\mathit{ilt}}\le M^{\prime }\times {\eta }_l\forall l\in L$$ 18

$$\sum _{l=1}^L{Q}_{\mathit{jlt}}\le {\tilde{\beta }}_t\times {\mathit{Ih}}_{j(t-1)}\forall j\in J,\forall t\in T$$ 19

$$\sum _{t=1}^T\sum _{j=1}^J{Q}_{\mathit{jlt}}\le M{}^{\prime }\times {\eta }_l\forall l\in L$$ 20

$$\sum _{l=1}^L{E}_{\mathit{klt}}\le {\tilde{\theta }}_t\times d_{\mathit{kt}}\forall k\in K,\forall t\in T$$ 21

$$\sum _{t=1}^T\sum _{k=1}^K{E}_{\mathit{klt}}\le M{}^{\prime }\times {\eta }_l\forall l\in L$$ 22

$$\sum _{l=1}^L{B}_{\mathit{mlt}}\le {\tilde{\gamma }}_t\times ({\tilde{d}{}^{\prime }{}^{\prime }}_{\mathit{ft}}+{{\tilde{d}}^{{}^{\prime }{}^{\prime }{}^{\prime }}}_{\mathit{et}})\forall m\in M,\forall t\in T$$ 23

$$\sum _{t=1}^T\sum _{m=1}^M{B}_{\mathit{mlt}}\le M{}^{\prime }\times {\eta }_l\forall l\in L$$ 24

$$\sum _{m=1}^M{D}_{\mathit{jmt}}\le {\tilde{\beta }{}^{\prime }}_t\times {\mathit{Ih}}_{j(t-1)}\forall j\in J,\forall t\in T$$ 25

$$\sum _{t=1}^T\sum _{j=1}^J{D}_{\mathit{jmt}}\le M{}^{\prime }\times {\zeta }_m\forall m\in M$$ 26

$$\sum _{m=1}^M{N}_{\mathit{kmt}}\le {\tilde{\theta }{}^{\prime }}_t\times {\tilde{d}}_{\mathit{kt}}\forall k\in K,\forall t\in T$$ 27

$$\sum _{t=1}^T\sum _{k=1}^K{N}_{\mathit{kmt}}\le M{}^{\prime }\times {\zeta }_m\forall m\in M$$ 28

$$\sum _{m=1}^M{Y}_{\mathit{imt}}\le {\tilde{\alpha }}_t\times {\lambda }_{\mathit{it}}\forall i\in I,\forall t\in T$$ 29

$$\sum _{t=1}^T\sum _{i=1}^I{Y}_{\mathit{imt}}\le M{}^{\prime }\times {\zeta }_m\forall m\in M$$ 30

$$(\sum _{i=1}^I{Y}_{\mathit{imt}}+\sum _{j=1}^J{D}_{\mathit{jmt}}+\sum _{k=1}^K{N}_{\mathit{kmt}})\times \tilde{\omega }=\sum _{f=1}^f{R}_{\mathit{mft}}+\sum _{e=1}^E{R}_{\mathit{met}}\forall m\in M,\forall t\in T$$ 31

$$\sum _{f=1}^F{R}_{\mathit{mft}}+\sum _{e=1}^E{R}_{\mathit{met}}\le {\widetilde{\lambda p}}_{\mathit{mf}}+{\widetilde{\lambda p}}_{\mathit{me}}\forall m\in M,\forall t\in T$$ 32

$$\sum _{m=1}^M{R}_{\mathit{mft}}\le {\tilde{d}"}_{\mathit{ft}}\forall f\in F,\forall t\in T$$ 33

$$\sum _{m=1}^M{R}_{\mathit{met}}\le {\tilde{d}}_{\mathit{et}}^{″\prime }\forall e\in E,\forall t\in T$$ 34

$$\left(\sum _{i=1}^I,V_{\mathit{ilt}},+,\sum _{i=1}^I,W_{\mathit{ilt}},+,\sum _{j=1}^J,Q_{\mathit{jlt}},+,\sum _{m=1}^M,B_{\mathit{mlt}},+,\sum _{k=1}^K,E_{\mathit{klt}}\right)\times \tilde{\phi }=\sum _{o=1}^O{F}_{\mathit{lot}}\forall l\in L,\forall t\in T$$ 35

$$\sum _{o=1}^O{F}_{\mathit{lot}}\le {\widetilde{\lambda r}}_l\forall l\in L,\forall t\in T$$ 36

$$\sum _{l=1}^L{F}_{\mathit{lot}}\le {\tilde{d}{}^{\prime }}_{\mathit{ot}}\forall o\in O,\forall t\in T$$ 37

$$\sum _{p=1}^P{\mathit{SR}}_{\mathit{pit}}\times \sum _{p=1}^P{\sigma }_{\mathit{pit}}\le \sum _{p=1}^P{\widetilde{\mathit{MSR}}}_{\mathit{pt}}\forall i\in I,\forall t\in T$$ 38

$$\sum _{j=1}^J{X}_{\mathit{ijt}}+\sum _{k=1}^K{S}_{\mathit{ikt}}+\sum _{m=1}^M{Y}_{\mathit{imt}}+\sum _{i=1}^I{V}_{\mathit{ilt}}+\sum _{i=1}^I{W}_{\mathit{ilt}}\le \sum _{p=1}^P{\widetilde{\mathit{MP}}}_{\mathit{it}}\forall j\in J,\forall k\in K,\forall m\in M,\forall l\in L,\forall t\in T$$ 39

$${\vartheta }_j,{\eta }_l,{\zeta }_m,{\sigma }_{\mathit{pit}}\in \left\{0,,,1\right\}\forall j\in J,\forall l\in L,\forall m\in M,\forall p\in P,\forall i\in I$$ 40

$$X_{\mathit{ijt}},S_{\mathit{ikt}},Y_{\mathit{imt}},V_{\mathit{ilt}},W_{\mathit{ilt}},U_{\mathit{jkt}},D_{\mathit{jmt}},Q_{\mathit{jlt}},N_{\mathit{kmt}},E_{\mathit{klt}},R_{\mathit{mft}},R_{\mathit{met}},B_{\mathit{mlt}},F_{\mathit{lot}}\ge 0\forall i\in I,j\in J,k\in K,m\in M,l\in L,f\in F,e\in E,t\in T$$ 41

$${\mathit{Ih}}_{\mathit{jt}}\ge 0,{\lambda }_{\mathit{it}}\ge 0\forall i\in I,\forall j\in J,\forall t\in T$$ 42

Each of the constraints delineated above plays a certain role in the mathematical model. Constraint (9) shows that the producers minus wastes equal the transfer from those producers to the distributors, target markets and factories. Constraint (10) is related to constraint (9) and emphasizes that shipments to potential distributors will occur if the site is established. Constraint (11) indicates that the amount of the product manufactured by smaller producers is equal to their maximum capacity. Limitation (12) ensure that the distributor’s inventory in each time equals the inventory in the previous time minus the waste of the previous time plus the new products entering the storage minus the outgoing products from the storage to the processing and packaging lines and factories. Constraint (13) states that the maximum inventory of distributors is smaller or equal to the storage capacity. Limitation (14) means that the market request value in each time is greater or equal to the products of the plants and distributors. Constraints (15) to (24) indicate that the amount of the waste in each section of a reverse flow is displayed if recycling centers are established. Constraints (25) to (30) denote that the inputs from such sectors as production, distribution, fruit markets and customers are sent to factories including manufacturers as well as food and energy industries.Constraint (31) signifies that the total input sent from the production and distribution centers as well as fruit markets and customers to the factory multiplied by the conversion rate of the processed products in the factory is equal to the total products sent to the market from food and energy production plants.

Constraint (32) implies that the total output of factories sent to the customer or market is less than or equal to the production volume of food and energy plants. Constraints (33) and (34) assume that all the products of a factory sent to a smaller factory or market are equal to customer demand and the market capacity of the food and energy factory. Accordingly, constraint (33) is related to the capacity of food factories as well as their market demand and customers. Constraint (34), however, is about the capacity of energy factories as well as their market demand and customers. Constraint (35) shows the total waste sent to recycling centers from producers, distributors, customers and factories, multiplying the conversion of waste products to compost. It is equal to the total recycled product sent to markets and compost customers. Constraint (36) states that all the compost sent to the compost market and smaller customers is equal to the production capacity of recycling centers. Limitation (37) marks that all the compost sent to the compost market and smaller customers is equal to the demand of the compost user. Limitation (38) assumes that the risk of providing the product from the supplier is multiplied if it is less than or equal to the maximum risk allowed to produce the product. Constraint (39) states that the total output of the orchards that goes to the distribution centers, fruit markets or customers, factories and compost is smaller than the maximum products of the orchards. Equations (40) to (42) represent the last constraints. They include the signs of free variables (zero and one for variables) and their positivity.

Numerical problems

The results of the current study are described in 5 steps. First, the necessary details for the mathematical model are provided. Then, all numerical parameters used in this research are introduced. In the third stage, the deterministic and meta-heuristic finding of the model are shown. In the following, meta-heuristic methods are compared with each other. Finally, a sensitivity analysis on the most influential parameters of the model is presented. In general, the present study is a multi-objective localization modeling. To solve this problem, a number of variables that accept zero or one value have been used. These binary variables cause complexity in the model (Tan et al., 2019). Generally, realistic case study have large dimensions, and the corresponding models become NP-hard as those dimensions grow. According to the complexity of the presented model in this study, exact methods such as the epsilon constrained method are not efficient enough (Saleem et al., 2020). To solve the mathematical model in this study, a meta-heuristic algorithm is used as the solution method, and Pareto solutions are found by NSGA-II as a multi-objective non-dominated sorting genetic algorithm. This research has also sought to design a multi-level closed-loop network including producers, distributors, customers, compost centers, compost customers, and factories, their customers in the field of food and energy production. The designed network serves to supply pomegranates to the cities and provinces in Iran. Figure 6 shows the map of the cities and provinces with the most pomegranate orchards.

Fig. 6.

Map of Iran: the provinces with the largest areas under pomegranate cultivation

For this purpose, the production capacities of different regions of Iran are examined, the number of the distribution centers in each region is identified, and their relationship with consumption centers is established. Distance, product quality, cost, and region capacity effectively determine the number of distribution centers and their relation to consumption centers. As a result, identifying the regions and determining the number of centers and the required distribution and transportation routes are the most important problems facing this research. In this study, 12 problems in different dimensions have been dealt with. The data on the dimensions of the problems are shown in Table 3.

Table 3.

Dimensions of the problem

p	o	L	e	f	M	k	j	i	Problem number
4	2	4	2	3	4	3	4	3	1
4	2	4	3	4	4	4	5	3	2
5	3	5	4	5	5	6	5	4	3
5	4	5	4	6	5	8	6	4	4
6	5	6	5	6	6	10	6	5	5
8	5	6	5	7	6	10	7	5	6
10	6	6	6	8	7	11	7	6	7
12	7	7	6	8	7	11	7	7	8
14	7	8	7	9	8	12	9	7	9
16	8	8	7	9	8	12	9	8	10
18	8	9	8	10	9	13	10	8	11
20	9	9	8	10	9	13	10	9	12

The first sub-problem is the realistic study in current research, which is used to show model and method validation. The cities selected for each location in this case are mentioned in Table 4.

• The values of φ, ρ and Mʼ are considered to be 1.1, 0.6 and 10¹⁵.

• The value of cp’ is assumed to be equal to 180.

Table 4.

Selected cities for each indicator

Indicators	I	J	K	L	O	m	f	e	P
Cities	Behshahr	Taft	Firoozkooh	Gilan	Behshahr	Ardakan	Ardestan	Ardakan	Behshahr
	Saveh	Neyriz	Ardestan	Meybod	Neyriz	Behshahr	Natanz	Behshahr	Kashan
	Neyriz	Mehriz	Kuhdasht	Neyriz		Shahreza	Julfa
		Mahvelat		Mahvelat		Saveh

The rest of the model parameters are presented in Table 5.

Table 5.

The values of the remaining model parameters

Parameter	Value	Unit
T	3	Period (months)
t'	3	Period (months)
Ω	1.12	Percent
1-ρ	0.4	Percent
λcit	Uniform ~ [0, 95]	Ton
fj	Uniform ~ [0, 585714]	$
fl	Uniform ~ [0, 398060]	$
fm	Uniform ~ [0, 42745]	$
cht	Uniform ~ [58, 72]	Dollar/ton
cpt	Uniform ~ [84, 104]	Dollar/ton
crt	[86, 137]	Dollar/ton
dkt	Uniform ~ (Jabbarzadeh et al., 2016; Mangla et al., 2018)	Ton
λhj	10 or 20 or 30	Ton
λpme	Uniform ~ (Barbedo, 2016; Borodin et al., 2016)	Ton
λpmf	Uniform ~ (Borodin et al., 2016; Carvajal et al., 2019)	Ton
λrt	Uniform ~ (Borodin et al., 2016; Dhande et al., 2021)	Ton
d'ot	Uniform ~ (Jabbarzadeh et al., 2016; Mangla et al., 2018)	Ton
αt	[0, 0.15]	%
αʼt	[0, 0.8]	%
βt	[0, 0.05]	%
βʼt	[0, 0.05]	%
θt	[0.02, 0.05]	%
θʼt	[0.02, 0.049]	%
γt	[0, 0.05]	%
SRpit	[0.02, 0.05]	%
MSRpt	[0.09]	%
MPpit	Uniform ~ [35, 119]	Ton/hectares
RCpit	Uniform ~ [42, 73]	Dollar/ton
RVi,t	uniform(Jabbarzadeh et al., 2016; Tan et al., 2019)	range
Rφp,t	0.003	Percent/intensity
Rθi,t	10	Abundance rate

Computational results

Solving numerical problems

After the parameters are specified in the numerical problems, the NSGA-II and multi-objective PSO algorithms are implemented and the solution results of the generated problems are compared. Thus, the 12 problems introduced in the previous section are examined first by the GAMS software and then by the NSGA-II and MOPSO algorithms. Table 6 presents the objective function values and the CPU time of the experimental examples conduct by GAMS®. GAMS can solve small-size problems. When the size of the problems is increased, however, it cannot achieve acceptable solutions within reasonable time. In this case, only numerical problems 1 to 6 can be solved and yield optimal solutions. From problem 7 on, GAMS is not capable of solving, and the solutions are zero.

Table 6.

Computational results for 12 problems in the GAMS software

Problem	CPU time (Ms)	OF 1	OF 2	OF 3
1	412	1,869,908.962	2172.638	65.926
2	656	1,837,811.506	2178.582	65.849
3	803	1,898,706.611	2185.266	63.746
4	938	1,982,916.427	2522.612	61.856
5	1139	2,113,917.000	2593.624	58.92
6	1154	2,324,197.569	2748.065	56.889
7	-	0	0	0
8	-	0	0	0
9	-	0	0	0
10	-	0	0	0
11	-	0	0	0
12	-	0	0	0

Since the GAMS software can only solve low-size problems, more powerful solution methods have been needed in this study. In this regard, the NSGA-II algorithm has served the purpose very well. The feasible solution obtain finding are shown in Table 7. As indicated, the bigger the problem, the higher the total cost and risk, and the lower the accountability (e.g., objective function 3). It also takes more time to solve bigger-size problems, which is denoted as the CPU time.

Table 7.

Computational results for 12 problems by NSGA-II

Problem	CPU time	OF 1	OF 2	OF 3
1	31.3076	438,115.27	3162.55	117.37
2	35.9681	607,336.50	3163.52	114.7
3	34.8851	613,106.24	3165.11	106.35
4	36.7549	942,209.29	3161.54	96.21
5	41.6321	1,162,369.14	3169.14	95.47
6	59.6989	1,283,678.78	3214.78	87.14
7	72.1453	1,456,989.14	3285.14	84.63
8	95.4157	1,861,456.85	3298.79	83.88
9	108.5704	1,984,714.32	3354.63	82.47
10	129.4719	2,111,456.87	3441.73	82.06
11	138.6548	2,499,874.72	3486.42	81.72
12	156.1427	2,873,647.46	3523.21	80.11

In addition, to compare the performances of different solution methods here, multi-objective PSO is used as another meta-heuristic algorithm. Table 8 presents the computational results for different problems solved by the MOPSO algorithm. As it can be seen, raising in the size of a problem leads to raise in the total cost and risk. Furthermore, the bigger a problem, the lower its objective function of accountability and the higher the CPU time.

Table 8.

Computational results for 12 problems by MOPSO

Problem	CPU Time	OF 1	OF 2	OF 3
1	68.2543	696,927.53	3163.15	97.96
2	64.3393	1,150,280.74	3162.21	92.48
3	64.2156	1,414,827.13	3161.62	87.59
4	66.4515	1,689,628.78	3211.16	86.32
5	73.1416	1,723,619.48	3286.73	85.64
6	76.1468	1,897,469.14	3658.44	83.47
7	79.6936	1,953,665.29	3665.75	83.16
8	83.7569	2,063,626.19	3717.69	77.96
9	89.4665	2,183,646.31	4236.11	75.97
10	92.1454	2,667,646.86	4869.46	76.78
11	95.3693	2,841,997.64	5139.44	72.33
12	99.1767	2,978,616.73	5536.49	65.99

The results yielded by different solution algorithms are compared in Fig. 7. It can be concluded that the NSGA-II algorithm involves lower costs than the other two methods. The next ranks in this respect are for the MOPSO algorithm and the GAMS software respectively.

Fig. 7.

Objective function 1 for the numerical problems by three solution methods

The solution algorithms applied in this study are also compared in Fig. 8. As it can be concluded, the lower-size problems solved by the GAMS software emerge to have lower supply risks than conducted by the NSGA-II and MOPSO algorithms. Moreover, for bigger-size problems, NSGA-II has proved to perform better than the MOPSO algorithm.

Fig. 8.

Objective function 2 for the numerical problems by three solution methods

By comparing the three solution methods, one understands that solving problems by the NSGA-II algorithm results in higher accountability. The next two ranks in this respect belong to MOPSO and GAMS, respectively (Fig. 9).

Fig. 9.

Objective function 3 for the numerical problems by three solution methods

In terms of the running time, as Fig. 10 shows, it takes more CPU time to solve low-size problems in the GAMS software than by the other solution methods. For big-size problems, however, the MOPSO algorithm takes less CPU time than the NSGA-II algorithm.

Fig. 10.

CPU time for the numerical problems in the NSGA-II and GAMS®

Sensitivity analysis

In this section, the validity and treatment of the model are examined through changing different parameters including the maximum supply risk (${\widetilde{\mathit{MSR}}}_{\mathit{pt}}$), period of natural disasters or disruption from the supplier (${\widetilde{\mathit{RV}}}_{\mathit{it}}$), severity of disruption or natural disasters for the crop $({\widetilde{R\phi }}_{\mathit{pt}}$), and frequency rate of crop failure or natural disasters for the product $({\widetilde{R\theta }}_{\mathit{pit}}$).

As the graphs in Fig. 11 show, a rise in the maximum supply risk leads to a rise in the total cost and risk but a reduction in the accountability of the network.

Fig. 11.

The treatment of the objective functions versus the maximum supply risk

Graphs in Fig. 12 indicate that an increase in the disruption period results in increased total cost and risk of the network, while it first increases and then decreases the accountability.

Fig. 12.

The treatment of the objective functions versus the increased disruption period

According to Fig. 13, once the severity of disruption is increased in the OF1, the total cost of the network is increased too, and the total risk rises at a constant rate. In contrast, the accountability is first reduced, then increased, and finally reduced linearly.

Fig. 13.

The treatment of the objective functions versus the increased severity of disruption

As for results in Fig. 14, an increase in the frequency rate of failure, namely OF1 keeps the total cost of the network stable but then increases it. The total risk is also increased linearly at a constant rate. The accountability, however, rises at first and then decreases.

Fig. 14.

The treatment of the objective functions versus the increased frequency of failure

Conclusion

In current study, a MOO modeling is presented for a sustainable CLSC. The reverse logistic operations in the chain, such as waste recycling, are intended to reduce the energy consumption and cost of the network. To test and validatef the designed problem, a case study has been conducted on pomegranates in Iran. As designed in the supply chain, the fruit is turned into food, medicine, and concentrate after supply and distribution. Through reverse logistics, the pomegranate waste is also converted into recycled products including ethanol as an automotive fuel and a renewable source of energy and compost as an organic fertilizer. The image processing approach can be useful as a fast, economic, non-contact method (without human intervention) and with a much higher accuracy than visual and manual inspection methods, in controlling the quality of pomegranate production lines. Therefore, the costs of the chain are reduced to a great extent because automation reduces transportation costs and personnel wages. The use of picture processing in the diagnosis of pomegranate quality ensures compliance with international standards. Also, defective products and waste have always been an important challenge for manufacturers. Pomegranate recycling and reprocessing and reuse of products not only reduce harmful effects for the environment, but also improve their competitive sittuation in the market. Therefore, in current study, considering the CLSC, it is possible to significantly reconstruct the supply chain networks and maximize the economic benefits. This structure can achieve the economic, social and environmental goals of the organization at the same time. Therefore, the finding of this study can be useful for organizations and managers who deal with crops, food, and in general, products that have the possibility of failure. Considering image processing to determine the relationships among pomegranate appearance, color and size, cost of using artificial intelligence for pest control in gardens and the cost of maintaining the processed products and risk and uncertainty in pomegranate SC are the unique contributions of current study. The basis goal of current modeling is to decrease the cost and risk of the SC and increase the profits of gardeners and investors in the public and non-profit agriculture sectors in Iran. To solve the problem of the mathematical model, NSGA-II and MOPSO are used as two meta-heuristic algorithms. The problem is also solved with the GAMS software, and the finding of the three solutions are compared. By the analysis of the computational findings, it is reported that the NSGA-II algorithm has better mechanism than the other solution methods to reduce the total cost and the supply risk and enhance the accountability. Finally, to examin the behaviour of modeling, a series of sensitivity analyses are conducted on the parameters of maximum supply risk, disruption period, frequency of failure and disruption risk. Results show that once the severity of disruption is increased in OF1 total cost of the network is increased too, and the total risk rises at a constant rate. In contrast, the accountability is first reduced, then increased, and finally reduced linearly. Also an increase in the frequency rate of failure, namely the first objective function, keeps the total cost of the network stable but then increases it. The total risk is also increased linearly at a constant rate. The accountability, however, rises at first and then decreases.

Momenitabar et al. (2022a) presented sustainable CLSC network considering lateral resupply. They using fuzzy system is used to predict the demand in the system. Considering image processing to determine the relationships among pomegranate appearance, color and size, and cost of using artificial intelligence for pest control in gardens are the contribution of current research compared with the one done by Momenitabar et al. (2022a).

Babaeinesami et al. (2022) presented a CLSC combination considering environmental effects. They used a self-adaptive NSGA-II algorithm to solve the proposed problem. Considering risk and uncertainty and image processing to determine the relationships among pomegranate appearance, color and size are the contribution of current research compared with the one done by Babaeinesami et al. (2022).

Safaei et al. (2022) presented a new 2EMPCLSC network. In this paper, demand forecasting is done using time series modeling. They run the model using genetic algorithm. Considering the cost of using artificial intelligence for pest control in gardens and the cost of maintaining the processed products and image processing to determine the relationships among pomegranate appearance, color and size are the contribution of current study compared with the one done by Safaei et al. (2022).

Current study has some limitatation follow as:

• The solutions in NSGA-II and MOPSO algorithms depend on the type of definition of chromosomes as well as the initialization of their parameters. If there is a change in their value, the results of the research will change.

• Due to the fact that there was no database to record documents and data, questions have been designed to collect transportation costs in each route. Then, the collected data is used in the mathematical model.

Also, some suggustion for future study are follow as:

• Considering the fuzzy set system to predict the amount of uncertain demand for pomegranate

• Considering the proposed model as a scenario-based model and solving it with a robust optimization approach

• Considering a cooperative game between pomegranate producers and the government using a game theory approach

Data Availability

Data will be made available upon request.