Abstract
In this chapter, we construct a food supply chain network model under oligopolistic competition and perishability, with a focus on fresh produce. The model handles food spoilage through arc multipliers, with the inclusion of the discarding costs associated with disposal. We allow for product differentiation due to such relevant issues as product freshness and food safety and include alternative technologies associated with various supply chain activities. A case study, in which we analyze different scenarios prior to, during, and post a foodborne disease outbreak, based on the cantaloupe market, illustrates the modeling and computational framework.
1 Motivation and Overview
This chapter begins Part II of the book, which focuses on game theoretical supply chain networks for perishable products. Unlike the single-organization optimization models in Part II, the models in this part are multiple decision-maker competitive equilibrium models. These supply chain network problems are formulated under product/brand differentiation to capture the reality of real-world competition and to provide the appropriate analytics.
Food supply chains have several characteristic differences from other product supply chains and, hence, merit individual consideration and treatment. The fundamental difference between food supply chains and other supply chains is the continuous change in the quality of food products throughout the entire supply chain. This is especially important for fresh produce supply chains where increasing attention is being placed on both product freshness and safety. Moreover, growing demand for such products is supported by statistics from the United States Department of Agriculture (USDA 2011), which suggests that the consumption of fresh vegetables has increased at a much faster pace than the demand for traditional crops.
Today’s food supply chains are complex global networks that create pathways from farms to consumers, involving production, processing, distribution, and even the disposal of food (see Ahumada and Villalobos 2009). Consumers’ expectation of year-around availability of fresh food products has encouraged the globalization of food markets. With growing global competition and the greater distances between food production and consumption locations, there is increasing pressure for the integration of food production and distribution. This results in new challenges for food supply chain modeling, analysis, and solutions.
Given the thin profit margins in food industries, product differentiation strategies are increasingly being used with product freshness considered as one of the major differentiating factors. Retailers now realize that food freshness can be a competitive advantage (Lütke Entrup 2005; see also Aiello et al. 2012).
Moreover, the high perishability of food products results in immense food waste/loss, further stressing food supply chains. While some food waste and loss are inevitable on food supply chain network links, it is estimated that approximately one third of the global food production is wasted or lost annually (Gustavsson et al. 2011). Food products often need special handling, transportation, and storage technologies. Furthermore, the quality of fresh food products decreases with time, despite the use of the most advanced processing, handling, and shipping methods.
In this chapter, we formulate, analyze, and solve food supply chain network problems, under product differentiation and competition, including food deterioration/spoilage. We focus on fresh produce items, such as vegetables and fruits, that require simple or limited processing and whose lifespans can be measured in days. The fresh produce supply chain network oligopoly model developed in this chapter is distinct from other studies on perishable food products in that:
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It captures the deterioration of fresh food along the entire supply chain from a network perspective.
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It handles the exponential time decay through the introduction of arc multipliers.
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It captures oligopolistic competition with product differentiation.
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It includes the disposal of the spoiled food products, along with the associated costs.
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It allows for the assessment of alternative technologies for each supply chain activity.
This chapter is organized as follows. In Sect. 4.2, we construct the food supply chain network oligopoly model and derive variational inequality formulations. We also provide some qualitative properties. In Sect. 4.3, we present the computational algorithm, which we then apply to a case study focused on fresh produce in Sect. 4.4. We summarize our results and present our conclusions in Sect. 4.5. We conclude with a Sources and Notes section.
2 The Food Supply Chain Network Oligopoly Model
In this section, we consider a food supply chain network with a finite number of I food firms, with a typical firm denoted by i. The food supply chain network activities include production, processing, storage, distribution, and the disposal of the food products. The firms will typically be vertically integrated, which, as a strategy, has become increasingly important as food systems become more consumer-driven (see Bhuyan 2005).
The individual food firms compete noncooperatively in an oligopolistic manner. We allow for product differentiation by consumers at the demand markets, due to, for example, product freshness and food safety concerns that may be associated with a particular firm. This means that the fresh food products of a given type are not necessarily homogeneous.
Food firms are represented by supply chain networks of their economic activities as depicted in Fig. 4.1. Each firm seeks to determine its optimal product flows. Each food firm i possesses n M i production facilities, n C i processors, and n D i distribution centers to satisfy the demands at n R demand markets. Let \(\mathcal{G} = [N,L]\) denote the graph consisting of the set of nodes N and the set of links L in Fig. 4.1, where \(L \equiv \cup _{i=1,\ldots,I}{L}^{i}\) and L i denotes the set of directed links corresponding to the sequence of activities associated with firm i.
The first set of links connecting the top two tiers of nodes corresponds to the food production at each of the production units of firm i. Production may involve seasonal operations such as soil agitation, sowing, pest control, nutrient and water management, and harvesting. The multiple possible links connecting each top tier node i with its production facilities, \(M_{1}^{i},\ldots,M_{n_{M}^{i}}^{i}\), represent alternative production technologies that may be associated with a given facility.
The second set of links from the production facility nodes is connected to the processors of each firm i which are denoted by \(C_{1,1}^{i},\ldots,C_{n_{C}^{i},1}^{i}\). These links correspond to the shipment links between the production units and the processors. The multiple shipment links denote different possible modes of transportation, characterized by varying time durations and environmental conditions.
The third set of links connecting nodes \(C_{1,1}^{i},\ldots,C_{n_{C}^{i},1}^{i}\) to \(C_{1,2}^{i},\ldots,C_{n_{C}^{i},2}^{i}\) denotes the processing of fresh produce. The major food processing activities include cleaning, sorting, labeling, and simple packaging. Different processing activities and technologies may result in different levels of quality degradation.
The next set of nodes represents the distribution centers. Thus, the fourth set of links connecting the processor nodes to the distribution centers is the set of shipment links. Such distribution nodes associated with firm i are denoted by \(D_{1,1}^{i},\ldots,D_{n_{D}^{i},1}^{i}\). There are also multiple shipment links in order to capture different modes of transportation.
The fifth set of links, which connects nodes \(D_{1,1}^{i},\ldots,D_{n_{D}^{i},1}^{i}\) to \(D_{1,2}^{i},\ldots,D_{n_{D}^{i},2}^{i}\), are the storage links. Since fresh produce items may require alternative storage conditions, we represent these alternative conditions through multiple links at this tier.
The last set of links connecting the two bottom tiers of the supply chain network corresponds to the distribution links over which the fresh produce items are shipped from the distribution centers to the demand markets. Here we also allow for multiple modes of transportation.
In addition, the curved links in Fig. 4.1 joining the top-tiered nodes i with the processors, which are denoted by \(C_{1,2}^{i},\ldots,C_{n_{C}^{i},2}^{i}\), capture the possibility of on-site production and processing.
Most of the fresh produce items reach their peak quality at the time of production and then deteriorate over time (Blackburn and Scudder 2009). Microbiological decay is one of the major causes of food quality degradation, especially for fresh produce (Fu and Labuza 1993). Therefore, food deterioration usually follows a first-order equation with exponential time decay (see Labuza 1982, Nahmias 1982, Tijskens and Polderdijk 1996, Blackburn and Scudder 2009, Nga 2010, and Rong 2011). Exponential time decay is a special case of random lifetime perishability, which means that, while the time to spoilage of an individual item is uncertain, the percentage of products that are spoiled at a given time can be predicted (Nahmias 1982; see also Van Zyl 1964). It also has been recognized that the decay constant is highly dependent on the temperatures and other environmental conditions. Food supply chains can be grouped into three types based on various temperature requirements: frozen, chilled, and ambient. The normal temperature of the frozen chain is − 18 ∘ C, while temperatures range from 0 ∘ C for fresh fish to 15 ∘ C for potatoes and bananas in the chilled chain. There is no required temperature control in an ambient chain.
In the existing literature on perishability, exponential time decay has been utilized to describe either the decrease in quantity or the degradation in quality. The decrease in quantity, which has been discussed in studies on perishable inventory, represents the number of units of decayed products such as vegetables and fruits. On the other hand, the degradation in quality emphasizes that all the products deteriorate at the same rate simultaneously, which is more relevant to meat, dairy, and bakery products. Since our model focuses on fresh produce items, we adopt exponential time decay to capture the discarding of spoiled products associated with all the postproduction activities. We will highlight how the model can be modified/adapted to handle degradation in quality.
2.1 The Underlying Chemistry
Food products deteriorate over time even under optimal conditions. We assume that the temperature and other environmental conditions associated with each postproduction activity/link are known and fixed. Following Nahmias [1982], we assume that each unit has a probability of e − λt to survive another t units of time, where λ is a positive parameter known as the decay constant. Let N 0 denote the quantity at the beginning of the time interval (link). Then, the expected quantity surviving at the end of the time interval (specific link), denoted by N(t), can be expressed as
As in Chaps. 2 and 3, we assign a multiplier to each link in the supply chain network. Here, as in Chap. 3, we are interested in capturing the decay in the number of units. Hence, we can represent the arc multiplier α a for a postproduction link a as
where λ a and t a are the decay constant and the time duration associated with the link a. Both λ a and t a are given and fixed. We assume that the value of α a for a production link is equal to one.
In rare cases, food deterioration follows the zero-order reactions with linear decay (Tijskens and Polderdijk 1996 and Rong 2011). Then, \(\alpha _{a} = 1 - \lambda _{a}t_{a}\) for a postproduction link.
For definiteness and since we are focusing on the fresh produce application, we express the flows, the multipliers, and their relationships in this context.
As noted in Chaps. 2 and 3, f a denotes the (initial) flow of product on link a, and f a ′ denotes the final flow on link a, that is, the flow that reaches the successor node. Therefore,
The number of units of the spoiled produce on link a is the difference between the initial and the final flow, f a − f a ′, where
Associated with the food deterioration is a total discarding cost function, \(\hat{z}_{a}\), which is a function of flow on the link, f a , that is,
\(\hat{z}_{a}\) is assumed to be convex and continuously differentiable.
In developed countries, the overall average percentage loss of fruits and vegetables during postproduction supply chain activities is approximately 12% of the initial production. In developing regions the percentage loss is greater. It is imperative to remove the spoiled fresh food products from the supply chain network because the presence of spoiled products may, in many cases, increase the spoilage rate. Here, we focus mainly on the disposal of the decayed food products at the processing, storage, and distribution stages (see also Thompson 2002).
As in Chaps. 2 and 3, x p represents the (initial) flow of product on path p joining an origin node with a destination node, but the supply chain network is for all firms as in Fig. 4.1. Hence, the origin node i associated with firm i is node i, with a destination node, reflected by the demand market k, being R k . Since we are dealing with real physical products, the path flows must be nonnegative:
where \(\mathcal{P}_{k}^{i}\) is the set of all paths joining the origin node i with destination node R k , that is, the origin/destination (O/D) pair w k i = (i, R k ).
The arc–path multiplier, α ap , which is the product of the multipliers of the links on path p that precede link a on path p is defined analogously in (2.13).
Hence, the relationship between the link flow, f a , and the path flows can, again, be expressed as
Recall that μ p denotes the multiplier corresponding to path p, defined as the product of all link multipliers on links comprising that path, that is,
The demand for firm i’s product at demand market R k is a variable and is denoted by d ik . It must be equal to the sum of all the final flows—subject to perishability—on paths connecting O/D pair w k i:
The consumers may differentiate the fresh food products due to food safety and health concerns. We group the demands d ik ; \(i = 1,\ldots,I;k = 1,\ldots,n_{R}\) into the I ×n R -dimensional vector d.
2.2 Competitive Behavior and Cournot–Nash Equilibrium
We denote the demand price associated with food firm i’s product at demand market R k by ρ ik and assume that
Note that the price of food firm i’s product at a particular demand market may depend not only on the demands for its product at that and the other demand markets but also on the demands for the other substitutable food products at all the demand markets. These demand price functions are assumed to be continuous, continuously differentiable, and monotone decreasing.
To address the competition among the firms for resources used in the production, processing, storage, and distribution of the fresh produce, we assume that the total operational cost on link a may, in general, depend on the product flows on all the links, that is,
where f is the vector of all the link flows. The total cost on each link is assumed to be convex and continuously differentiable.
Let X i denote the vector of path flows associated with firm i where \(X_{i} \equiv \{\{ x_{p}\}\vert p \in {\mathcal{P}}^{i}\} \in R_{+}^{n_{{\mathcal{P}}^{i}}}\), \({\mathcal{P}}^{i} \equiv \cup _{k=1,\ldots,n_{R}}\mathcal{P}_{k}^{i}\), and \(n_{{\mathcal{P}}^{i}}\) denotes the number of paths from firm i to the demand markets. Thus, X is the vector of all the food firms’ strategies, that is, \(X \equiv \{\{ X_{i}\}\vert i = 1,\ldots,I\}\).
The profit function of a food firm is defined as the difference between its revenue and its total costs (operational and discarding). Each firm i seeks to maximize its profit. The statement of the maximization of profits for firm i, in link flows, is
In view of (4.9), we may redefine the demand price functions (4.10) as \(\hat{\rho }_{ik}(x) \equiv\rho _{ik}(d)\), \(i = 1,\ldots,I;\,k = 1,\ldots,n_{R}\). The statement equivalent to (4.12), but purely in path flows, is
where the path total operational costs and path total discarding costs in (4.13) are defined as in (3.19) [see also (3.20)] in Chap. 3, but with the paths corresponding to those in the food supply chain network in Fig. 4.1.
We let U i (X) denote the profit expression for firm i, as in (4.13), with the I-dimensional vector U being the vector of the profits of all the firms as follows:
In the Cournot–Nash oligopolistic market framework, each firm selects its product path flows in a noncooperative manner, seeking to maximize its own profit, until an equilibrium is achieved.
Definition 4.1 (Supply Chain Network Cournot–Nash Equilibrium).
A path flow pattern \({X}^{{_\ast}}\in K = \prod\limits _{i=1}^{I}K_{i}\) constitutes a supply chain network Cournot–Nash equilibrium if for each firm i, \(i = 1,\ldots,I\):
where \(\hat{X}_{i}^{{_\ast}}\equiv(X_{1}^{{_\ast}},\ldots,X_{i-1}^{{_\ast}},X_{i+1}^{{_\ast}},\ldots,X_{I}^{{_\ast}})\) and \(K_{i} \equiv \{ X_{i}\vert X_{i} \in R_{+}^{n_{{\mathcal{P}}^{i}}}\}\).
In other words, an equilibrium is established if no firm can unilaterally improve upon its profit by changing its product flows in its supply chain network, given the product flow decisions of the other firms.
Next, we provide the variational inequality formulations of the Cournot–Nash equilibrium for the fresh produce supply chain network under oligopolistic competition satisfying Definition 4.1, in terms of both path flows and link flows (see Cournot 1838, Gabay and Moulin 1980, Nash 1950, 1951, and Nagurney 2006).
Theorem 4.1 (Variational Inequality Formulations).
Assume that, for each food firmi, the profit functionU i (X) is concave with respect to the variables inX i and is continuous and continuously differentiable. ThenX ∗ ∈ Kis a supply chain network Cournot–Nash equilibrium according to Definition 4.1 if and only if it satisfies the variational inequality:
where ⟨ ⋅, ⋅⟩ denotes the inner product in the corresponding Euclidean space and \(\nabla _{X_{i}}U_{i}(X)\) denotes the gradient ofU i (X) with respect toX i . Variational inequality (4.16), in turn, for our model, is equivalent to the variational inequality: determine the vector of equilibrium path flowsx ∗ ∈ K 1 such that
where \({K}^{1} \equiv \{ x\vert \,x \in R_{+}^{n_{\mathcal{P}}}\}\), and for each pathp, \(p \in \mathcal{P}_{k}^{i}\); \(i = 1,\ldots,I\); \(k = 1,\ldots,n_{R}\),
Variational inequality (4.17) can also be reexpressed in terms of link flows as follows: determine the vector of equilibrium link flows and the vector of equilibrium demands (f ∗ , d ∗ ) ∈ K 2 such that
whereK 2 ≡ { (f, d) | ∃x ≥ 0 and (4.7) and (4.9) hold}.
Proof. Variational inequality (4.16) follows directly from Gabay and Moulin [1980]. Note that
For each path p; \(p \in \mathcal{P}_{k}^{i}\), we have
In particular, we have
Therefore, variational inequality (4.17) is established. Also, using Eqs. (4.7) and (4.9), variational inequality (4.19) then follows from (4.17). □
Variational inequalities (4.17) and (4.19) can be put into standard form (2.41). We have already defined X as the vector of path flows (strategies). We now define
and \(\mathcal{K}\equiv{K}^{1}\) then (4.17) can be reexpressed as (2.41). Similarly, for the variational inequality in terms of link flows, if we define the vectors X ≡ (f, d) and F(X) ≡ (F 1(X), F 2(X)), where
and \(\mathcal{K}\equiv{K}^{2}\), then (4.19) can be rewritten as (2.41).
Since the feasible sets K 1 and K 2 are not compact, we cannot obtain the existence of a solution simply based on the assumption of the continuity of F. However, the demand d ik for each food firm i’s product at every demand market R k may be assumed to be bounded, since the total demand for these products is finite (although it may be large). Consequently, we have that
where b > 0 and x ≤ b means that x p ≤ b for all \(p \in \mathcal{P}_{k}^{i}\); \(i = 1,\ldots,I\), and \(k = 1,\ldots,n_{R}\). Then \(\mathcal{K}_{b}\) is a bounded, closed, and convex subset of K 1. Thus, the following variational inequality
admits at least one solution \({X}^{b} \in \mathcal{K}_{b}\), since \(\mathcal{K}_{b}\) is compact and F is continuous. Therefore, following Kinderlehrer and Stampacchia [1980] (see also Theorem 1.5 in Nagurney 1999), we have the following theorem:
Theorem 4.2 (Existence).
There exists at least one solution to variational inequality (4.17) (also to (4.19)), since there exists ab > 0 such that variational inequality (4.26) admits a solution in \(\mathcal{K}_{b}\) with
In addition, we now provide a uniqueness result.
Theorem 4.3 (Uniqueness).
With Theorem 4.2, variational inequality (4.17) and, hence, variational inequality (4.19) admit at least one solution. Moreover, if the functionF(X) of variational inequality (4.19), as defined in (4.24), is strictly monotone on \(\mathcal{K}\equiv{K}^{2},\) that is,
then the solution to variational inequality (4.19) is unique, that is, the equilibrium link flow pattern and the equilibrium demand pattern are unique.
It is more reasonable that we would have, given the supply chain network topology, uniqueness of an equilibrium in link flows than in path flows.
Our proposed supply chain network model can also be applied to other fresh food supply chain oligopoly problems under quality competition. In such cases, the food products get delivered to the demand markets with distinct levels of quality degradation. Thus, the arc multiplier for a postproduction link, α a , captures the corresponding food quality degradation associated with that link, instead of the fraction of unspoiled products. We refer to Labuza [1982] and Man and Jones [1994] for thorough discussions about food quality deterioration.
3 The Algorithm
To computationally solve the supply chain network problem, we utilize the Euler method as given in (2.59). Its realization in the context of our competitive food supply chain network problem, governed by the variational inequality formulation (4.17), yields subproblems, at each iteration, that can be computed explicitly in closed form (as was also the case for the blood supply chain network problem in Chap. 2).
3.1 Explicit Formulae for the Euler Method Applied to the Fresh Produce Supply Chain Network Oligopoly Variational Inequality (4.17)
The elegance of this procedure for the computation of solutions to the fresh produce supply chain network oligopoly problem can be seen in the following explicit formulae. In particular, we have the following closed form expression for the path flow on each path \(p \in \mathcal{P}_{k}^{i}\); \(i = 1,\ldots,I\); \(k = 1,\ldots,n_{R}\), at iteration τ + 1:
It is important to emphasize that the Euler method, which, as noted in Chap. 2, is induced by the general iterative scheme of Dupuis and Nagurney [1993], can also be interpreted as a discrete-time adjustment process. Indeed, the general iterative scheme was devised to not only compute the stationary points of projected dynamical systems (which coincide with the solutions to the associated variational inequality problems) but also provide a means of tracking the associated trajectories over time. Hence, (4.29) may be interpreted as a discrete-time adjustment process with the iteration corresponding to a time period. The firms update their product path flows at each time step by reviewing the product path flows in the previous time period along with the marginal revenue associated with that path. Convergence is achieved to the desired tolerance level when the respective subsequent path flow iterates lie sufficiently near one another, and hence, a stationary point (at which there is essentially no change) has been reached. For a discussion of convergence, see Chap. 2.
In the next section, we solve fresh produce supply chain network oligopoly problems using the above algorithmic scheme.
4 A Case Study
This case study focuses on the cantaloupe market in the United States. Most of the cantaloupes consumed in the United States are grown in either California, Mexico, or Central America. We assume that there are two firms, Firm 1 and Firm 2, which may represent, for example, one firm in California and one firm in Central America. Each firm has two production sites, one processor and two distribution centers, and serves two geographically separated demand markets, as depicted in Fig. 4.2. The production sites and the processor of Firm 1 are located in California. The production sites and the processor of Firm 2 are located in Central America, with, typically, lower operational costs. However, all the distribution centers and the demand markets are located in the United States. We let D k i denote the k-th distribution center of firm i. Hence, as described in Sect. 4.2, D k, 1 i denotes the beginning of the storage activity at D k i and D k, 2 i denotes the end of the storage activity at D k i. The first distribution centers of both firms, D 1 1 and D 1 2, are located closer to their respective production sites than their second distribution centers, D 2 1 and D 2 2. The demand market R 1 is located closer to the firms’ first distribution centers, D 1 1 and D 1 2, whereas the demand market R 2 is closer to their second distribution centers, D 2 1 and D 2 2.
Typically, cantaloupes can be stored for 12–15 days at 2. 2 ∘ –5 ∘ C (36 ∘ –41 ∘ F). Their decay may result from postproduction diseases such as Rhizopus, Fusarium, and Geotrichum, depending on the season, the region, and the handling technologies utilized (see Suslow et al. 1997 and Sommer et al. 2002). As discussed in Sect. 4.2, we captured the food deterioration through the arc multipliers. The values of the decay constants and the time durations, although hypothetical, were selected so as to reflect the various supply chain activities. The values of the arc multipliers were calculated using (4.2). For example, we assume that Firm 1 utilizes more effective cleaning and sanitizing techniques for its processing activities which results in relatively higher operational costs but lower decay constants associated with the successive supply chain activities.
We implemented the Euler method (cf. (4.29)) for the solution of variational inequality (4.17), using MATLAB. We set the sequence \(\{a_{\tau }\} =.1\big{(}1, \frac{1} {2}, \frac{1} {2},\ldots \big{)}\). The convergence tolerance was 10 − 6. In other words, convergence was assumed to have been achieved when the absolute value of the difference between each path flow in two consecutive iterations was less than or equal to this tolerance.
Scenario 1.
In Scenario 1, we assumed that consumers at the demand markets are indifferent between the cantaloupes of Firm 1 and Firm 2. Furthermore, consumers at demand market R 2 are willing to pay relatively more than those at demand market R 1. The corresponding demand price functions were
where d 11, d 12, d 21, and d 22 denote the demands for cantaloupes, per day.
The arc multipliers, the total operational cost functions, and the total discarding cost functions are reported in Table 4.1, as are the decay constants (/day) and the time durations (days) associated with all the links. These cost functions have been constructed based on the data of the average costs available on the web (see, e.g., Meister 2004a,b). Table 4.1 also provides the computed equilibrium product flows on all the links in Fig. 4.2.
The computed equilibrium demands for cantaloupes were
The incurred equilibrium prices were
Furthermore, the profits of two firms were
Since consumers do not differentiate between the cantaloupes of the two firms, the prices of these two firms’ cantaloupes at each demand market are identical. Due to the difference in consumers’ willingness to pay, the price at demand market R 1 is lower than the price at demand market R 2. Consequently, the distribution links 21 and 25, connecting Firm 1 and Firm 2 to demand market R 1, respectively, have zero product flows. In other words, there is no shipment from the distribution centers D 2 1 and D 2 2 to demand market R 1. In addition, the volume of the product on distribution link 22 (or link 26) is higher than that on distribution link 20 (or link 24), which indicates that it is more cost-effective to provide fresh fruits from the nearby distribution centers. As a result of its lower operational costs, Firm 2 dominates both of the demand markets, leading to a substantially higher profit.
Scenario 2.
In Scenario 2, we considered that the Centers for Disease Control (CDC) reports a multistate cantaloupe-associated disease outbreak. Due to food safety and health concerns, many of the regular consumers of cantaloupes switch to other fresh fruits. The demand price functions are now given by
The longer shipment times associated with links 13 and 14 in Table 4.2 represent the need for inspections of imported food by the US government. Therefore, the values of arc multipliers associated with links 13 and 14 in Table 4.2 are lower than those in Table 4.1, which implies more cantaloupes will spoil during these links. The other arc multipliers and the total operational and the total discarding cost functions are the same as in Scenario 1, as shown in Table 4.2. The new computed equilibrium link flows are also reported in Table 4.2.
The computed equilibrium demands for cantaloupes were
The incurred equilibrium prices were:
Furthermore, the profits of two firms were:
The demand for cantaloupes is decreased by the associated outbreak, significantly decreasing the demand prices at demand markets. Both firms experience dramatic declines in their profits. In addition, additional distribution links 20, 21, 24, and 25 have zero product flows (as compared to Scenario 1), since the extremely low demand price cannot cover the costs associated with long-distance distribution.
Scenario 3.
Given the severe shrinkage in the demand, Firm 1 has realized the importance of regaining consumers’ confidence in its own product after the outbreak. Thus, Firm 1 has the label on its cantaloupes redesigned to incorporate a guarantee of food safety. This causes additional expenditures associated with its processing activities. The arc multipliers and the total operational and the total discarding cost functions are the same as in Scenario 2, except for the total operational cost function associated with the processing link 9. Please refer to Table 4.3.
The demand price functions corresponding to the two demand markets for cantaloupes from these two firms are now given by
The computed values of the equilibrium link flows are given in Table 4.3.
Note that links 21, 24, and 25 have zero flow. Hence, as we did for the medical nuclear supply chain network case study in Chap. 3, we depict the final network topology for this scenario with only the links with positive equilibrium flows displayed in Fig. 4.3. Note that the analogous final topology for Scenario 2 would be that given in Fig. 4.3 but with link 20 removed. As for the final topology for Scenario 1, it would be as in Fig. 4.2 but with only links 21 and 25 removed, since those have zero flow at the equilibrium solution.
The computed equilibrium demands for cantaloupes were
The incurred equilibrium prices at the demand markets were
Furthermore, the profits of two firms were
Consumers differentiate cantaloupes due to food safety and health concerns in Scenario 3. With the newly designed label, Firm 1 has managed to increase the consumption of its cantaloupes at both demand markets, whereas the demands for Firm 2’s cantaloupes are even lower than those in Scenario 2. Because of the cantaloupe-associated foodborne disease outbreak, the firms do not return to the same profit level as in Scenario 1. A comparison of the results in Scenario 2 and Scenario 3 suggests that practicing product differentiation may be an effective strategy for a food firm to maintain its profit. The demand for Firm 1’s product at demand market R 1 in Scenario 3 is even higher than that in Scenario 1, which is likely due to the decrease in the price coupled with the introduced guarantee of food safety.
It is also interesting to note that, due to food deterioration, Firm 1 produced an amount \(f_{1}^{{_\ast}} + f_{2}^{{_\ast}} = 73.56\) for the total demand of \(d_{11}^{{_\ast}} + d_{12}^{{_\ast}} = 63.98\), with roughly 13% of the cantaloupes spoiled and discarded during the postproduction supply chain activities. Firm 2 produced an amount of \(f_{3}^{{_\ast}} + f_{4}^{{_\ast}} = 11.87\) for the total demand of \(d_{21}^{{_\ast}} + d_{22}^{{_\ast}} = 9.39\).
As we did in Chap. 3, we now also present the computed equilibrium path flows. In Table 4.4, we report both nonzero (positive) and zero path flows since there are fewer paths than in the Scenario 3 example in Chap. 3. Links are numbered as in Fig. 4.2. Only two of the available four paths connecting each of the O/D pairs, w 1 1, w 1 2, and w 2 2, are used at the equilibrium solution. Only O/D pair w 2 1 has all its paths used.
Remark.
We emphasize that, with appropriate modifications, the model can handle the degradation in quality, which is more relevant to meat, dairy, and bakery products, and even quality competition. For example, the path multipliers μ p ; \(p \in \mathcal{P}\) can be used as quality parameters in the demand price functions. The conservation of flow equations would need to be then modified, since the arc and path multipliers would not measure quantity losses. For a dynamic network oligopoly model with quality competition and product differentiation, see Nagurney and Li [2012].
5 Summary and Conclusions
This chapter focused on food spoilage/deterioration between production and consumption locations, which poses unique challenges for food supply chain management. In particular, we presented a food supply chain network model under oligopolistic competition and perishability, with a focus on fresh produce, such as vegetables and fruits. Each food firm is involved in such supply chain activities as the production, processing, storage, distribution, and even the disposal of the food products and seeks to determine its optimal product flows, in order to maximize its own profit.
We captured the exponential time decay of food in the number of units through the introduction of arc multipliers, which depend on the time duration and the environmental conditions associated with each postproduction supply chain activity. We also incorporated the discarding costs associated with the disposal of the spoiled food products at the processing, storage, and distribution stages. Moreover, the competitive model allows consumers to differentiate food products at the demand markets due to product freshness and food safety concerns. In addition, the flexibility of the supply chain network topology enables decision-makers to evaluate alternative technologies involved in various supply chain activities.
We derived the variational inequality formulations of the food supply chain network Cournot–Nash equilibrium and presented some qualitative properties of the equilibrium pattern. We also provided the explicit formulae for each step of the iterative scheme, the Euler method, which was also the algorithm used for the blood supply chain network model in Chap. 2. We illustrated the model and the algorithm through a case study of the cantaloupe market in the United States. The results of the case study suggest that product differentiation may be an effective strategy for financial resiliency, especially in times of foodborne disease outbreaks.
6 Sources and Notes
This chapter is based on the paper by Yu and Nagurney [2013]. Here, we integrate food supply chains into the broader context of perishable product supply chain networks, the focus of this book, report additional output data, and provide an expanded discussion. As noted by Van der Vorst [2006], it is imperative to analyze food supply chains within the full complexity of their network structure, as we have done here. Monteiro [2007] claimed that the theory of network economics (cf. Nagurney 1999) provides a powerful framework in which the supply chain can be graphically represented and analyzed, which we also use as a foundation. He utilized the theory to study the traceability. Blackburn and Scudder [2009] suggested a cost minimization model for specific perishable product supply chain design, capturing the declining value of the product over time. The authors noted that product value deteriorates significantly over time at rates that highly depend on temperature and humidity. Rong [2011] presented a mixed integer linear programming model for food production and distribution planning (see also Kopanos 2012) with a focus on quality.
Several other contributions that integrated and synthesized two or more processes associated with food supply chains are worth mentioning. Zhang et al. [2003] studied a physical distribution system in order to minimize the total cost for storage and shipment with the product quality requirement fulfilled. Widodo et al. [2006] developed models dealing with flowering-harvesting and harvesting-delivering problems of agricultural products by introducing a plant-maturing curve and a loss function to address, respectively, the growing process and the decaying process of the products. Ahumada and Villalobos [2011] discussed the packing and distribution problem of fresh produce, with the inclusion of perishability.
Numerous challenges have underlined the need for the efficient management of food supply chains. The review by Lowe and Preckel [2004] focused on farm planning. Lütke Entrup [2005] discussed how to integrate shelf life into production planning in three sample food industries (yogurt, sausages, and poultry). Akkerman et al. [2010] outlined quantitative operations management applications in food distribution management. The survey by Lucas and Chhajed [2004] presented applications related to location problems in agriculture and recognized the challenges of strategic production-distribution planning problems in this industry. Due to the added complexity of perishability, there are fewer articles related to perishable food products than those related to nonperishable ones and even fewer models developed for fully integrated supply chain systems, the topic of this chapter.
As noted in Chaps. 1 and 2, our approach to product perishability utilizes generalized network concepts for nonlinear networks, with ideas based on Nagurney and Aronson [1989], who presented a multiperiod spatial price network equilibrium model. Spatial price equilibrium problems reflect perfect competition. In this book, we focus on a finite number of decision-makers, who have control over their supply chains. Liu and Nagurney [2012] proposed (cf. Fig. 4.4) a multiperiod oligopolistic network equilibrium model where perishability of the product over a fixed lifetime was captured through appropriate path definitions, reflecting that a product would not be consumed after a certain finite time period. The authors allowed for competition among the manufacturers and among the retailers, over the finite time horizon. Such supply chain network equilibrium models, also formulated as variational inequality problems, were introduced by Nagurney et al. [2002].
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© 2013 Anna Nagurney, Min Yu, Amir H. Masoumi, Ladimer S. Nagurney
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Nagurney, A., Yu, M., Masoumi, A.H., Nagurney, L.S. (2013). Food Supply Chains. In: Networks Against Time. SpringerBriefs in Optimization. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6277-4_4
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