Optimization Strategies for the Integrated Managementof Perishable Supply Chains: A Literature Review

Purpose: The main purpose of this article is to systematically review the papers published in the period 2005-2020 about the integration of production, inventory and distribution activities in perishable supply chains. Design/methodology/approach: The proposed research methodology is based on several steps. First, database and keywords are selected, with the aim to search and collect the main papers, dealing with the integration of production, inventory, distribution activities in perishable supply chains. Then, a bibliometric analysis is carried out, to detect: the main publishing sources, the chronological distribution, the most used keywords, the featured authors, about the selected papers. A five-dimension classification framework is proposed to carry out a content analysis, where the papers of the literature review are classified and discussed, according to: supply chain structure, objective, perishability type, solution approach, approach validation. Findings: Interest in the application of optimization models for integrated decision-making along perishable supply chains is strongly growing. Integrating multiple stages of the supply chain into a single framework is complex, especially when referring to perishable products. The vast majority of the problems addressed are then NP-Hard. Only a limited quantity of the selected papers aims to solve real-life case studies. There is a need for further research, which is capable of modeling and quantitatively improving existing supply chains. The potentials of Industry 4.0 are currently little explored. Originality/value: Based on the analysis of the papers published, this article outlines the current state of the art on the optimization strategies for the integrated management of perishable supply chains, which are very complex to be managed. Research trends and gaps are discussed, future challenges are presented.


Introduction
In its most traditional form, a supply chain (SC) is characterized by four main actors or stages: suppliers, manufacturers, distributors and customers (Kumar, Ganapathy, Gokhale & Tiwari, 2020). Variety and heterogeneity of stakeholders, who often have conflicting objectives, make the design and management of supply chains an extremely difficult task (Diabat, Abdallah & Le, 2016;Hammami, Frein & Bahli, 2017). According to Bank, Mazdeh and Heydary (2020), supply chain management (SCM) is the process that aims to efficiently integrate the different stages of the supply chain with the aim of delivering the right number of products at the right time to end users. SCM usually means making decisions at different levels: strategic, tactical, and operational, based on the temporal impact (Diabat & Theodorou 2015;Miller, 2002). Strategic decisions have a long-term impact, usually years, in fact they cannot be easily changed as they involve significant investments (e.g., location-allocation decisions). Tactical decisions, on the other hand, have mid-term effects (i.e., months) and most often concern inventory management. Finally, operational decisions usually have a daily or weekly impact and frequently concern scheduling and routing decisions (Hiassat, Diabat & Rahwan, 2017;Rafie-Majd, Pasandideh & Naderi, 2018). decisions are extremely critical and challenging (Palak, Eksioglu & Geunes, 2018;Rohmer, Claassen & Laporte, 2019). Perishability affects multiple fields. In the health sector, kidney or heart transplants are strongly influenced by the perishable nature of the organ (Zahiri, Tavakkoli-Moghaddam & Pishvaee, 2014), in the pharmaceutical field, chemical composition of medicines determines the period of time within which they are still effective (Chung & Kwon, 2016), in hospitals a correct blood bank management is crucial for patient health (Najafi, Ahmadi & Zolfagharinia, 2017). However, the agri-food sector is probably the one in which the perishable nature of goods affects decision-making the most. The agri-food supply chains are often characterized by frequent customer orders of small quantities, tight time windows for deliveries, yield and demand uncertainty. Therefore, the related optimization models are often extremely complex to solve through exact methods, from the computational point of view, and heuristic quantitative approaches are necessary in order to determine good solutions in a reasonable time. In this context, real-time information sharing between the players in the chain becomes a key-factor to ensure an adequate service level while minimizing costs. The lack of coordination in the food supply chains mainly causes (i) unharvested fruits and/or vegetables upstream, while (ii) unsold products downstream. In both cases, bad operations management causes waste, which must be disposed of with consequent increase in costs and pollution, due to further transport.
The main purpose of this study is to systematically review the papers published in the 2005-2020 period about the integrated management of perishable supply chains (IMPSCs). Our review aims to answer the following research questions: • RQ1. How much research has been conducted, in the last years, on the IMPSCs?
• RQ2. Which journals and authors have contributed most to the literature on the IMPSCs? • RQ3. What aspects characterize perishable supply chains, when they are addressed according to an integrated perspective?
• RQ4. What are the most recurring objectives within the IMPSCs?
• RQ5. What are the most used approaches for solving problems within the IMPSCs and how are they validated?
• RQ6. How is the concept of perishability modeled and treated within the IMPSCs?
• RQ7. What are the research gaps and future challenges in the field of IMPSCs?
With the aim to find exhaustive answers to these questions, our paper critically reviews the existing issues in IMPSCs literature, which is discussed in the subsequent sections. In particular, an original five-dimension classification framework is proposed, in order to categorize all the reviewed papers.

Other Literature Reviews and our Contribution
In this section, we summarize the other literature reviews, in which supply chain integration is discussed, in order to make our contribution as clear as possible. Bilgen and Ozkarahan (2004) provide a literature survey on supply chain management at the strategic, tactical and operational levels. In particular, they focus their attention on models concerning the problems of production and distribution. The goal is to provide a classification according to the solution approach used: optimization-based methodologies, meta-heuristic-based models, information-technology-driven models, hybrid models. Finally, the most significant research trends are revealed. Mula, Peidro, Diaz-Madronero and Vicens (2010) propose a literature review on mathematical programming models for supply chain production and transport planning. The 44 reviewed papers, covering a period of 20 years (i.e., 1989-2009), are classified and discussed on the basis of some criteria (see Table 1). A detailed review of tactical optimization models for integrated production and transport routing planning decisions is presented by Diaz-Madronero, Peidro and Mula (2015). The authors analyze 22 papers over 20 years  and propose a classification framework, based on the following criteria: production, inventory, and routing aspects, objective function, solution approach. Soto-Silva, Nadal-Roig, Gonzalez-Araya and Pla-Aragones (2016) propose an interesting survey about the application of operational research models to the fresh fruit supply chain. The selected papers are classified according to: decision level (i.e., strategic, tactical, operational), modeling approach, purpose, applicability, novelty. The literature review by Moons, Ramaekers, Caris and Arda (2017) focuses on the integration of production scheduling and vehicle routing decisions at the operational level. The 33 selected papers are first divided into 3 main groups (i.e., single-machine environment, parallel-machine environment, other machine environments) and then classified according to production, inventory, and distribution features, objective, solution approaches.
In Table 1, we summarize the main features of the other literature reviews. In particular, we highlight the reference period and the number of papers reviewed. Furthermore, we show according to which dimensions the collected papers have been classified. As it can be seen, there is no other survey focused on the integration of supply chain activities, in which the papers are jointly classified on the basis of the five dimensions of our framework: supply chain structure, objective function, solution approach, approach validation, perishability type.
Basically, we propose a literature review, where papers dealing with perishable items are collected and classified. One of the main goals is to highlight the most used approaches to address perishability and limit food waste, in order to make supply chains more sustainable. Perishability significantly affects revenue, as goods subject to deterioration are usually sold in accordance with discount policies (Chen, Wu, Wang & Li, 2019), but also costs, as it is necessary to take into account additional requirements regarding production (e.g., restricted time constraints), storage (e.g., refrigerated inventories), and transportation activities (e.g., refrigerated vehicles) (Meneghetti & Monti, 2015;Meneghetti & Ceschia, 2020;Castro & Jaimes, 2017;Giallombardo, Mirabelli & Solina, 2021). Furthermore, the main consequences of decay are lost sales and disposal costs. According to Rohmer et al. (2019), in today's competitive markets, the quality and freshness of food products significantly influence customer satisfaction (Guido et al. 2020), therefore they are fundamental aspects for the survival of each business. Moreover, we review 54 articles, which is quite a significant number when compared to many past literature reviews. It should be noted that there are also other literature reviews on perishable (or in general agrifood) supply chains, but they are not focused on the integration of production, inventory, distribution activities (Siddh, Soni & Jain, 2015;Ganesh-Kumar, Murugaiyan & Madanmohan, 2017;Vrat, Gupta, Bhatnagar, Pathak & Fulzele, 2018).
The remainder of this paper is organized as follows. In Section 2, we give all the details on our research methodology. Section 3 presents a bibliometric analysis about the selected papers. Section 4 is a content analysis, where our classification framework is applied. In Section 5, we outline the main current research gaps, the possible future challenges. Section 6 shows some conclusions.

Research Methodology
In this section, we describe the research methodology used to carry out our literature review. It is characterized by several steps in sequence, as shown in Figure 1. First of all, it is necessary to select one or more databases within which to retrieve the records useful for conducting the survey. Several scientific databases exist. We have decided to query Scopus, which is currently one of the most recognized and comprehensive databases. It contains more than 20,000 peer reviewed journals related to different publishers (e.g., Elsevier, Taylor & Francis, IEEE) and is much more exhaustive than others (e.g., Web of Science, IEEE Explore). Scopus was queried at the end of 2020. The second step of the methodology concerns the keywords selection to carry out the database queries. The main purpose of our study is to collect research works, where quantitative approaches to integrate decisions related to production, storage and distribution activities, are proposed. Therefore, we have decided (i) to combine the keywords linked to these 3 areas (i.e., production, storage and distribution), and (ii) to consider in any case the distribution issues and the perishable nature of the products. Altogether, six queries were conducted, the results of which are shown in Table 2. We have used specific search criteria (see Table 3).

Search criterion Selection
The result of this first search was a set of 366 documents, which were reduced to 88 after reading their title and abstract. Then, after reading the full-text, it was decided to include only 54 documents, as shown in Table 4. Basically, the following exclusion criteria were applied: • Articles not based on quantitative approaches (intended mainly as optimization models).
• Articles not relevant to the topic.
• Articles, where there is no explicit integration between 2 or more activities (i.e., production, inventory, distribution) of the perishable supply chain.

Bibliometric Analysis
In this section, we show the results of our bibliometric analysis, which was partly supported by the free software VOSviewer (1.6.13 version) (Van Eck & Waltman, 2009). We refer only to the 54 selected papers.

Publishing Sources
In Table 5, the distribution of the 54 reviewed articles across the journals is shown. The top 9 journals have published around 60% of the articles (i.e., 32), and the International Journal of Production Economics is the most prolific, with 6 papers. It is followed by

Chronological Distribution
With the aim to have an idea about the interest over the years in supply chain coordination strategies in the case of perishable products, in Figure 2 we show a graph relating to the temporal distribution of the selected papers with reference to the period 2005-2020. As it can easily be seen, interest in the topic of this literature review has grown considerably in recent years. This trend is quite expected for several reasons. In recent years, the development of global markets and higher customer expectations have forced the supply chain players to better coordinate and integrate their plans, in order to maintain high levels of performance and be competitive on the market. Today, in fact, companies compete not only on product price or quality, but also on the reliability and timeliness of deliveries. Therefore, academic interest in models capable of jointly optimizing the production, storage and distribution of perishable goods has significantly grown recently.

Statistics on Keywords
It is also very interesting to highlight the keywords most frequently used by the authors, in order to identify the most recurring research areas. Since the VOSviewer software is unable neither to distinguish between singular and plural nor between words that have the same root and meaning, similar keywords have been merged. In Table 6, for each overall keyword, the list of keywords from which it derives, and the overall number of occurrences, are shown. A hyphen means that the related overall keyword has not changed from the original one.

Original keywords Overall keyword # Occurrences
Perishable good, Perishable goods, Perishable product, Perishable products Perishable good(s)/product(s) 38 Based on the keywords combination used to carry out the search (i.e., in each query, the word "perishable" is present), as expected, the perishability-related keywords are the most frequently used. However, from the above analysis, some important research trends emerge. First of all, it is possible to state that the distribution stage is often addressed and modelled through a vehicle routing problem (VRP). In its most traditional form, a VRP aims to find an optimal set of routes for a vehicle fleet to serve a certain number of customers, while minimizing the total traveling cost (Vidal, Laporte & Matl, 2020). In this context, it is important to underline that the high number of occurrences of the overall keywords "Inventory routing (problem)" and "Vehicle routing (problem)" was certainly influenced by a couple of combinations used to conduct the queries: "production routing" and "inventory routing". The overall keyword "Time window(s)" suggests that the requirement to serving customers within delivery time windows is also taken into account. Moreover, many papers deal with an inventory routing problem (IRP), which is an extension of the VRP, where inventory control and routing decisions are combined (Rafie-Majd et al., 2018). The occurrence of the overall keywords "Genetic algorithm(s)" and "Metaheuristic(s) (algorithm)" is not a surprise because when different supply chain stages are integrated into a single monolithic problem, non-exact approaches are often necessary to find good solutions in a reasonable time. In fact, the large set of decisions and factors to be jointly considered often result in computationally intractable problems (Neves-Moreira et al., 2019).

Featured Authors
In Table 7, the information about the top contributing authors and their relative institution and country, is shown. Almada-Lobo B. is the most prolific author with 4 publications. He has co-authored three of them with Amorim P. (Amorim et al., 2012(Amorim et al., , 2013Belo-Filho et al., 2015). Their research area has mainly focused on addressing the integrated production and distribution planning with perishable products, from an operational point of view. A quite similar topic has been treated by Ghoreyshi S.M. and Seyedhosseini M. in their co-authored papers (Seyedhosseini and Ghoreyshi, 2014a, 2014b. Chu F. and Li Y. has instead co-authored 3 articles, mainly focused on modelling and solving some interesting variants of the production-inventory-routing problem with perishable goods (Li et al., 2016(Li et al., , 2020. The peculiarity of the three Ahumada's papers concerns the study of agri-food supply chains (i.e., pepper and tomato), with a lot of attention to the harvesting stage and the possibility of multiple transportation modes (e.g., trucking, railroad, air) for what concerns the distribution stage (Ahumada & Villalobos, 2011a, 2011bAhumada et al., 2012). Diabat, A. has considered both the location-inventory-routing problem and the inventory-routing problem in his studies (Le et al., 2013;Diabat et al., 2016;Hiassat et al., 2017).
In Table 8, we show the 10 countries, which have the highest number of documents, based on the origin of the authors. Iran and United States are the most prolific. Moreover, we can state that the challenge of integrating decision-making in perishable supply chains has been taken up to a quite similar extent in 3 continents, namely Europe, Asia, America. As mentioned, globalization has involved companies and supply chains all over the world, therefore it is not possible to identify a specific country for which these problems are much more acute than others.
With the aim to better understand the influence of the various authors on the topic of this paper, in Figure 3 we provide a citation network analysis regarding the authors of the 54 selected papers, performed through VOSviewer.
Almada-Lobo B. and Ahumada O. are the most cited authors, but the first one, as it can be easily seen from Figure 3, is much more connected with the other selected papers. Diabat A., Laporte G. and Coelho L.C. are the three authors with the highest "Total link strength", which means their papers are very influential for what concerns the topic addressed in this manuscript.

Content Analysis
In this section, the content of the 54 selected papers is widely discussed. A five-dimension classification framework, as shown in Figure 4, is proposed. Such a framework was built inductively. Basically, by reading and studying the selected articles, the authors identified 5 macro-areas to better classify them and derive the main research trends and possible future challenges as regards optimization strategies for the integrated management of perishable supply chains: supply chain structure, objective, perishability type, solution approach and approach validation.

Supply Chain Structure
In this subsection, we report some statistics on the supply chain structure of the 54 selected papers. First, with the aim to give an idea about the complexity of each addressed problem, we define a very critical indicator, which is the number of modeled supply chain stages. As shown in Figure 5, 72 % of the papers concern a two-level structure. The remaining papers address a more complex supply chain structure. Wu et al. (2015) tackle a problem characterized by three main levels: a single supplier, a set of potential facility locations, a set of retailers. One of the main contributions is the modeling of the facilities as cross-docking points. Similarly, Dolgui et al. (2018) address a supply chain, made up of multiple plants, multiple cross-docks, multiple markets. Cross-docking is a logistics concept, which aims to coordinate as much as possible the arrival of goods and their next shipment in order to minimize storage times and optimize distribution to end customers. This approach favors economies of transportation and, above all, enables hub-and-spoke networks which replace the more traditional point-to-point structures (Stephan & Boysen, 2011). A similar concept is used by Hu et al. (2018), who consider a set of growers, a consolidation center, and a set of retailers/wholesalers. Basically, the perishable product is transferred from local growers to a consolidation center via short-haul routing, while long-haul routing is necessary for the shipment from the consolidation center to geographically dispersed nodes. Vahdani et al. (2017) formulate and solve a mathematical programming model for simultaneous scheduling of production and delivery of perishable products to customers. The production system is divided into two stages, each of them having several production sites. The last level includes, instead, a set of retailers, that place the different orders. Rafie-Majd et al. (2018) consider a three-echelon supply chain, characterized by a supplier, a set of distribution centers, a certain number of retailers, within an integrated inventory-location-routing problem. This structure is very similar to that by Chao et al. (2019), where instead the suppliers/manufacturers are multiple. Rohmer et al. (2019) consider an intermediate depot between a single supplier and multiple customers. Violi et al. (2020) take into account a real agriculture supply chain in Italy, where a single perishable product is moved from a set production plants to a set of retailers, through a supplier located in the middle. Only two papers consider a four-echelon supply chain. Bortolini et al. (2016) place two intermodal hubs between multiple producers and multiple retailers. Biuki et al. (2020) take into account suppliers, manufacturers, distribution centers, and customers. Lastly, the three papers written by Omar-Ahumada and Rene Villalobos Vilallobos, 2011a, 2011b;Ahumada et al., 2012), with the aim to replicate as much as possible the behavior of the fresh agricultural supply chains, model explicitly the following stages: harvesting, packing, warehousing, distribution centers, customers. Then, we have defined some parameters related to the 3 main areas of this literature review: • Production area: # suppliers/plants/production facilities, # products.
• Inventory Area: inventory modelling at suppliers/plants/production facilities, inventory modelling at retailers/customers.
• Distribution area: # retailers/customers, # vehicles, nature of the fleet of vehicles.
We specify that when we talk about # suppliers/plants/production facilities and # retailers/customers, we are referring respectively to the first and last level of each modelled supply chain. In Figures 6-11, the main statistics about the supply chain structure of the analyzed papers are summarized. The suppliers / plants / production facilities are in most cases (i.e., 68 %) single, as well as the number of products (i.e., 57 %). It should be noted that 2 papers speak of jobs rather than products (Marandi & Zegordi, 2017;Bank et al., 2020). The retailers/customers are always multiple, except for the research work by Zanoni and Zavanella (2007), where a single-vendor-to-single-buyer model is addressed. Inventory is modeled at least upstream or downstream in most papers. Instead, it is completely absent in 22 % of cases, especially when authors model production-distribution problems, in which it is not necessary to store finished products before their distribution. The vehicles are multiple and homogeneous in the vast majority of cases. It should be noted that Lacomme et al. (2018) refer to both a single vehicle case (i.e., literature instances) and a multiple vehicle case (i.e., new instances). The number of vehicles is not explicitly specified in some papers, because the possibility of multiple transportation modes is taken into account (e.g., trucking, railroad, air) (Ahumada & Villalobos, 2011a, 2011bAhumada et al., 2012;AriaNezhad et al., 2013;Bortolini et al., 2016) or third-party logistics service providers (3PL) are considered for what concerns the distribution phase (Amorim et al., 2012).

Objective
In Tables 9-11, the 54 selected papers are classified according to their objective. For what concerns the mono-objective minimization, we refer to production cost (PC), inventory cost (IC), distribution cost (DC). The term "other" indicates cost items, which do not explicitly belong to one of the aforementioned categories.

PC IC DC Other
Zanoni & Zavanella (2007 Table 9, only when explicitly considered; in fact, in many cases, it is implicitly included into the inventory costs. Only one paper introduces a risk measure within the objective function (Violi et al., 2020). They give a mean-risk structure to the objective, that is written as a convex combination of two terms: the expected value of the overall costs and a risk measure, that is the conditional value at risk (CVaR) (Rockafellar & Uryasev, 2000). Except for one case (Viergutz & Knust, 2014), the maximization of the objective function is always about profit. In only five papers, the multi-objective optimization is addressed. In this case, some sustainability-oriented objectives are considered, such as carbon foot print minimization (Bortolini et al., 2016;Rahimi et al., 2017;Chan et al., 2020), environmental efficiency of the logistics chain network and social sustainability maximization (Biuki et al., 2020).

Perishability Type
Supply chain management in the case of perishable items is quite complicated and time-critical (Biuki et al., 2020;Alkaabneh, Diabat & Gao, 2020). The main aim of this part is to provide a picture of the ways in which the different authors have addressed perishability. In the literature, the most recognized classification of perishability concerns two types of items: (i) fixed-lifetime and (ii) age-dependent. Fixed-lifetime goods have a well-defined expiration date, beyond which they perish (e.g., dairy products, pharmaceuticals). Age-dependent goods, instead, are subject to deterioration, then they lose value over time (e.g., agricultural products); although they have an expiration date, it is not predetermined (Coelho & Laporte, 2014;Palak et al., 2018). The 54 selected papers are classified according to such a taxonomy. Figure 12 shows how perishability issues are dealt in the literature. Fixed shelf-life and deterioration are almost equally popular as a way of dealing with perishable conditions. Some papers use even both assumptions, while others deal with perishability implicitly, without therefore specifying whether the good has a well-defined expiration date or loses its quality over time.  (Osvald & Stirn, 2008); basically, they take into account explicitly the loss of product quality over time. Coelho and Laporte (2014) define a discrete set for the product age, and they study the impact of item age on revenue and inventory holding costs. They claim that their approach can also be used in the case of products subject to deterioration. Amorim et al. (2012) formulate two different models to consider respectively the cases of fixed and loose shelf-life. In this latter case, the authors link the shelf-life to the knowledge of predictive microbiology, then to the stocking temperature. The use of a multi-objective framework allows to take into account, in both cases, the freshness of the product, therefore the customers' willingness to pay. S.M. Seyedhosseini and S.M. Ghoreyshi claim that, although their optimization models were designed for perishable products with fixed lifetime, they can be extended to deteriorating goods that decrease their value throughout the lifetime. In this case, the quality loss of goods should be included in the inventory costs (Seyedhosseini & Ghoreyshi, 2014b;Seyedhosseini & Ghoreyshi, 2015). In some papers, reaching the expiration date is penalized in the objective function through the waste cost (Soysal et al., 2015;Soysal et al., 2018;Rafie-Majd et al., 2018). With the aim to limit the amount of unsold products, Rahimi et al. (2017) introduce a step-wise nonlinear holding cost. They take into consideration that non-fresh products need extra-inspections before they are carried to the next period. The proposed holding cost function replicates the price discounts for non-fresh products, and guarantees a trade-off between economic, service level and environmental criteria. Li et al.
(2020) take into consideration different shelf-lives, depending on the packaging. Their article is mainly based on the following assumption: the same item can have a different shelf-life depending on the packaging used. Basically, innovative food packaging can significantly lengthen the expiration date of products, lowering the decay rate (Rizzo & Muratore, 2009;Li, Yu, & Wu, 2017). The authors investigate the tradeoff between packaging costs and shelf-life benefits. One of the main contributions of Biuki et al. (2020) is instead to consider both the shelf-life of raw materials and finished products, within the proposed Mixed-Integer Programming (MIP) model.

Deterioration
In 25 papers, the shelf-life is not fixed and known a priori, but the aspects related to deterioration and loss of quality are emphasized.
A set of research works traces explicitly the food quality throughout the supply chain, by using a quality level index (Li et al., 2016;Ghasemkhani et al., 2019;Chan et al., 2020). They stress the assumption that food quality significantly affects selling price and customer demand.
Most papers propose a fixed decay rate, which implies a linear deterioration of the product quality (Chen, 2009;Ahumada & Villalobos, 2011a;Ahumada et al., 2012;Farahani et al., 2012;Rohmer et al., 2019;Violi et al., 2020;Dai et al., 2020). It must be noted that Chao et al. (2019) take into account both the deterioration due to the transport time and that due to a break during the transportation process (e.g., turning on/off frequently the door of a truck).
Ahumada and Villalobos (2011b) focus their attention on the loss of value of agri-products, once harvested. They use a post-harvest color function (Hertog, Lammertyn, Desmet, Scheerlinck & Nicolai, 2004), assuming to store the products at a constant temperature. Basically, in the objective function of the proposed optimization model, an expected cost is used, derived from rejected or discounted shipments, which depend on the color of the product when it reaches the customer. Deterioration is addressed by Mirzaei and Seifi (2015), by considering lost sale as a linear or exponential function of the inventory age. A couple of papers assume exponential deterioration rate (Azadeh et al., 2017;Dolgui et al., 2018). This assumption takes into account the growth of micro-organisms within the product, which in many cases follows an exponential behavior. Although Accorsi et al. (2017) consider a set of shelf-life values in their model, they explicitly take into account the quality decay of perishable products. In particular, they refer to kinetic models, based on the Arrhenius equation and the accelerated aging factor (Lee, Yam & Piergiovanni, 2008;Tsironi, Dermesonlouoglou, Giannaglou, Gogou, Katsaros & Taoukis, 2017). Common thermodynamic models are instead considered to replicate the heat transfer mechanism in refrigerated storage rooms and vehicles. In some models, the deterioration rate is time-dependent (Wu et al., 2015;Vahdani et al., 2017;Qiu et al., 2019;Wei et al., 2020;Sinha & Anand, 2020). Onggo et al. (2019) consider the case of inventory of products with different ages. They address product perishability, by considering multiple degradation speed levels. Manoucheri et al. (2020) use the Gompertz equation (Gil, Miller, Brandao & Silva, 2011), with the aim to estimate microbial growth.

Solution Approach
In this section, we group the 54 selected papers, according to the approach used, to solve the addressed problem.

Exact Approaches
In many research works, the proposed solution approach is the main contribution. The models that integrate the planning of production, storage and distribution activities are generally very complex from a computational point of view, therefore they require heuristic approaches (i.e., not exact), to find a good solution in reasonable time. For this reason, in a few papers the solution is determined solely through an optimal approach by using a software (e.g., CPLEX, LINGO, MATLAB) (Soysal et al., 2015;Accorsi et al., 2017;Soysal et al., 2018;Ghasemkhani et al., 2019). Some authors do not use heuristic approaches, but impose an optimality gap, with the aim to limit the computational time (Ahumada & Villalobos, 2011a, 2011bAmorim et al., 2013). The branch-and-cut is adopted within different papers (Coelho & Laporte, 2014;Shaabani & Kamalabadi, 2016;Qiu et al., 2019;Li et al., 2020;Wei et al., 2020), while Viergutz and Knust (2014) use the branch-and-bound algorithm. In many other cases, the optimal or sub-optimal solutions are determined, only with the aim to demonstrate the efficiency, in terms of computational time, of the proposed novel algorithm, through a specific comparison. Figure 13 summarizes some statistics about number of papers, using exact approaches among those selected in this comprehensive literature review. Figure 13. Statistics on the use of Exact Approaches

Genetic Algorithms
Genetic algorithms aim to mimic biological evolutionary processes and have been successfully used for solving many optimization problems (Godinho-Filho, Barco & Tavares Neto, 2012;Lee, 2018). Naso et al. (2007), with the aim to solve the problem of coordinating the production and distribution planning within a network of independent supply centers, propose a hybrid metaheuristic approach, where genetic algorithms and constructive heuristics are integrated. Amorim et al. (2012) formulate models for the integrated production and distribution planning, where perishable goods have fixed or loose shelf-life. For the second case, a hybrid genetic heuristic is proposed. A genetic algorithm, coded in MATLAB, is designed by AriaNezhad et al.

Neighborhood Search
Some meta-heuristics exploit the concept of solution neighborhood: variable neighborhood search (VNS) and large neighborhood search (LNS) are among the most used approaches in this context. VNS is based on two main steps: a descent phase to find a local optimum and a perturbation to "escape" from the corresponding valley (Hansen, Mladenovic, Brimberg & Moreno Perez, 2010). About LNS, instead, an initial solution is gradually improved through some phases of destruction and reparation (Pisinger & Ropke, 2010). Farahani et al. (2012) design a novel approach, that integrates short-term production and distribution planning, within an iterative scheme. The distribution problem is solved through an LNS algorithm. Belo-Filho et al. (2015) develop an adaptive LNS for solving the operational integrated production and distribution problem with perishable products. The adaptiveness of the approach is related to the destroy and repair operators, that are chosen adaptively. The authors demonstrate that the adaptive LNS outperforms exact approaches and the fix-and-optimize method. An adaptive LNS metaheuristic is also developed by Rohmer et al. (2019) for tackling an inventory-routing problem. An LNS-based heuristic is used by Vahdani et al. (2017) for a production-inventory-routing problem. Such an approach appears quite promising, when compared with LINGO and a water cycle algorithm (Eskandar, Sadollah, Bahreininejad & Hamdi, 2012). The hybrid search algorithm proposed by Manouchehri et al. (2020) combines VNS and SA with good results for a production routing problem. Li et al. (2020) use an improved LNS, which outperforms a genetic algorithm, in solving an integrated production and distribution problem with a minimum total order weighted delivery time.
Seyedhosseini and Ghoreyshi (2014a) study an integrated model for production and distribution planning of perishable products. While the production submodel is solved by LINGO, a PSO-based heuristic is proposed with good results for the distribution part. Marandi and Zagordi (2017) design and apply an improved PSO to deal with a production-distribution scheduling problem. The contribution mainly lies in the use of additional operators (i.e., 1-exchanged and 2-opt), with the aim to prevent the premature convergence of the algorithm. Chan et al. (2020) propose a modified multi-objective particle swarm optimization algorithm with multiple social structures. With the aim to efficiently solve an integrated location-routing-inventory problem, Biuki et al. (2020) propose two hybrid metaheuristics as parallel and series combinations of GA and PSO. The computational experience shows that the parallel approach is better than the series one.

Simulated Annealing
Simulated Annealing (SA) is a technique that has gained a lot of popularity in recent years in solving optimization problems. It is based on an analogy with the behavior of physical systems during the cooling process (Suman & Kumar, 2010). Mirzaei and Seifi (2015) combine SA and Tabu Search (TS) for an inventory routing problem. Shaabani and Kamalabadi (2016) present a population-based simulated annealing algorithm (PBSA) for a multi-product multi-retailer perishable inventory routing problem. The PBSA is compared with SA and GA, to show its superiority in terms of efficiency. Bank et al. (2020) propose a low-level co-evolutionary hybrid algorithm to solve an integrated production-distribution problem. The approach combines SA and GA, because the features of GA are applied in the local search process of SA.

Tabu Search
For what is known, Tabu Search was proposed by Glover (1986) and is nowadays one of the most used heuristic methods for combinatorial optimization. Viergutz and Knust (2014) propose model extensions with reference to the research work by Armstrong et al. (2008), which addresses an integrated production-distribution scheduling problem. They solve some of them by a tabu search approach. Diabat et al. (2016) design a hybrid tabu search, which outperforms a column generation approach in solving a periodic distribution inventory problem.

Approach Validation
Each proposed solution approach needs to be properly validated. In this subsection, the selected papers are classified according to the nature of the instances through which the goodness of each algorithm has been demonstrated, see Table 12. Some authors use instances already known in the literature. Others, introducing completely new problems, are forced to randomly generate new data. While, in some cases, real data are used, derived from specific case studies. Furthermore, the supply chain type is reported; obviously, it is specific, only when referring to a case study (real or hypothetical).

Case Study
Zanoni & Zavanella (2007  Only a couple of papers refer to non-food products. Naso et al. (2007) focus on the production of ready-mixed concrete for construction engineering and refer to a supply network in Northern Europe. The instances replicate the typical workdays of the different nodes of the network. The authors claim that the perishability features of ready-mixed concrete have many similarities with agri-food products. Sinha and Anand (2020) refer, instead, to an Iranian glass industry, whose main information is reported in (Hajiaghaei-Keshteli and Fathollahi Fard, 2019).
10 papers are about fruit and/or vegetables. The three articles by Omar Ahumada and J. Rene Villalobos (Ahumada & Villalobos, 2011a, 2011bAhumada et al., 2012) refer to a hypothetical producer of peppers and tomatoes, based in Mexico. Farahani et al. (2012) develop a main set of instances, based on the real setting of a catering company in Denmark. Additional test sets are included, in order to validate the solution approach under different problem sizes. Some parameters are randomly generated, while others are derived from the Solomon instances (Solomon, 1987). AriaNezhad et al. (2013) consider a manufacturing company of conserved wax bean and jam. Soysal et al. (2015) apply their model to a real supply chain located in Turkey, where a distribution center provides fresh tomatoes to 11 supermarkets. Several Key Performance Indicators (KPIs) are introduced to highlight the benefits, coming from the proposed mathematical approach: average vehicle load, number of vehicles used, total emissions, total driving time, total fuel cost, total inventory cost, total waste cost, total cost. Bortolini et al. (2016) address the distribution of six types of fruits and vegetables (i.e., potatoes, apples, pears, Brussels sprouts, oranges, tomatoes) cultivated by a set of Italian producers to supply a set of European retailers. They consider and compare three different transportation modes (i.e., truck, train, airplane). The proposed expert system, called Food Distribution Planner, can effectively manage product perishability and limit CO2 emissions. Accorsi et al. (2017) consider an Italian supply chain of fresh cherries, characterized by 4 production plants aimed at processing raw cherries and packaging, and 3 warehouses. The application of the proposed Mixed-Integer Linear Programming (MILP) model can significantly impact on economic and environmental aspects of the considered cold chain. Soysal et al. (2018) prove the benefits of horizontal collaboration between two real suppliers, which produce figs and cherries, respectively. The data used by Violi et al. (2020) come from an Italian supply chain, where a medium agri-food company supplies tangerines to a set of retailers.
4 papers refer to meat products. A very challenging case study is addressed by Neves-Moreira et al. (2019): 175 multiple meat products are manufactured by a meat processing center and delivered to 185 meat stores, by using 35 heterogeneous vehicles. The proposed integrated approach can guarantee a cost saving of 21.73 %, compared to the company's solution. Qiu et al. (2019) consider a Chinese chain, where a food company provides fresh meat products to some stores. The proposed approach ensures a decrease of 12 % in the total cost. Manouchehri et al. (2020) refer to a company in Iran, that supplies packed chicken to a set of customers, by refrigerated vehicles. The developed optimization model returns the production lot size, inventory levels and optimal routes for each vehicle. The Chinese supply chain considered by Chan et al. (2020) is made by a company, which provides a single fresh meat product to 40 retail stores. 3 homogeneous vehicles are used for the distribution phase, under a planning horizon of one week. The model designed by the authors provides a 15.51 % cost reduction.
Three papers concern respectively dairy products (Azadeh et al., 2017), cut flowers (Hu et al., 2018), fresh seafood (Chao et al., 2019). All the remaining papers randomly generate new data, or refer to some well-known literature instances, as specified in Table 12.

Discussion: Research Trends and Future Challenges
In this review paper, a framework for classifying scientific articles, which address the coordination of activities in perishable supply chains, via optimization strategies, has been proposed. This framework contains five dimensions: objective, perishability type, solution approach, approach validation, supply chain structure. Some important research trends can be detected: • Most of the publishing journals belong to the areas "management science and operations research", "industrial and manufacturing engineering".
• Interest in the application of optimization models for integrated decision-making along perishable supply chains is strongly growing. More than 50 % of the articles published in the last 15 years refer to the period 2016-2020.
• A careful analysis of the selected documents shows a strong interest in the following main problems, as shown in Table 13: the integrated production-distribution problem, the inventory-routing problem, the production-routing and the production-inventory-routing problem, the location-inventory-routing problem.  Table 13. Most recurrent problems

Reference
• Integrating multiple stages of the supply chain into a single framework is complex, especially when referring to perishable products. The vast majority of the problems addressed are then NP-Hard, therefore non-exact approaches are needed to find a sub-optimal solution in reasonable time. Currently, evolutionary algorithms (e.g., genetic algorithms) are among the most preferred by researchers.
• A very high percentage of the reviewed research works deal with two-level supply chains, i.e., supplier(s)-to-customer(s). When the supply chain is 3-tier, a cross-docking or consolidation node is often placed halfway.
As a consequence, there are some significant challenges that need to be addressed in the next future: • In more recent years, the research branch concerning electric vehicles is spreading significantly (Macrina, Di Puglia Pugliese, Guerriero & Laporte, 2019;Zhen, Xu, Ma & Xiao, 2020). Erdogan and Miller-Hooks (2012) defined the green vehicle routing problem, where the fleet is composed of alternative fuel vehicles, which have a very positive environmental impact. Currently, this topic is very little explored within the perishable supply chains. Furthermore, only a few of the reviewed research works use CO2 emissions as a supply chain KPI.
• As for the distribution phase, very few papers evaluate the possibility of using transportation modes different to truck (e.g., train or airplane) and even intermodal ones. This research line deserves to be further investigated because it could make perishable supply chain more efficient and guarantee a higher service level to the end customer.
• Only one of the reviewed papers (Soysal et al., 2018) deals with horizontal collaboration as regards the distribution of goods with limited shelf-life. This topic deserves to be better developed in the coming years because it is very sustainability-oriented. Sharing vehicles for shipping means limiting CO2 emissions and then protecting environment (Pan, Trentesaux, Ballot & Huang, 2019;Giallombardo et al., 2021).
• The potential of Industry 4.0 (Ivanov, Tang, Dolgui, Battini & Das, 2021) is little explored. Blockchain is an emerging technology, which has a significant impact on efficiency and sustainability of supply chains (Saberi, Kouhizadeh, Sarkis & Shen, 2019;Saurabh & Dey, 2020;Astarita, Giofrè, Mirabelli & Solina, 2020). It would be very interesting to understand how improving the quality of information exchange between the various nodes of the chain can impact on the coordination of activities. Basically, blockchain could better guarantee the demand-supply matching, which would result in the reduction of perished products (i.e., waste). Moreover, many of the documents examined deal with production planning, but the features of smart manufacturing appear very little exploited until now.
• There is a significant need to give impetus to the development of multi-objective models, in order to pursue not only economic, but also environmental and social sustainability.
• Only one of the reviewed papers explores the relationship between product shelf-life and packaging used (Li et al., 2020). More research is needed addressing this emerging topic. More expensive packaging is usually also more sustainable and extends the product lifetime. In this context, it would be very useful to have more investigations on the impact that different packaging can have on the economic and environmental sustainability of the perishable supply chains.
• Only 35% of the selected papers aim to solve real-life case studies. The reasons behind this low number could be linked to poor collaboration between academia and industry. There is a need for further research, which is capable of modeling and quantitatively improving existing supply chains. In fact, it is very important to have useful approaches available to practitioners and entrepreneurs, in order to create a stronger link between academia and industry.

Conclusions
The analysis of the collected documents shows that there is a growing interest in the topic of integrated management of perishable supply chains. At the moment, most of the papers concern 2-level supply chains and the proposed optimization models are mainly validated theoretically. Regarding the concept of perishability, there is a balance regarding the number of papers concerning fixed shelf-life products and subject-to-deterioration goods. Currently, the main gap in the literature concerns the limited number of case studies, probably due to a poor academia-industry connection; furthermore, the potential of Industry 4.0 still seems little explored.
The main limitation of this research consists in having used a limited number of keywords for the collection of documents, and in having excluded all those papers not focused on the adoption of optimization models, as a means of managing supply chain in an integrated manner.
As this literature review collects the main scientific articles published between 2005 and 2020, related to the integrated management of production, storage and distribution activities in perishable supply chains, it can be a starting point for all scholars and practitioners who want to study and deepen such a topic in the coming years.