Shelf life modelling for first-expired-first-out warehouse management

Abstract

In the supply chain of perishable food products, large losses are incurred between farm and fork. Given the limited land resources and an ever-growing population, the food supply chain is faced with the challenge of increasing its handling efficiency and minimizing post-harvest food losses. Huge value can be added by optimizing warehouse management systems, taking into account the estimated remaining shelf life of the product, and matching it to the requirements of the subsequent part of the handling chain. This contribution focuses on how model approaches estimating quality changes and remaining shelf life can be combined in optimizing first-expired-first-out cold chain management strategies for perishable products. To this end, shelf-life-related performance indicators are used to introduce remaining shelf life and product quality in the cost function when optimizing the supply chain. A combinatorial exhaustive-search algorithm is shown to be feasible as the complexity of the optimization problem is sufficiently low for the size and properties of a typical commercial cold chain. The estimated shelf life distances for a particular batch can thus be taken as a guide to optimize logistics.

1. Introduction

The globalization of supply networks makes the task of supply chain management more and more challenging and often requires strategic shifts to continue to meet market demands. In the supply chains for perishable products, such as processed and fresh food products, the partners have a shared responsibility of minimizing quality losses to deliver high-quality products to the end users. In spite of their efforts, a large portion of what is produced is never consumed. In the case of fresh fruits and vegetables, the combined post-harvest loss and waste can reach as high as 50% of the produced volume [1].

Food loss refers to the decrease in edible food mass at the production, post-harvest and processing stages of the food chain owing to processes such as weight loss, microbial rots, diseases and insect damage. Food waste, a symptom merely characteristic of developed countries' consumerist lifestyles, refers to the discard of products not meeting set quality standards, waste generated during processing, surpluses during catering and consumption, and unsold volumes running out of shelf life owing to a mismatch between supply and demand. Some of these factors are inherent to the perishable character of the food products while others are clearly economically and socially determined. To be able to sustain a growing world population with enough food within the restriction of limited land resources, the global supply chains for perishable products, such as processed and fresh food products, should above all focus on reducing existing food loss and waste by intelligent food logistics [2].

By understanding the behaviour of our food products in response to the handling conditions of the supply chain, logistics can be further improved. To reach a successful integrated chain management approach, several international efforts have been developed focusing on the various relevant aspects, such as sensor technology to monitor the logistic conditions [3], radio frequency identification (RFID) and GPS technology to enable fast communication throughout the supply chain [4,5], improved transport modalities to guarantee better climate control [6], new warehouse management approaches dedicated to perishable products [7], shelf life models to predict the product's behaviour [8] or a combination of one or more of these aspects [9]. The ultimate path to optimizing perishable food logistics would engage all of the aspects mentioned above while taking the product's requirements as the central instruction leaflet to shape the supply chain around it.

This contribution focuses on the use of prediction models describing product quality changes during handling and transport and on how this information can be incorporated into warehouse management systems, moving emphasis from the classical first-in-first-out (FIFO) towards a first-expired-first-out (FEFO) strategy. By implementing such a model-based approach, the flow of perishable goods can be optimized by taking into account the expected shelf life of the products. By doing so, unnecessary losses throughout the supply chain can be prevented, thus minimizing economic losses as well, while clients can be served better by providing them with product meeting their requirements and will also help match the product-holding versus demand ratios [10].

2. Quality in the supply chain of perishables

(a). Concepts of shelf life and keeping quality

The quality of a horticultural product is largely based on the subjective consumer evaluation of a complex of quality attributes (such as taste, texture, colour and appearance), which are based on specific product properties (such as sugar content, volatile production and cell wall structure) [11,12]. What constitutes quality largely depends on the social and economic background of the consumer and the intended usage of the product. Quality can be seen resulting from the concerted action of several quality attributes, each based on its own physiological or physical product property. These product properties generally change over time, as part of the normal metabolism of the product.

In general, quality decreases with time. In spite of all efforts, post-harvest handling will not improve the quality of a product; it can only delay the process of quality loss. Only in some exceptional cases, one might interpret the changes as an improvement in quality such as in the case of fruit ripening. Depending on the position of the product in the supply chain, this might be interpreted as a gain (offering ready-to-eat fruit in a supermarket) or a loss (fruit becoming too ripe to be shipped to distant markets).

This raises the concepts of shelf life and keeping quality [13]. Keeping quality refers to the time it takes under real-life supply chain conditions before quality falls below some limit, making the product unacceptable, while shelf life is the keeping quality under well-defined storage conditions (e.g. air storage at 18°C and 80% relative humidity). Generally, the limiting factor with regard to consumer acceptance can be pinpointed to a single quality attribute. To predict keeping quality of a product, monitoring of this single attribute suffices but does not necessarily give a complete picture of the quality. For this, a more elaborate compound quality index is required.

(b). Environmental factors affecting product quality

The quality of perishable products is not a static parameter but is a highly dynamic variable. Depending on the supply chain conditions, quality will change over time at varying rates. One of the premises in the perishable food industry is that the physiological, microbial and (bio)chemical processes responsible for quality loss can be suppressed by manipulating the conditions under which the produce is stored, packaged and transported. Generally, most emphasis is placed on the control of temperature followed by humidity, and the levels of oxygen and carbon dioxide. To understand the mode of action of these environmental factors, a good understanding of how relevant product properties depend on storage conditions is required.

Temperature is the main factor affecting all (bio)chemical processes through its effects on activation enthalpy and entropy of the underlying reactions [14]. This is valid for both enzymatic and non-enzymatic reactions and therefore applies to a wide range of fresh and processed food products. While low temperatures are often required to extend shelf life, some products (e.g. most tropical fruits) are sensitive to low-temperature decay, resulting in a reduced shelf life [15]. In addition, mechanical cooling goes hand in hand with drying of the air, inducing saleable weight loss and often also affecting the product's appearance through wilting or shrivelling [16]. For this reason, relative humidity is considered the second most important factor affecting quality. To minimize water loss, additional humidification or proper packaging is required. At the same time, too high humidity can induce moistening of, for instance, dried products and microbial rot of many fresh and processed food products. Hence, relative humidity plays an important role in the conservation of both dry and water-rich fresh and processed food products. Given the importance of temperature control in the supply chain of perishable food products, one often tries to maintain proper low temperatures throughout the supply chain. Such a temperature-controlled supply chain is often referred to as a cold chain.

Many food products are derived from living plant or animal parts, and in an unprocessed or minimally processed form these products continue to exhibit an active metabolism required to maintain the biological integrity of the tissue. By proper control of the levels of oxygen (O₂) and carbon dioxide (CO₂) in the storage atmosphere, the product's metabolism can be controlled; by reducing the levels of O₂ and increasing the levels of CO₂, the metabolic rate can be suppressed to the bare minimum, reducing the energy requirements and maximizing the product's shelf life [17,18]. In addition, high CO₂ is known to inhibit microbial growth and thus, from a microbial point of view, contributes to the extension of shelf life [19]. Temperature strongly interacts with the effects of O₂ and CO₂ on both the energy-demanding and -producing processes. Some quality-degrading processes are affected more than others owing to the way they depend on the composition of the storage atmosphere [17].

The majority of the sensor applications in the supply chain focus on measuring and logging the supply chain conditions, across time to be used for first-, second- and third-order supply chain logistics (figure 1). First-order logistics use the raw data for compliance issues, to see whether the conditions remained within their prescribed ranges throughout the supply chain. Going one step further, second-order logistics involve processing the monitored data into more useful information such as product quality and remaining shelf life. Finally, third-order logistics use this derived product quality and remaining shelf life data for smart supply chain decisions such as FEFO strategies.

Figure 1.

First-, second- and third-order logistics in a monitored cold chain. (Online version in colour.)

3. Warehouse management for perishables

(a). Different strategies

Warehouse management may be viewed as the ability to coordinate both incoming and outgoing goods to limit waste product arriving or leaving the warehouse. At individual warehouses or distribution centres (DCs), which serve as a hub to other warehouses, this process has often been mastered and tailored to the individual product and/or asset being handled. That is, the strategy adopted will have to consider (i) the product deterioration rate and (ii) product demand [20]. Irrespective of the strategy adopted, the primary aim is to deliver efficiencies across a number of business processes, including a reduction in product lead times and also a reduction in product quality losses. Collectively, these systems aim at reducing the cost of business operations and adding value to the supply chain. Such systems mainly function by facilitating the flow of information in parallel with the flow of product, increasing supply chain transparency. However, these in-house systems often function in isolation and may all too often be incompatible with other systems across trading partners across a supply network. This level of incompatibility will lead to disjointed and inaccurate transfers of information across the supply network, resulting in an inability to determine the quality and integrity of many of the incoming goods at an individual DC.

The possibilities to address such issues depend on the level of ownership across the supply chain. If an organization assumes complete ownership across primary production, secondary processing and distribution to retail, this facilitates overall process control and makes the task of information sharing much easier. It allows information-sharing channels to be created across the full supply network where product flows in parallel with its quality and integrity information. This adds an element of transparency both internally within one's own warehouse and also across trading partners, as recommended in the global reporting initiative [21]. However, in reality complete supply chain control and, more importantly, asset visibility across all stages is all too often lacking. When evaluating warehouse management strategies, it is important to consider that each warehouse is part of a wider supply chain spanning across countries, nations and often continents, and that for each warehouse it is necessary to consider the product's history providing an appreciation of the product.

(b). First-in-first-out and first-expired-first-out

Common warehouse management and supply chain strategies aimed at efficient product management across the distribution chain include FIFO and FEFO. FIFO is the more commonly adopted approach as it seems to be a logical choice towards asset rotation, ensuring stock is shipped out based on its arrival date at each individual DC. This approach requires each individual warehouse or DC to first ship products that have spent most time on site irrespective of their remaining shelf life and their final destination [22]. This approach makes the often-criticized assumption that all products arriving on a particular date have the same shelf life potential, which all too often is not the case.

A FEFO approach makes different assumptions in terms of a product's shelf life. FEFO will only ship products depending on their shelf life potential in relation to their end destination. It will only ship goods when their expiry date is known, thus ensuring only high-quality products arrive at their destination and eliminating product loss during transport. The transition to a strategy of FEFO requires the implementation of information-sharing highways across supply chains between trading partners. This enables a data-driven supply network that will give the DC manager more information about the integrity (shelf life) of all incoming goods and, as a result, the DC manager may then choose to distribute goods based on their remaining shelf life. In these cases, DCs will now need to coordinate a ‘days to destination’ approach to logistics. Also, DC managers will be able to view the complete history of a particular product across primary production, secondary processing and distribution, which goes beyond a one-step-back, one-step-forward approach.

(c). Inter-company relations and data exchange

Global financial trading platforms rely on the ability to capture, interpret and transmit data across the globe in real time, without which the global financial market would not survive. Similarly, commodity supply chains, which once adopted a local approach towards trading, have expanded exponentially and nowadays operate on a global scale across time zones and national and international markets as well. To this end, all related activities, including sourcing, logistics, processing, storage and distribution, need to be adapted to meet such a global scale. The key to success is to ensure that the physical product and its corresponding information travel in synchrony across primary production, storage and distribution [23].

To implement these information-sharing highways, it is important to develop a ‘cyber-highway’ infrastructure that will relay product information across the full supply network. It will bridge the traditional cyber–physical gap between the flow of product and the corresponding flow of information. This cyber–physical link will objectively and accurately provide the essential pre-requisites of a responsive, fully flexible global supply chain essential to address modern-day food security issues and to reduce post-harvest food losses.

Information resources come in a variety of formats from a variety of sources both internal within the organization and outside from trading partners or competitors. As a consequence, organizations need to synchronize the information traded using similar languages, formats, structuring and information and make this information available in the correct way at the correct time [24]. Only then can trading partners begin to understand their supply chain as this valuable data source rich in management information will help identify value-adding and non-value-adding processes. From here, one can develop decision-supported algorithms to feed information into diagnostic systems, which directly improve the operational efficiency of the supply chain. When used correctly information resources can (i) provide full chain transparency, (ii) form the architecture of an early identification system and (iii) provide an invaluable recourse in the decision-making process at both strategic and exception management.

4. Modelling approaches for warehouse management

A wide range of mathematical models have found their application in the wider food area [25]. Within the framework of developing models for warehouse management, three approaches of increasing complexity are considered: (i) statistical process control (SPC), where conditions are monitored and controlled to stay within limits defined by statistical concepts, (ii) generic shelf life models, where the shelf life of a product is modelled as a function of the conditions in the logistic chain, and (iii) specific quality attribute models, which describe a specific quality-related property of a specific product (e.g. avocado ripening, strawberry spoilage and mushroom browning) as a function of the measured logistic chain conditions. The SPC approach forms the foundation for first-order logistics while the generic shelf life model and the specific quality attribute model focus on second-order logistics, eventually enabling the development of third-order logistics strategies.

(a). Statistical process control

SPC is about monitoring process variables to make sure that the process stays constant within certain well-defined specifications [26]. Even though SPC has been widely applied in the area of batch processing [27], the application in food production processes has been limited [28–31] and the application to supply chain logistics is almost non-existent [32]. Applied to the supply chain of perishable food products, the ‘process’ refers to ‘climate control in the supply chain’ while the ‘variables’ are the climate conditions realized. The dimensionality of the control problem is by definition limited, as the number of variables is often limited to the two main factors, temperature and humidity, only.

The control limits are defined relative to the targeted climate conditions, being either optimal conditions based on the product's requirements or conditions desirable from a managerial point of view. Together, they define the range for which the process of climate control is considered to be ‘in control’. The control limits can be defined either based on expert knowledge on what variation is deemed acceptable to the product or based on the inherent variation of the climate conditions during normal operation of the available climate control systems. Both the targeted conditions and the enclosing control limits can remain constant throughout the supply chain or might vary with the position of the food product in the supply chain and the typical requirements and/or limitations in that particular part of the supply chain. Only when control limits are specified based on the inherent climate variation observed for when the process is ‘in control’ can the approach be referred to as a true SPC. If the control limits are defined based on expert knowledge, the approach becomes a kind of pseudo-SPC, as the statistical description of stability of the climate conditions is missing. However, from a product point of view, the expert-based control limits might be more relevant than statistically based control limits.

SPC provides a simple generic approach focusing on quality of the climate control more than on controlling the quality of a specific food product. Therefore, SPC can add value to the supply chain management of perishable food products in situations where specific product knowledge is lacking but where there is an urgent need to guarantee strict climate control. The limited application to date to supply chain management might be due to the fact that the supply chain is owned by various players, hampering the data exchange essential to proper SPC.

When applying SPC for warehouse management, batches of food products can be differentially handled based on their climate history incurred to date relative to the targeted climate conditions and the control limits. In this approach, products having been exposed to more extreme conditions can be considered the first to expire.

(b). Generic shelf life models

Shelf life models are models where the shelf life of a product is modelled as a function of the conditions in the logistic chain taking into account the overall acceptance by the end users, focusing on the product's suitability for subsequent marketing. Model predictions give an appreciation of the quality of the incurred logistic handling chain by translating the impact of the logistic conditions on product quality in terms of the days of remaining shelf life. This generic model approach builds on and integrates earlier published work originally focusing on temperature [33] and O₂ and CO₂ [34]. Within the framework of the PASTEUR project, this approach was extended to include the effect of relative humidity as well [8]. When quality is interpreted as the additive result of several quality attributes which non-interfering parallel processes degrade, keeping quality (t_KQ) at constant environments can be described as

4.1

The exact form of the quality function f(Q₀,Q_lim) depends on the underlying physiological mechanism and is the function of the initial quality (Q₀) and the defined threshold value (Q_lim). The quality function can be either complex or simple but it has been shown that, for the concept of keeping quality, the actual mechanism does not play a role [33]. For this reason, it is convenient to assume zero-order reaction kinetics. The rate constants k_i (with generally i≤3) define the rates of change for the involved quality attributes which in the basic concept [33] were assumed to depend on temperature only, following Arrhenius's law [14]. In the extension to include the effect of gas composition (O₂ and CO₂) on the overall metabolic rate of the product, relative respiration rate (R_rel) was introduced [34]. In this approach, relative respiration was calculated as the ratio between respiration under possibly modified gas conditions applied during transport and storage relative to the respiration under regular air, with respiration being modelled using one of the available Michaelis–Menten-based respiration models [35]. If quality degradation is unaffected by modified atmosphere conditions, the effect of respiration can be ignored (R_rel=1). If quality loss either relies on multiple quality breakdown processes (i>1) or not depending on the modified atmosphere conditions, the effect of respiration can be adapted accordingly by using either the actual R_rel for R_rel,i or by using R_rel,i=1.

The final extension was to include the effect of relative humidity on quality loss. Relative humidity was assumed to affect quality through the absolute water vapour pressure deficit (e_def), calculated as the difference between the saturated and actual vapour pressure [36]. The effect of humidity was incorporated similar to R_rel, assuming e_def is directly affecting the rate constants of quality loss. Again, the different quality breakdown processes might be differently affected by humidity. For those quality-degrading reactions not influenced by humidity, e_def,i=1; for those reactions that are affected by humidity, e_def,i equals e_def as calculated for the current supply chain condition.

Under dynamic supply chain conditions, loss of quality is described using a simple differential equation assuming zero-order reaction kinetics following

4.2

thus taking into account the combined effects of temperature, O₂, CO₂ and relative humidity. From its moment of creation up to the moment a decision-maker has to decide on the next step towards the product's future, the change in quality can be calculated following equation (4.2). At each of these decision moments, the estimated remaining shelf life can be calculated assuming the product (starting from its current quality Q_s) would, from that moment on, be kept at some constant standard shelf life condition under regular air. Following the assumed zero-order reaction kinetics, the expected remaining shelf life (t_SL) is calculated as

4.3

with k_1…n depending on the constant shelf life temperature and assuming shelf life takes place under regular air conditions (R_rel=1). This assumes that, during shelf life, the product's respiration rate is no longer inhibited by modified atmosphere gas conditions, but that the humidity during shelf life still plays a role for those processes sensitive to it. The original keeping quality model, which only incorporates the temperature effect, can be seen as a special case of the final model described by equations (4.2) and (4.3). The final model version is the most versatile, given that all the relevant supply chain conditions are covered and can be applied to a wide range of perishable products (either fresh or processed, food or non-food).

(c). Specific quality attribute models

Specific quality attribute models describe the evolution of a specific quality attribute for a specific product as a function of the supply chain handling conditions. Which quality attributes are of interest depends on the product under study. While the generic shelf life models do not intend to model the product's physiology but only the time for which a perishable product will remain acceptable to a consumer, the specific quality attribute models do pretend to provide a description of the (relevant) processes going on inside the perishable product that result in the observed change in quality. By definition, such a model will be based on a simplification of the food product and, therefore, will never be ‘true’ as the only true model is the product itself. The aim of modelling food quality attributes is, however, not to develop true models but to develop valid models; that is, models that are consistent with the current knowledge level and that contain no known or detectable flaws of logic [25]. Also, models should be detailed enough for the intended purpose but at the same time simplified enough to give robust manageable models. The basic strategy to develop a suitable model is to apply a systematic process of problem decomposition, dissecting the problem into its basic building blocks and then reassembling them leaving out the unnecessary detail. What is essential and what is redundant depends largely on the intended application of the model. In the end, the models are to be used to provide an appreciation of the quality of the logistic handling chain and to translate this into the impact the logistic conditions have on product quality attributes.

Given the dynamic conditions affecting food quality attributes, the models should be based on differential equations that can deal with dynamic inputs. Kinetic models [37] are most suitable to generically describe complex quality attributes, such as fruit mealiness [38], starting from the underlying (bio)chemical, physiological and physics mechanisms. Over the past 20 years in post-harvest, where one deals with highly perishable products in relatively complex supply chains, an increasing awareness has been observed of the need for proper mechanistic models to support an integrated quality management [39–41]. While initially models focused on describing the specific product quality, increased emphasis has been placed on developing product quality models applicable to dynamic supply chain conditions [42–45]. The quality of food products is characterized by an inherent large amount of biological variance. To be able to manage the supply chain, clear insight is required in the propagation of such variance during post-harvest. In this context, increasing effort has gone towards including biological variation as an integral part of the quality models ([46]; for an extensive review on this topic see [47]).

5. Shelf-life-based cold chain optimization

The different modelling methods described above can be applied to optimize cold chain management, reducing waste and quality loss. Sufficient insight into a perishable supply chain (or cold chain) enables optimization of the third-order logistics (figure 1) in deciding where to ship which product to minimize both waste and quality loss. The discussion on how to define supply chain management compared with a more traditional definition of logistics has been hotly debated since the 1990s but has not yet been truly applied to special requirements of perishable food logistics. Cooper et al. [48] believe that the level of coordination between different organizations in the supply chain in terms of all activities and processes should extend beyond traditional logistics. In order to successfully manage and coordinate these activities and processes, some researchers have tried to categorically define the performance components, such as planning and control, organizational structure, product and information flow [49]. Based on the definitions of some of these components, other researchers have laid the theoretical groundwork for analysing supply chain management performance from different perspectives, such as a resource- or knowledge-based view, systems theory and the more recent stakeholder theory [50,51].

(a). Cold chain management

As monitoring technologies advanced over time, many application possibilities opened up to track the temperature of perishable products through storage and transportation in the cold chain [3,4]. Wireless sensor front-ends, such as active or semi-passive RFID systems, enable accumulation of high-resolution environmental data, such as the temperature at pallet or product level, which were otherwise absent in the decision dynamics for cold chain management. On the other hand, back-end systems, such as servers storing the data, run algorithms on the recorded data, such as shelf life prediction, to create actionable knowledge on the quality of products in the cold chain. Finally, such knowledge is used to alter decision dynamics to create a better optimized logistics scenario in the form of distribution algorithms such as FEFO as opposed to FIFO. For instance, an inventory management model proposed in [52] considers the use of RFID front-end technology to monitor temperature and replace the static shelf-life-based low-resolution models in the literature with a higher resolution (unit level) dynamic model where the demand is quality driven similar to FEFO. However, the authors of that article make it clear that their purpose was not to develop optimal solutions but rather to show that using their model provides a ‘good solution’, which can be operationalized. Other researchers have looked into ways to quantify the logistic performance in a perishable supply chain [53]. The generally used logistic network parameters such as consumer and retail demand, shrinkage, transport, etc. are described from a perishable point of view. Simulation results showed a significant increase in food safety, which is only possible by having access to up-to-date product quality information in terms of microbial growth.

Since 2001, more than 200 articles have been published discussing how to model and control perishable inventory control [20]. When considering inventory models for perishables, two types that focus on deterioration (as either fixed, age- or inventory-dependent lifetime) and demand (as either stochastic or deterministic) come into view. The majority of these models consider one or more of the following important performance parameters: price increase or discount based on shelf life, shortages and backordering, single or multi-item control, etc. Optimization for such models can be prescribed as an ordering policy, and for perishables different types of ordering policies were proposed in the literature. For example, Haijema [54] presents an optimal ordering policy as a Markov decision problem; however, only last-in-first-out (LIFO) and FIFO are considered as two different types of demand, neither of which is similar to FEFO, which is based strictly on the current quality of the product rather than the time it was received. In addition, the performance parameter of this optimal ordering policy is the expected average cost per ordering period, which may or may not correlate well with the ratio of wasted perishables per ordering period.

Other researchers looked at the improvement in opportunity losses when dynamic expiry dates are used instead of static ones [55]. Simulation results showed a decrease of approximately 80% with a dynamic expiry date scenario where the expiry dates of the products are determined based on transport and initial microbial conditions for better pricing or management, though mainly with store-level optimization.

In creating an optimization strategy to manage a perishable supply chain, one has to make the following observations. First, in [10], where the authors claim that different segments of the perishable supply chain might require different strategies, they describe a performance parameter called a product's marginal value of time (MVT), which shows the rate at which a product loses value over time in the supply chain. Based on the behaviour of the MVT, they show that not only a hybrid model is the best option for minimizing lost value, but also that the segments of the perishable supply chain are only slightly linked, which means that separate optimization processes can more easily achieve value maximization. The second observation is the fact that less than 10% of the perishable inventory control studies in the literature investigate two warehouses for controlling perishable inventory with others focusing on single-warehouse scenarios [20]. Hence, an optimization algorithm running on such models, no matter what the cost function is, will have to mostly operate on single-warehouse scenarios.

Based on these observations, we will now explore the feasibility of a combinatorial exhaustive-search algorithm from a DC or warehouse management point of view. In a standard supplier–distributer–retailer supply chain model, given a set of finite inventory, shelf-life-based optimization can be defined as:

• — minimize the number of wasted products and

• — maximize the average quality of products delivered to the store.

There are different ways to include an additional performance indicator, such as product quality or shelf life, in generic supply chain optimization. In general terms, this is a combinatorial optimization problem, where given a finite set of states the goal is to find the optimal state [56]. The optimal state for a general supply chain is given by a list of key performance indicators (KPIs), which are mostly related to traditional supply chain management goals such as reduced travel times and transport costs or product replenishment frequencies [57]. The KPIs are multiplied by weighting factors to result in the cost function. The process to go from traditional supply chain planning to food-specific planning can be described as adding new, shelf-life-related KPIs to the cost function, such as remaining shelf life and product quality.

(b). Optimization search algorithm

The trivial and perfect solution to such a problem is to find and calculate the outcome for all possible combinations, commonly called an exhaustive search. However, in many optimization problems, this type of categorical search is simply not feasible owing to computational requirements. In this section, it is shown that an exhaustive-search algorithm for shelf-life-based optimization is possible and the complexity is sufficiently low for the size and properties of a typical commercial cold chain.

In defining the optimization search algorithm for a cold chain, the following information is assumed to be available to create a shelf life inventory at each node in the supply chain:

• (i) a shelf life model for the transported commodity that operates on recorded supply chain handling conditions (e.g. temperature, humidity, etc.) and

• (ii) average supply chain handling condition profiles of transportation lanes between points in the supply chain.

In an exhaustive-search algorithm, a performance variable such as remaining shelf life is calculated for every possible shipment scenario. In order to evaluate the feasibility of applying the exhaustive search in this case, one needs to look at the number of combinations possible given a finite set of retailers, DCs and product requests.

If there are N number of DCs connected to any given store, for n number of requested products, there are P(n, N) number of unique possible ways to divide the number of shipments between the connected DCs. Here, P is called the partition function, which outputs the number of different ways one can write an integer number as a sum of positive integers [58]. In this case, a modified version is proposed of the partition function, P(n, N), which is the number of different ways one can write n as a sum of only up to N positive integers. The partition function does not have a closed-form solution and instead is either calculated via computer methods or approximated by other mathematical expressions, the discussion of which is beyond the scope of this text [59]. Furthermore, as a set of unique numbers of product requests can be distributed in different ways to different DCs, there is an additional probability multiplier, which can be calculated as follows:

5.1

where N_m is the number of possibilities given N warehouses and m unique integers in making the sum of total products requested, whereas m_k denotes the number of times the kth unique integer appears in the summation. Hence, for n number of product requests, the total number of all possible scenarios can be calculated as follows:

5.2

where

5.3

and N_m(i) is the number of possibilities for the ith element of the partition function with its own set of unique integers in making up the sum n as shown in equation (5.1). As shown here, the complexity of the exhaustive-search algorithm is directly correlated to the complexity of the partition function. There have been studies in number theory to show the asymptotic behaviour of the partition function going back to 1917—such as the one by Hardy & Ramanujan [60]— which states

5.4

This number grows exponentially with increasing number of orders, creating a complexity problem for the exhaustive-search approach. However, a key difference is the fact that the specific exhaustive-search algorithm for a cold chain uses a modified version where n is written as a sum of unique integers only up to the number of DCs. This observation results in a significant reduction in the number of combinations an order can be placed through serving DCs; although multiple warehouses might serve a single retailer, they have different inventories limiting the number of available DCs when creating the combinatorial table. The number of DCs serving the same perishable product to the retail store thus defines the limiting constraint on the partition function. Figure 2 shows how computationally intensive the search algorithm becomes as a function of the number of DCs and product requests.

Figure 2.

Complexity of the exhaustive-search algorithm for cold chain optimization. (Online version in colour.)

As expected from Hardy and Ramanujan's findings of asymptotic behaviour, one can observe that a linear increase in the number of DCs serving a particular store results in an exponential increase in the number of combinations. For instance, in the case of a single DC serving the retail store, no matter what the number of requested products is, there is only one way to distribute them to the store. In comparison, the number of combinations increases to 5151 when there are three DCs serving the store for the same number of product requests. However, in a real-life scenario, assuming pallet-level monitoring (where a product equals a pallet), daily order volume for a retail store for a specific perishable product is significantly lower (one or two pallets), as is the number of DCs which can deliver that product to that store (one or two DCs), thus resulting in a sufficiently low number of combinations to make a full exhaustive search possible.

Once all the combinatorial possibilities are identified, a front-end distribution logic, such as

• (i) FIFO—ship the products in the order that they are received at the DC,

• (ii) FEFO—ship the products based on their dynamic expiry dates calculated from application of the shelf life prediction algorithm to recorded temperature, and

• (iii) any other distribution logic such LIFO or variations of FEFO with local constraints,

is applied exhaustively on all shipment configurations, shelf life inventories are updated at each node in the cold chain and performance indicators, such as the number of wasted products or average product quality, are calculated to form the global optimization matrix. As previously mentioned, in a real-life scenario, there are other KPIs to consider but the optimization algorithm is flexible in that exclusive indicators for perishables, such as shelf life and quality, can supplement the former criteria to create an optimization matrix calculated across all combinations. In fact, an optimal point can be found if a certain weighting function, which defines the relative importance of such criteria with respect to one another, is used.

6. Conclusion

Many current commercial warehouse management systems offer fragmented solutions and lack the ability to adopt a holistic perspective to supply chain integrity owing to an absence of information-sharing channels and the necessary tools to obtain relevant environmental data. Given a global need to increase transparency and responsiveness, reduce lead times and enhance security in the perishable food chain, the shift from FIFO to FEFO strategies has gathered significant traction. Most conventional systems estimate shelf life based on an onsite approach and a handover of information which, in many cases, is not based on the actual supply chain history to which the product was exposed.

This article presents the base for an integrated approach in which front-end sensor technologies enable the use of generic shelf life modelling approaches taken from the post-harvest research area to alter the decision dynamics in a cold chain with state-of-the-art algorithms. By combining this with systems for real-time monitoring of supply chain conditions, the perishable specific supply chain optimization algorithms can be used to form a strategic response management system optimizing product flows by taking into account the shelf life inventories and estimated shelf life distances between different nodes in the supply chain. The estimated shelf life distances for a particular batch can be taken as a guide to evaluate the potential of the given batch for all the possible transportation scenarios, while proper confidence intervals should still be considered to account for the omnipresent biological variation limiting the accuracy of the shelf life prediction models.

Although the technical means are available to implement such data-driven model-based optimization approaches in practice, its success will largely depend on the chain-wide willingness to participate in and contribute to the information-sharing highways.

Funding statement

The authors acknowledge support from the European CATRENE office (CT204-PASTEUR) and financial support from the Flemish Agency for Innovation by Science and Technology (IWT, project 090633 – PASTEUR).