# A supply chain network design problem for fruits and vegetables

Salima Kendi, Mohammed Said Radjef, Abdelhakim Hammoudi, Fodil Laib

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Salima Kendi, Mohammed Said Radjef, Abdelhakim Hammoudi, Fodil Laib. A supply chain network design problem for fruits and vegetables. 2025. hal-05316615

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A supply chain network design problem for fruits and vegetables

Salima Kendi<sup>a</sup>, Mohammed Said Radjef<sup>b</sup>, Abdelhakim Hammoudi<sup>c</sup>, Fodil Laib<sup>d</sup>

<sup>a</sup> LaMOS Research Unit (Modeling and Optimization of Systems), Faculty of Natural Sciences and Life, University of Bejaia, Bejaia, Algeria (salima.kendi@univ-bejaia.dz) (Corresponding Author).

<sup>b</sup> LaMOS Research Unit (Modeling and Optimization of Systems), Department of Operational Research, Faculty of Exact Sciences, University of Bejaia, Bejaia, Algeria (mohammedsaid.radjef@univ-bejaia.dz).

<sup>c</sup> Paris-Saclay University, INRAE, UR ALISS, Ivry-sur-Seine, France (ahhammoudi@yahoo.fr).

<sup>d</sup> CEVITAL Group, Illot D, No. 6 ZHUN, Garidi II, Kouba 16005 – Algiers, Algeria (fodil.laib@cevital.com).

**Abstract**

Food products reach the consumer through a complex network, involving stages of pro-
duction, bundling, sorting, aggregation, distribution and retailing. The supply and distribu-
tion system for these products is an integral part of road transport in urban, interurban and
peri-urban areas, from the place of production to the consumer. Consolidation sites (collection
centers, warehouses, wholesale markets) constitute an important part of the very complex logis-
tics associated with these systems. Their locations can promote or, on the contrary, negatively
impact the efficiency of supply chains by notably reducing the costs of transporting goods and
the food losses recorded during the transport of products to the final market. From this point of
view, our study contributes to the design of an efficient supply and distribution network while
guaranteeing the satisfaction of sustainability criteria : reduction of food losses and minimiza-
tion of the carbon footprint due to the transport of goods. We propose a model for locating
wholesale markets and multi-product collection centers with the objective of satisfying both
retailer demand and sustainability criteria at the lowest possible cost. The model is a mixed
integer linear program (MILP). The solution method of the proposed model is implemented in
the CPLEX 12.6 solver.

**Keywords**

Distribution network design, Fruits and vegetable supply chain, Food safety, Distribution cen-
ters, Facility location, Mixed integer linear programming.

# 1 Introduction

Food security is a fundamental issue for all countries and particularly for developing countries. This issue is exacerbated by the deterioration of climatic conditions which require more than ever, not only a review of production methods but also of product delivery to minimize waste of resources and produced goods. The stakes are all the more important when it comes to perishable products such as fruits and vegetables. Furthermore, meeting demand in volume and quality through the fight against waste in the supply stages must also today be accompanied by environmental concern, that is to say a minimization of the carbon footprint in transport operations of goods from places of production to places of marketing. Thus, the fight against food loss and food waste worldwide is often associated with the survival of the world's population (FAO, 2015; Bruinsma, 2009; Koester, 2014) and with ecological issues through the associated effects in terms of carbon footprint and water waste (Lundqvist et al., 2008). FAO (2015; 2013) estimated that the carbon and water footprint of this significant amount of food loss and waste at 4.4 gigatons (or 8% of the world's total) of CO2 equivalent and 250 km3 of blue water.

The factor which has a structural impact on waste and the performance of supply chains is, first of all, the geographical location of the infrastructures (collection/storage/packaging centers, distribution centers, etc.) which are involved in the transportation of products from their places of production to places of sale. The places of marketing and production and consumption of fruits and vegetables are spatially separated and production is generally located in rural areas and consumption in urban areas. The interface between the latter two (marketing phase), which allows products to move from an area of surplus to an area of need (Tracey-White, 1991) is mainly made up of a network of wholesalers, intermediaries and retailers. These players rely on complex marketing circuits to ensure the collection and redistribution of fresh products. The growth in demand for fresh food products, due in particular to the increase in population, accentuates the organizational challenges of supply chains. Wholesale markets (WM) are located within or near to cities (urban areas). These centers may be supplied by collection centers (CC) in the rural areas or directly from farms, particularly those in peri-urban areas. The supply is either from agents, traders or by the farmers themselves. The produce, after its arrival in an urban area, often passes through a number of intermediaries, including retailers before it reaches consumers (Tracey-White, 2005).

CCs are concentrated in rural areas. These are generally large outdoor ponds, where products collected from production fields are cleaned and consolidated for transport to the WM. Their
role is to facilitate transactions between producers and actors operating downstream of the sup-
ply chain. The MW manages considerable volumes of food. In the particular case of developing
countries, and not only, the inadequate distribution of these markets across national territories
leads to enormous losses and costs in wasted products and resources (Tracey-White, 2005).
More generally, the FAO estimates that significant quantities of fruits and vegetables intended
for human consumption around the world are lost or wasted. Thus, a quarter of all food produ-
ced in the world is lost, with from 25The aim of our work is to contribute to the efficient design
of fruit and vegetable distribution networks. The originality of our contribution compared to
previous contributions consists in the introduction of practical parameters : deterioration rates
of commodities in transportation and consolidation facilities (CC and WM), retailer demand
satisfaction rates that we defined by a supply strategy in the case of supply limitation, and the
proximity factor of WM candidate site on main roads in order to avoid. The introduction of
these parameters responds more to current issues and generalizes the work in this area.

We start notably from the reality of developing countries, namely the inadequacy of existing
fruit and vegetable supply distribution networks, and its negative consequences on transport
costs and ultimately on the final price of products and, on deterioration and microbiologi-
cal contamination of fresh products with its impact on consumer health. Inadequate locations
of collection and distribution networks lead to congestion problems, difficulties in accessing
markets, reduced distance traveled and consequently, gas emissions and carbon footprint. The
design of efficient supply chain infrastructures for fruits and vegetables, known for their high
perishability rate, depends on the location of consolidation sites or hubs (WM and CC) in rela-
tion to the location of main roads, of farms and points of sales. A good location of a WM or CC
allows for better access, reduced transport costs and the availability of healthy products. How
should we locate these different infrastructures to respond, within the framework of a multi-
criteria perspective, to all of the objectives cited above : minimization of transport costs and
the carbon footprint due to transport, minimization of products 'perishability and ultimately
product waste ? To answer this question, we develope a mathematical location model (a mixed
integer linear program (MILP)) for the optimal design of agricultural raw material distribution
networks, making it possible to determine the number of CC and WM, their location and the
optimal capacities of retail outlets centers. The model takes into account the specificity of fresh
products by defining a exogeneous product perishability factor which is a function of the great
diversity of the species and their physiological state after harvest (Fredriksson and Liljestrand,
2014), the travelled distance, the product handling at the CC and WM, the road conditions
and congestion, the truck characteristics (refrigerated or not), etc.

The purpose of our work is to contribute to the efficient design of production-distribution
networks of fruits and vegetables, adding more realistic parameters and constraints by reference to the anterior works. The model we propose has some similarities with other models of distribution networks design referenced in the literature. However, our model has distinctive characteristics (parameters mentioned above). Most studies of supply chains focus on the location of distribution centers for specified or unspecified products in general (see Kumari et al., 2023) : vehicles (see Colovic and Mayrhofer, 2011), pharmaceutical products (see Susarla and Karimi, 2012); agricultural and perishable products for a single product such as milk (see Song and Wu, 2023; Gebresenbet and Ljungberg, 2001; ...), meat (in abattoir chains) (see Soysal et al., 2014; Gebresenbet et al., 2011a; Sohos, 2008, ...), flowers (see Karpun, 2020), etc.; products from agri-food companies (see Onggo et al., 2019; Kendi et al., 2020). To our knowledge, Etemadnia et al. (2015) are the only ones who have studied the fruit and vegetable supply chain with a mathematical location model. The paper presents a mixed integer linear programming (MILP) model for finding optimal hub locations within a national logistics system. The model is developed to minimize the total transportation costs across two possible transportation modes, a standard low cost mode (land transfers) and a high cost mode (air transfers).

Some features distinguish our model from that of Etemadnia et al. (2015). In our model we studied CC, which is more realistic and more efficient, adopted the strategy of meeting demand for insufficient supply, modeled the products perishability factor and integrated it into the decision making process in order to evaluate the remaining supply for consumption and ameliorate food security, and took into account the impact of main roads on wholesale markets locations which is the most crucial criterion for choosing these sites (Tracey-White, 2005). Our contributions will appear also, in the experimental study by studying the impact of different parameters on the trade-off between the different kinds of costs and objectives.

The model and its associated program constitute decision-making tools for public authorities keen to better supervise the location of the network of the supply system for perishable products such as fruit and vegetables. The model is also an extension of previous work (Etemadnia et al. (2015)) by explicitly integrating the dimension of sustainability (carbon footprint and waste) and quality of the final product. The model shows how all the variables associated with these dimensions interact in the search for the optimal solution.

We have developed a mathematical location model (a mixed integer linear program (MILP)) for the optimal design of agricultural commodity distribution networks, making it possible to find the optimal number, locations and capacities of collection and wholesale centers, given the locations agricultural fields, retail outlets and major roads. The model takes into account the specificity of fresh products (availability of supply, product perishability factor, etc.). Through
the study of different distribution network structuring problems, encountered and motivated in
real life, the model makes it possible to suggest strategic solutions to decision makers. We have
demonstrated how certain parameters affect the final decision and lead to a series of recom-
mendations so that the various objectives set are achieved when designing fruit and vegetable
distribution networks.

## 2 Literature review

Historically, the foundations of the economic theory of location applied to agriculture were traced by Von Thünen (1826). The idea is to explain the location of agricultural activities, land use and the formation of rent by highlighting the role of the distance between production areas and market places (and ultimately the costs of transport associated with it). Later, a certain number of economic theories of location were developed, which differ essentially in the nature of the specific questions studied and the body of hypotheses taken into account : environment of the actors (typologies of markets, structure of production systems and demand, transport costs, ...), behavior of actors (price makers or takers, typology of interaction games on the markets, ...). The model of Geoffrion and Graves (1974) was the first distribution model to incorporate the location of distribution centers and the allocation of products and customers to them in order to minimize the total cost of the system.

For an optimal design of the fresh products supply chain, the concept of hubs is employed.
Campbell and O'Kelly (2012) define a hub as an intermediate node when the direct link bet-
ween each pair (Origin / Destination) is difficult or costly (travelers, airline passengers, farm
products, electronic mail in telecommunication networks, etc). A food hub in a food supply
chain system is defined by (Etemadnia et al., 2015) as an intermediate node that connects with
more than one production location and more than one consumption ones to consolidate product
flows. For the fruits and vegetables supply chain, hubs represent CC and WM because they
are intermediate nodes that consolidate product flows and allow them to be aggregated and
disaggregated.

The hub location problem is first posed by O'Kelly (1986, 1987). Given a set of origin desti-
nation pairs with positive flow, the hub location problem involves the decisions on the locations
of the hubs and the allocations of the demand nodes to these hubs. O'Kelly (1987) proposes
a cost minimizing formulation of the problem, which may be considered as the first model in
the hub location literature. Campbell (1994) classifies hub location problems according to their
optimization criteria : (i) minimization of the total transportation cost, p-hub median problem,
(ii) minimization of the total transportation cost and the fixed cost of establishing hubs, unca-
pacitated/capacitated hub location problem, (iii) minimization of the maximum transportation
cost, p-hub center problem, and (iv) minimization of the number of hubs while serving each
pair within a predetermined bound, hub covering problem. The majority of the literature is
devoted to the first two problems (i) and (ii) (see, for example, Campbell (1996); Ernst and
Krishnamoorthy (1996, 1998), O'Kelly et al. (1996); Sohn and Park (1998)).

Several case studies in the field of location and food distribution networks have been carried
out on the efficiency of food distribution networks, we quote : the multi-criteria location model
of Yildiz et al. (2019) for the selection of retail locations with illustration on food stores in Tur-
key; the stochastic location model of Baghalian et al. (2013) for the design of a multi-product
supply chain network and its application for the design of a network in the rice industry of
a Middle Eastern country. In Wang et al. (2017), the authors developed a location model of
refrigerated distribution centers to study a large supermarket chain in China. The efficiency of
food distribution networks (fresh or processed foods) was analyzed in the works of Bosona et al.
(2011a, 2011b, 2013a, 2013b), using Geographic Information System (GIS) and Route LogiX
software, in the goal of increasing the sustainability of the local food system in sweden.

3 Problem statement

For fruits and vegetables supply chain, the wholesale market is facing large quantities of
commodities. Naturally, the most general case is thus, the one where a market has no interest
in negotiating with an infinity number of producers, but with a reduced number of collectors.
Consequently, we consider in the current study, for the supply chain, producers, collectors, WM
and retailers. Figure 1 shows a schematic representation of this scenario.

Collection centers allow the grouping of products from different agricultural fields to be
consolidated into larger shipments destined for WM. The large size of the transported lots
induces the transition to larger transport units (big trucks), therefore the cost of the tonne-
kilometer decreases (economies of scale). And as CC's are generally in rural areas, close to
agricultural production fields and WM are in urban areas, close to consumers, the distance
between the two is considerable. As a result, transport costs are reduced over a long distance
and therefore a significant reduction in total transport costs is expected. Thus, even if the
establishment of CC generates costs (long-term decision) and can increase the total distance
travelled, the total cost of daily transportation in the network can be significantly reduced.

The design of efficient fruits and vegetables (known for their high rate of perishability) sup-
ply chain infrastructures is directly related to the location of CC and WM (hubs) according to
<figure><img src="image_2.png" /><figcaption>FIGURE 1 – Fruits and vegetables traceability</figcaption></figure>

those of farms, retail outlets and main roads. The location of these logistical bases can have
a big impact on the efficiency of a food system. Note that these design decisions are strategic
(long-term decisions) because they involve a huge capital investment.

Given a set of farms, retail outlets and candidate sites for locating collection and who-
lesale facilities (Figure 2). To design an optimal fruits and vegetables supply chain network, we
are asked to answer the following questions :

- How many facilities (CC and WM) should be installed ?

- What are the locations of the facilities to be installed ?

- What would be the size of each facilitie ?

- What will be the different allocations between supply, demand and the facilities ?

The answers to these questions will constitute critical decisions, making it possible to minimize
the total cost of the distribution network, which includes :

- constructing collection and wholesale facilities ;

- transporting commodities in the network;

- penalty cost of remoteness between the WM and the main roads.
<figure><img src="image_3.png" /><figcaption>FIGURE 2 – Fruits and vegetables from production to consumption</figcaption></figure>

## 4 Assumptions and problem formulation

We define below the parameters and the variables to be used (Tables : 1-5).

<table><caption>TABLE 1 – Sets and descriptions</caption><thead><tr><th>Sets</th><th></th></tr></thead><tbody><tr><td>I(|I| = n<sub>1</sub>)</td><td>Set of production fields</td></tr><tr><td>J<sub>1</sub>(|J<sub>1</sub>| = n<sub>21</sub>)</td><td>Set of main retail outlets</td></tr><tr><td>J<sub>2</sub>(|J<sub>2</sub>| = n<sub>22</sub>)</td><td>Set of secondary retail outlets</td></tr><tr><td>J = J<sub>1</sub> &cup; J<sub>2</sub>, J<sub>1</sub> &cap; J<sub>2</sub> = &emptyset; (|J| = n<sub>2</sub>)</td><td>Set of all retail outlets</td></tr><tr><td>H(|H| = m)</td><td>Set of candidate sites to locate CC</td></tr><tr><td>K(|K| = n)</td><td>Set of candidate sites to locate WM</td></tr><tr><td>L(|L| = p)</td><td>Set of commodities (fruits and vegetables)</td></tr></tbody></table>
<table><caption>TABLE 2 – Input data and descriptions</caption><thead><tr><th colspan="2">Input data</th></tr></thead><tbody><tr><td>S<sub>i</sub><sup>l</sup></td><td>Available supply quantity of commodity l at production field i, l &isin; L, i &isin; I</td></tr><tr><td>D<sub>j</sub><sup>l</sup></td><td>Requested demand quantity of commodity l at retail outlet j, l &isin; L, j &isin; J</td></tr><tr><td>d<sub>ih</sub><sup>l</sup></td><td>Distance between production field i and collection center candidate site h, i &isin; I, h &isin; H</td></tr><tr><td>d<sub>ik</sub><sup>l</sup></td><td>Distance between collection center candidate site h and wholesale market candidate site k, h &isin; H, k &isin; K</td></tr><tr><td>d<sub>kj</sub><sup>l</sup></td><td>Distance between wholesale market candidate site k and retail outlet j, k &isin; K, j &isin; J</td></tr><tr><td>d<sub>k</sub><sup>l</sup></td><td>Distance between wholesale market candidate site k and the nearest main road to it, k &isin; K</td></tr><tr><td>c<sub>1</sub></td><td>Unit cost of commodity transportation</td></tr><tr><td>c<sub>2</sub></td><td>Fixed construction cost of a collection center</td></tr><tr><td>c<sub>3</sub></td><td>Unit construction cost of a collection center</td></tr><tr><td>c<sub>4</sub></td><td>Fixed construction cost of a wholesale market</td></tr><tr><td>c<sub>5</sub></td><td>Unit construction cost of a wholesale market</td></tr><tr><td>c<sub>6</sub></td><td>Unit penalty cost of remoteness between a wholesale market candidate site and the nearest main road</td></tr><tr><td>a<sub>h</sub></td><td>Minimum capacity of the collection center to be constructed, where appropriate, at candidate site h, h &isin; H</td></tr><tr><td>b<sub>h</sub></td><td>Maximum capacity of the collection center to be constructed, where appropriate, at candidate site h, h &isin; H</td></tr><tr><td>e<sub>k</sub></td><td>Minimum capacity of the wholesale market to be constructed, where appropriate, at candidate site k, k &isin; K</td></tr><tr><td>f<sub>k</sub></td><td>Maximum capacity of the wholesale market to be constructed, where appropriate, at candidate site k, k &isin; K</td></tr><tr><td>&alpha; (0 &lt; &alpha; &lt; 1)</td><td>Discount factor reducing unit transportation cost of commodities between collection and wholesale facilities (hubs) : economies of scale from consolidated transportation</td></tr></tbody></table>

<table><caption>TABLE 3 – Decision variables and descriptions</caption><thead><tr><th colspan="2">Decision variables</th></tr></thead><tbody><tr><td>u<sub>ih</sub><sup>l</sup></td><td>Quantity of commodity l to be transferred from production field i to the collection center at site h, i &isin; I, h &isin; H, l &isin; L</td></tr><tr><td>v<sub>hk</sub><sup>l</sup></td><td>Quantity of commodity l to be transferred from the collection center at site h to the wholesale market at site k, h &isin; H, k &isin; K, l &isin; L</td></tr><tr><td>x<sub>h</sub></td><td>Boolean variable : x<sub>h</sub> = 1 if a collection center is to be located at candidate site h and 0 otherwise, h &isin; H</td></tr><tr><td>y<sub>k</sub></td><td>Boolean variable : y<sub>k</sub> = 1 if a wholesale market is to be located at candidate site k and 0 otherwise, k &isin; K</td></tr><tr><td>t<sub>kj</sub></td><td>Boolean variable : t<sub>kj</sub> = 1 if retail outlet j is to be supplied by the wholesale market at candidate site k and 0 otherwise, k &isin; K, j &isin; J</td></tr></tbody></table>
The underlying assumptions and formulations of the problem are detailed in the following subsections.

## 4.1 High products perishability

Fruits and vegetables are highly perishable products and due to the great diversity of the species and their physiological state after harvest, the marketing of these foodstuffs varies widely between countries and results huge losses and costs, notably in the storage and the transport (Fredriksson and Liljestrand, 2015). We have modeled this perishability factor by introducing fixed deterioration rates and a unit cost of wasting food, in order to integrate this aspect into the decision making process and to evaluate the remaining supply for consumption. We use average deterioration rates of commodities in transportation and others in collection and wholesale facilities :

· $\theta^l (0 < \theta^l < 1)$ : average deterioration rate of commodity $l$ at CC, $l \in L$.

· $\eta^l (0 < \eta^l < 1)$ : average deterioration rate of commodity $l$ at WM, $l \in L$.

· $\lambda^l (0 < \lambda^l < 1)$ : average deterioration rate of commodity $l$ during its transportation from production field to collection center, $l \in L$.

· $\mu^l (0 < \mu^l < 1)$ : average deterioration rate of commodity $l$ during its transportation from collection center to wholesale market, $l \in L$.

The average deterioration rate of goods in transportation between production fields and CC $\lambda^l$, is lower than that between CC and WM $\mu^l$, because the distance between these two last is more important, $l \in L$.

## 4.2 Retailers demand satisfaction strategy

In the literature, the majority of studies assumes that supply is greater than demand to assure the feasibility of the model. In this paper, we overlook this hypothesis and study all possible cases, that are : supply superior to demand, supply equal to demand and supply lower than demand. When supply is insufficient, which leads to the rise in prices of commodities, customers (retailers) demand are automatically not fully satisfied, so two classes of customers appears : a class of customers who supply their entire demand (priority customers), who we name "main retailers" and the rest of customers whose demand is not fully satisfied ("secondary retailers"). To define our demand satisfaction strategy, we modeled the situation by establishing some entities :

· $S^l = \sum_{i \in I} S_i^l$ : available supply quantity of commodity $l$ at all production fields, $l \in L$.
· $\bar{S}^l = (1 - \lambda^l)(1 - \theta^l))(1 - \mu^l)(1 - \eta^l))S^l : \text{remaining supply quantity of commodity } l \text{ after losses in transportation and sorting, } l \in L.$

· $\gamma_1^l = \frac{\bar{S}^l}{\sum_{j \in J_1} D_j^l}$ : demand rate of commodity $l$ that can be satisfied for main retailers (without taking in consideration secondary retailers), $l \in L.$

· $\tau_1^l = \min\{1, \gamma_1^l\}$ : to ensure that the satisfaction rate of main retailers, for the product $l$ will not exceed 100% (this case appears when the supply is greater than the demand), $l \in L.$

· $\gamma_2^l = \frac{\bar{S}^l - \tau_1^l}{\sum_{j \in J_2} D_j^l}$ : demand rate of commodity $l$ that can be satisfied for secondary retailers (after main retailers had been served), $l \in L.$

· $\tau_2^l = \min\{1, \gamma_2^l\}$ : to ensure that the satisfaction rate of secondary retailers, for the product $l$ will not exceed 100%, $l \in L.$

Main retailers are first served and the satisfaction rate $\tau_1^l$, of their requests for each product $l$, is equal to the availability rate of the total supply (remaining after subtraction of the quantities damaged from the production fields to the WM) in relation to their total demand. Secondary retailers are after served fairly of what remains (with the satisfaction rate $\tau_2^l$ for each product $l$), and if there are not $(\tau_2^l = 0)$, they are not served, $l \in L.$

## 4.3 Impact of main roads

From an economic viewpoint (Tracey-White, 1991, 2005), given that WM are generally located in urban areas, known by very frequent overcrowding, the most crucial criterion for choosing a wholesale market location is its proximity to a main road, preferably with more than one access point, to ensure fluidity and avoid congestion. For this important reason, we take into account the distance $d_k^4$ between each wholesale market candidate site $k$ and the nearest main road to it, $k \in K$. We integrate then the impact of the main roads into the decision making by affecting a penalty cost of remoteness between these last and the WM.
# 5 The model

According to the assumptions mentioned above, we propose below a mathematical model which assists in the design of the efficient fruits and vegetables supply chain networks :

$$f_{11}(u_{ih}^l; i \in I, h \in H, l \in L) = \sum_{i \in I} \sum_{h \in H} c_1 d_{ih}^1 \sum_{l \in L} u_{ih}^l.$$

$$f_{12}(v_{hk}^l; h \in H, k \in K, l \in L) = \sum_{h \in H} \sum_{k \in K} \alpha c_1 d_{hk}^2 \sum_{l \in L} v_{hk}^l.$$

$$f_{13}(t_{kj}; k \in K, j \in J) = \sum_{k \in K} \left( \sum_{j \in J_1} c_1 d_{kj}^3 \sum_{l} D_j^l t_{kj} \tau_l^1 + \sum_{j \in J_2} c_1 d_{kj}^3 \sum_{l \in L} D_j^l t_{kj} \tau_l^2 \right).$$

$$f_{21}(u_{ih}^l, x_h; i \in I, h \in H, l \in L) = \sum_{h \in H} \left( x_h c_2 + c_3 \sum_{i \in I} \sum_{l \in L} u_{ih}^l \right).$$

$$f_{22}(v_{hk}^l, y_k; h \in H, k \in K, l \in L) = \sum_{k \in K} \left( y_k c_4 + c_5 \sum_{h \in H} \sum_{l \in L} v_{hk}^l \right).$$

$$f_3(y_k) = c_6 \sum_{k \in K} d_k^4 y_k.$$

$$f_1(u_{ih}^l, v_{hk}^l, t_{kj}) = f_{11}(u_{ih}^l) + f_{12}(v_{hk}^l) + f_{13}(t_{kj})$$

$$f_2(u_{ih}^l, v_{hk}^l, x_h, y_k) = f_{21}(u_{ih}^l, x_h) + f_{22}(v_{hk}^l, y_k)$$

$$f(u_{ih}^l, v_{hk}^l, t_{kj}, x_h, y_k) = f_1(u_{ih}^l, v_{hk}^l, t_{kj}) + f_2(u_{ih}^l, v_{hk}^l, x_h, y_k) + f_3(y_k).$$

Minimize

$$f(u_{ih}^l, v_{hk}^l, t_{kj}, x_h, y_k) \qquad (1)$$
Subject to

$$ \sum_{h \in H} u_{ih}^{l} \leq S_{i}^{l} \quad \forall i \in I, \forall l \in L \quad (2) $$

$$ (1 - \lambda^l)(1 - \theta^l) \sum_{i \in I} u_{ih}^l = \sum_{k \in K} v_{hk}^l \quad \forall h \in H, \forall l \in L \quad (3) $$

$$ (1 - \mu^l)(1 - \eta^l) \sum_{h \in H} v_{hk}^l = \sum_{j \in J_1} D_j^l t_{kj} \tau_1^l + \sum_{j \in J_2} D_j^l t_{kj} \tau_2^l \quad \forall k \in K, \forall l \in L \quad (4) $$

$$ \sum_{i \in I} \sum_{l \in L} u_{ih}^l \leq x_h \cdot b_h \quad \forall h \in H \quad (5) $$

$$ \sum_{i \in I} \sum_{l \in L} u_{ih}^l \geq x_h \cdot a_h \quad \forall h \in H \quad (6) $$

$$ \sum_{h \in H} \sum_{l \in L} v_{hk}^l \leq y_k \cdot f_k \quad \forall k \in K \quad (7) $$

$$ \sum_{h \in H} \sum_{l \in L} v_{hk}^l \geq y_k \cdot e_k \quad \forall k \in K \quad (8) $$

$$ \sum_{k \in K} t_{kj} = 1 \quad \forall j \in J \quad (9) $$

$$ t_{kj} \leq y_k \quad \forall j \in J, \forall k \in K \quad (10) $$

$$ u_{ih}^l \geq 0 \quad \forall i \in I, \forall h \in H, \forall l \in L \quad (11) $$

$$ v_{hk}^l \geq 0 \quad \forall h \in H, \forall k \in K, \forall l \in L \quad (12) $$

$$ t_{kj} \in \{0, 1\} \quad \forall k \in K, \forall j \in J \quad (13) $$

$$ x_h \in \{0, 1\} \quad \forall h \in H \quad (14) $$

$$ y_k \in \{0, 1\} \quad \forall k \in K \quad (15) $$

The obtained mathematical model is a mixed integer linear programming problem (MILP). The objective function, given in (1), minimizes the total cost of the network design, which includes the total transportation cost in the network ($f_1$), the total construction cost of collection and wholesale facilities ($f_{21}$ and $f_{22}$ respectively), and the penalty cost of remoteness between the WM and the main roads ($f_3$).

Constraints (2) assure that production field $i \in I$ can't deliver more than it has for each product $l \in L$. Equations (3) assure that tonnage, received by collection center $h \in H$ from production fields minus the deteriorated quantity, should be transferred to WM for each commodity $l \in L$. In equations (4), main and secondary retailers are affected what they need of goods or what is available, according to the defined demand satisfaction strategy between main and secondary retailers (described in subsection 4.2). Constraints (5) (resp. (7)) represent the constraints of capacity limitation of CC sites, resp. WM sites (for budgetary or geographical limitations). Constraints (6) (resp. (8)) set lower bounds on CC (resp. WM). Constraints (9) assure that each retail outlet can procure his products from only one wholesale market, in or-
der to reduce his transportation cost; these constraints also guarantee that all WM have all the varieties of produced fruits and vegetables. Constraints (10) assure that retail outlets can procure their products from only opened WM. Finally, constraints (11)-(12) (resp. (13)-(15)) are non-negativity, resp. integrality constraints.

<figure><img src="image_4.png" /><figcaption>FIGURE 3 – Integrated supply chain network design problem for fruits and vegetables</figcaption></figure>

In practice, several situations (problems) are encountered :

## 5.1 Problem 1 : design of a fruits and vegetables supply network

After the introduction of the data (Table (2), the linear program with mixed variables (1)-(15)) allows to determine a solution to the problem of designing a new supply chain network
for fruits and vegetables.

A problem in a similar chain was dealt with by Lin et al. (2006), where the design decisions considered are : the number and locations of consolidation and distribution centers (CC & DC)), the inventory levels of the various products (products in general) to be held at the DC, and the routing of shipments.

## 5.2 Problem 2 : location of WM

This problem concerns the situation where an existing supply chain network for fruits and vegetables already has collection centres and decision-makers (the public authority) consider it is necessary to build wholesale markets, and the problem is to determine their number, their locations and capacities. To answer the problem, we use the model (1)-(15), where, the set of the candidate sites to set up the CC, $H(|H| = m)$ becomes the set of existing CC. Consequently, we delete the decision variables $x_h, h \in H$ in the model, leading so to the cancelation of constraints (14) and the replacement of constraints (5) and (6) by respectively the following inequalities :

$$ \sum_{i \in I} \sum_{l \in L} u_{ih}^{l} \leq b_{h}, \quad \forall h \in H, \qquad (16) $$

$$ \sum_{i \in I} \sum_{l \in L} u_{ih}^{l} \geq a_{h} \quad \forall h \in H, \qquad (17) $$

where $a_h$ (resp. $b_h$) becomes the minimum (resp. the maximum) capacity of existing CC, $h \in H$. We maintain the rest of variables and constraints in order to search for the efficient supply chain. By deleting the construction cost of CC, the objective function becomes :

$$ f(u_{ih}^{l}, v_{hk}^{l}, t_{kj}, \cdot, y_{k}) = f(u_{ih}^{l}, v_{hk}^{l}, t_{kj}, x_{h}, y_{k}) - f_{21}(u_{ih}^{l}, x_{h}) \qquad (18) $$

Etemadnia et al. (2015) have studied this problem of location WM on a local territory in USA, taking into consideration the distribution modes of developed countries (land and air transfers).

## 5.3 Problem 3 : location of CC

In this situation, we suppose that WM are already established and operational in the supply chain and the decision makers want to establish an efficient collection system, given the advantages of CC. The modeling of this problem leads us to also use the model (1)-(15). In this case, the set of candidate sites to install WM, $K(|K| = n)$, becomes the set of existing WM. We delete the decision variables of locating WM $y_k, k \in K$ in the model, and so the constraints (10) and (15) are canceled and the constraints (7) and (8) are respectively replaced by :

$$ \sum_{h \in H} \sum_{l \in L} v_{hk}^{l} \leq f_{k} \quad \forall k \in K, \qquad (19) $$
$$ \sum_{h \in H} \sum_{l \in L} v_{hk}^{l} \geq e_k \quad \forall k \in K, \qquad (20) $$

where $e_k$ (resp. $f_k$) becomes the minimum (resp. maximum) capacity of existing WM, $k \in K$. We maintain the rest of variables and constraints in the model.

We delete the construction cost of WM and the penalty cost of remoteness between WM and main roads, than the objective function becomes :

$$ f(u_{ih}^{l}, v_{hk}^{l}, t_{kj}, x_{h}, \cdot) = f(u_{ih}^{l}, v_{hk}^{l}, t_{kj}, x_{h}, y_{k}) - f_{22}(v_{hk}^{l}, y_{k}) - f_{3}(y_{k}). \qquad (21) $$

## 5.4 Problem 4 : location of CC for a single product

It concerns a subproblem of Problem 3. The model can be reduced to the situation of setting up a collection system for a single agricultural product. A product with a precious consumption status at the national and international level (for his a high nutritional values for example), which raise it to play a significant role in the national economy of producing countries (like dates and banana). Therefore, to ensure an efficient coverage of the product, it may be necessary to establish CC (or warehouses), dedicated especially and only for that product in order to transport it to all existing WM. To solve this problem, we use the model of the Problem 3, on which we study a single product $l^*$, $l^* \in L$. The model will be :

**Minimize**

$$ f(u_{ih}^{l*}, v_{hk}^{l*}, t_{kj}, x_{h}, \cdot) = f(u_{ih}^{l*}, v_{hk}^{l*}, t_{kj}, x_{h}, y_{k}) - f_{22}(v_{hk}^{l*}, y_{k}) - f_{3}(y_{k}). \qquad (22) $$
Subject to

$$ \sum_{h \in H} u_{ih}^{l^*} \le S_i^{l^*} \quad \forall i \in I, \qquad (23) $$

$$ (1 - \lambda^{l^*})(1 - \theta^{l^*}) \sum_{i \in I} u_{ih}^{l^*} = \sum_{k \in K} v_{hk}^{l^*} \quad \forall h \in H, \qquad (24) $$

$$ (1 - \lambda^{l^*})(1 - \eta^{l^*}) \sum_{h \in H} v_{hk}^{l^*} = \sum_{j \in J_1} D_j^{l^*} t_{kj} \tau_1^{l^*} + \sum_{j \in J_2} D_j^{l^*} t_{kj} \tau_2^{l^*} \quad \forall k \in K, \qquad (25) $$

$$ \sum_{i \in I} u_{ih}^{l^*} \le x_h \cdot b_h \quad \forall h \in H, \qquad (26) $$

$$ \sum_{i \in I} u_{ih}^{l^*} \ge x_h \cdot a_h \quad \forall h \in H, \qquad (27) $$

$$ \sum_{h \in H} v_{hk}^{l^*} \le r_k \quad \forall k \in K, \qquad (28) $$

$$ \sum_{h \in H} v_{hk}^{l^*} \ge q_k \quad \forall k \in K, \qquad (29) $$

$$ \sum_{k \in K} t_{kj} = 1 \quad \forall j \in J \qquad (30) $$

$$ u_{ih}^{l^*} \ge 0 \quad \forall i \in I, \forall h \in H, \qquad (31) $$

$$ v_{hk}^{l^*} \ge 0 \quad \forall h \in H, \forall k \in K, \forall l \in L \qquad (32) $$

$$ t_{kj} \in \{0, 1\} \quad \forall k \in K, \forall j \in J \qquad (33) $$

$$ x_h \in \{0, 1\} \quad \forall h \in H \qquad (34) $$

where $q_k$ (resp. $r_k$) is the minimum (resp. the maximum) quantity of the product $l^*$ to be transferred to WM $k$, $l^* \in L$, $k \in K$. We have not found a paper in the literature dealing with this type of problem, either on a theoretical or practical level.

# 6 Experimental study

The methods for solving location problems, that are generally modeled by mixed linear programs, are divided into two large families, exact methods (Branch and Bound method (BB) for small problems, mathematical programming, etc.) and approximate methods (heuristics and metaheuristics (see Mohammed & Duffuaa, 2020; Riahi et al., 2021)). The model we propose is a mixed integer linear programming problem (MILP). We implemented the models deriving from Problems (1-4) in MATLAB and used CPLEX 12.6 as a solver for the MILP with the default options. The calculations were performed on high performance computers, with 8 processors and an allocation memory of 60 gigabytes. We tested then the behaviour of our model on a randomly generated examples. The software solution interface is presented in Figure 4.
<figure><img src="image_5.png" /><figcaption>FIGURE 4 – Fruits and vegetables supply chain network design software</figcaption></figure>

## 6.1 Preparing data

We generated a random test example. Table 4 illustrates the input parameters values.
<table><caption>TABLE 4 – Generated distribution network example</caption><thead><tr><th>Parameter</th><th>Value</th><th>Parameter</th><th>Value</th></tr></thead><tbody><tr><td>n<sub>1</sub></td><td>1000</td><td>c<sub>3</sub></td><td>28500</td></tr><tr><td>n<sub>21</sub></td><td>500</td><td>c<sub>4</sub></td><td>950000000</td></tr><tr><td>n<sub>22</sub></td><td>400</td><td>c<sub>5</sub></td><td>50892</td></tr><tr><td>m</td><td>20</td><td>c<sub>6</sub></td><td>28</td></tr><tr><td>n</td><td>15</td><td>a<sub>h</sub></td><td>0, h ∈ H</td></tr><tr><td>p</td><td>10</td><td>b<sub>h</sub></td><td>changes, h ∈ H</td></tr><tr><td>S<sup>l</sup><sub>i</sub></td><td>U[500000, 600000], l ∈ L, i ∈ I</td><td>e<sub>k</sub></td><td>0, k ∈ K</td></tr><tr><td>D<sup>l</sup><sub>j</sub></td><td>U[0.5, 7], l ∈ L, j ∈ J</td><td>f<sub>k</sub></td><td>changes, k ∈ K</td></tr><tr><td>d<sup>l</sup><sub>ih</sub></td><td>U[20, 30], i ∈ I, h ∈ H</td><td>α</td><td>0.2, 0.4, 0.6, 0.8, 1</td></tr><tr><td>d<sup>2</sup><sub>hk</sub></td><td>U[80, 100], h ∈ H, k ∈ K</td><td>λ<sup>l</sup></td><td>U[0.5, 0.1], l ∈ L</td></tr><tr><td>d<sup>3</sup><sub>kj</sub></td><td>U[50, 70], k ∈ K, j ∈ J</td><td>λ<sup>l</sup></td><td>U[0.1, 0.15], l ∈ L</td></tr><tr><td>d<sup>4</sup><sub>k</sub></td><td>U[4, 20], k ∈ K</td><td>θ<sup>l</sup></td><td>U[0.15, 0.2], l ∈ L</td></tr><tr><td>c<sub>1</sub></td><td>14</td><td>η<sup>l</sup></td><td>U[0.1, 0.15], l ∈ L</td></tr><tr><td>c<sub>2</sub></td><td>285000000</td><td></td><td></td></tr></tbody></table>

## 6.2 Sensitive analysis and results

The critical decisions on locations, allocations and capacities vary based on several parameters changes (hub capacity limitation, transportation discount factor, route conditions, etc), and this has remarkable consequences on the decisions. For this reason, we conducted a sensitive analysis to examine the effect of these parameters changes on the final decisions.

### 6.2.1 Impact of transportation economies of scale

To assess cost advantages of economy of scale, we took α-value (discount factor) to be 0.2, 0.4, 0.6, 0.8 and 1. We applied on the model (1)-(15), then we observed the impact on transportation cost, total cost and optimal locations of WM and CC (Table 5).

<table><caption>TABLE 5 – Problem 1: Impact of economy of scale between CC and WM on final decisions</caption><thead><tr><th>α</th><th>Locations of CC</th><th>Locations of WM</th><th>Transportation cost (×10<sup>4</sup>)</th><th>% reduction in transportation cost</th><th>Execution time</th></tr></thead><tbody><tr><td>0.2</td><td>1, 2, 3, 4, 6, 7, 8, 10, 12, 13, 14, 15, 17, 18, 19, 20</td><td>1, 2, 5, 6, 7, 8, 10, 12, 13, 15</td><td>10885</td><td>44.16%</td><td>6m50.762s</td></tr><tr><td>0.4</td><td>1, 2, 3, 5, 6, 7, 8, 10, 12, 13, 14, 15, 17, 18, 19, 20</td><td>1, 2, 5, 6, 7, 8, 10, 11, 12, 13</td><td>13039</td><td>33.11%</td><td>6m54.727s</td></tr><tr><td>0.6</td><td>1, 2, 3, 5, 6, 7, 8, 10, 12, 13, 14, 15, 17, 18, 19, 20</td><td>1, 2, 5, 6, 7, 8, 10, 11, 12, 13</td><td>15193</td><td>22.06%</td><td>6m57.381s</td></tr><tr><td>0.8</td><td>1, 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 18, 19, 20</td><td>1, 3, 5, 6, 7, 8, 10, 11, 12, 14</td><td>17343</td><td>11.03%</td><td>6m48.061s</td></tr><tr><td>1</td><td>1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 19, 20</td><td>1, 3, 5, 6, 7, 8, 10, 11, 12, 14</td><td>19492</td><td>0%</td><td>6m49.646s</td></tr></tbody></table>
The optimal number of CC (resp. WM) is 16 (resp. 10) for all cases. We noticed that construction cost (equal to $22424 \times 10^6$) is not affected by the discount factor variation, this is because the number and total required capacities of CC and WM have not changed depending on $\alpha$ while total transportation cost is reduced for low values of the factor. We note that for $\alpha$ equal to 0.4 and 0.6, the optimal locations of CC (resp. WM) are the same. For the other cases, the locations of the CC (resp. WM) differ slightly. We note that with a good reduction in economy of scale, up to 44.16% of the total transportation cost in the network can be achieved and this reflects one of the major advantages of collection centres.

### 6.2.2 Impact of variation in the maximum capacity of CC and WM

Tables 6, 7 and 8 show the optimal number and locations of hubs (CC and/or WM) with the generated costs according to variable maximum hub capacities for Problem 1, Problem 2 and Problem 3 respectively. Figures 5, 6 and 7 schematize the curves of these results.

#### Problem 1

<table><caption>TABLE 6 – Problem 1 : Impact of CC and WM capacity limitation</caption><thead><tr><th>(CC/WM)<br>max capacity</th><th>Number of<br>(CC/WM)</th><th>Locations of CC</th><th>Locations of WM</th><th>Transportation<br>cost (×10<sup>4</sup>)</th><th>Construction<br>cost (×10<sup>6</sup>)</th><th>Execution<br>time</th></tr></thead><tbody><tr><td>(6000/7000)<br>(7000/7000)</td><td>(18/14)</td><td>Infeasible<br>1, 2, 3, 4, 5, 6, 7, 8, 10, 11<br>12, 13, 14, 15, 17, 18, 19, 20</td><td>1, 2, 3, 4, 5, 6, 7, 8<br>10, 11, 12, 13, 14, 15</td><td>17349</td><td>26794</td><td>6m55.329s</td></tr><tr><td>(7000/8000)</td><td>(18/12)</td><td>1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11<br>12, 13, 14, 15, 17, 19, 20</td><td>1, 2, 3, 5, 6, 7, 8, 10<br>11, 12, 13, 15</td><td>17356</td><td>24894</td><td>8m31.940s</td></tr><tr><td>(8000/10000)</td><td>(16/10)</td><td>1, 2, 3, 5, 6, 7, 8, 10, 11<br>12, 13, 14, 15, 18, 19, 20</td><td>1, 3, 5, 6, 7, 8, 10, 11<br>12, 14</td><td>17343</td><td>22424</td><td>6m48.061s</td></tr><tr><td>(10000/15000)</td><td>(13/7)</td><td>1, 2, 3, 5, 7, 8, 10<br>12, 13, 14, 15, 19, 20</td><td>1, 6, 7, 8, 10, 12, 13</td><td>17366</td><td>18719</td><td>7m22.018s</td></tr><tr><td>(15000/20000)<br>(20000/30000)<br>(30000/50000)</td><td>(9/5)<br>(7/4)<br>(5/2)</td><td>1, 2, 7, 8, 13, 14, 15, 19, 20<br>3, 6, 7, 8, 13, 14, 19<br>2, 8, 13, 15, 19</td><td>1, 5, 10, 12, 14<br>1, 5, 7, 12<br>1, 5</td><td>17419<br>17491<br>17739</td><td>15679<br>14159<br>11689</td><td>14m24.929s<br>7m43.632s<br>6m59.131s</td></tr></tbody></table>
<figure><img src="image_6.png" /><figcaption>FIGURE 5 – Problem 1 : Impact of CC and WM capacity limitation</figcaption></figure>

## Problem 2

<table><caption>TABLE 7 – Problem 2 : Impact of WM capacity limitation</caption><thead><tr><th>WM max capacity</th><th>Number of WM</th><th>Locations of WM</th><th>Transportation cost (×10<sup>4</sup>)</th><th>Construction cost (×10<sup>6</sup>)</th><th>Execution time</th></tr></thead><tbody><tr><td>6000</td><td></td><td>Infeasible</td><td></td><td></td><td></td></tr><tr><td>7000</td><td>14</td><td>1, 2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15</td><td>17248</td><td>18112</td><td>59m50.635s</td></tr><tr><td>10000</td><td>10</td><td>1, 2, 5, 6, 7, 8, 10, 12, 13, 14</td><td>17257</td><td>14312</td><td>7m57.126s</td></tr><tr><td>15000</td><td>7</td><td>1, 5, 6, 7, 10, 12, 14</td><td>17311</td><td>11462</td><td>8m32.606s</td></tr><tr><td>20000</td><td>5</td><td>5, 6, 10, 12, 14</td><td>17386</td><td>9562</td><td>15m13.475s</td></tr><tr><td>25000</td><td>4</td><td>1, 5, 6, 14</td><td>17452</td><td>8612</td><td>9m48.854s</td></tr><tr><td>40000</td><td>3</td><td>1, 5, 14</td><td>17555</td><td>7662</td><td>9m22.124s</td></tr><tr><td>50000</td><td>2</td><td>1, 5</td><td>17731</td><td>6712</td><td>9m27.559s</td></tr><tr><td>Unlimited (100000)</td><td>1</td><td>5</td><td>18107</td><td>5762</td><td>6m30.081s</td></tr></tbody></table>
<figure><img src="image_7.png" /><figcaption>FIGURE 6 – Problem 2 : Impact of WM capacity limitation</figcaption></figure>

### Problem 3

<table><caption>TABLE 8 – Problem 3 : Impact of CC capacity limitation</caption><thead><tr><th>CC max capacity</th><th>Number</th><th>Locations of CC</th><th>Transportation cost (×10<sup>4</sup>)</th><th>Construction cost (×10<sup>5</sup>)</th><th>Execution time</th></tr></thead><tbody><tr><td>6000</td><td rowspan="2">16</td><td rowspan="2">1, 2, 3, 4, 5, 7, 9, 10, 11, 13, 14, 15, 16, 17, 18, 20</td><td rowspan="2">17576</td><td rowspan="2">81326</td><td rowspan="2">3m14.472s</td></tr><tr><td>8000</td></tr><tr><td>10000</td><td>13</td><td>1, 3, 4, 5, 7, 9, 10, 11, 14, 15, 16, 17, 18</td><td>17524</td><td>72776</td><td>3m15.549s</td></tr><tr><td>15000</td><td>9</td><td>1, 3, 4, 7, 10, 11, 14, 15, 18</td><td>17489</td><td>61376</td><td>3m18.528s</td></tr><tr><td>20000</td><td>7</td><td>3, 4, 7, 10, 11, 15, 18</td><td>17457</td><td>55676</td><td>2m46.067s</td></tr><tr><td>25000</td><td>5</td><td>3, 4, 7, 10, 15</td><td>17438</td><td>52826</td><td>3m16.721s</td></tr><tr><td>100000</td><td>2</td><td>3, 7</td><td>17506</td><td>41426</td><td>3m15.508s</td></tr><tr><td>Unlimited (2000000)</td><td>1</td><td>7</td><td>17719</td><td>38576</td><td>3m11.840s</td></tr></tbody></table>
<figure><img src="image_8.png" /><figcaption>FIGURE 7 – Problem 3 : Impact of CC capacity limitation</figcaption></figure>

## 6.3 Results discussion

When the maximum capacity of the hubs (CC and/or WM) is reduced, the goal is to have more hubs distributed throughout the network to facilitate access to products for collectors and/or retailers and reduce the total distance traveled. However, despite the reduction in transportation cost, the total cost of the objective function increases due to the increase in the total cost of building hubs (figures 5, 6 and 7).

When the maximum capacity of the hubs is increased, the number of hubs is reduced (CC and/or WM) (figures 5, 6 and 7). Therefore, the construction cost decreases. At the same time, collectors and/or retailers travel greater distances to procure products. In this case the total distance traveled increases and therefore the total cost of transportation increases with, but the total cost of the objective function decreases due to the decrease of the total cost of constructing hubs.

The goal of this study is to apply the proposed model on a regional scale, in order to
be able to provide local food nearby. The decisions of locating hubs are strategic and heavy of
consequences because it is not possible to modify them over a long time period (several years).
On the other hand, the transportation of goods is daily. So, the investment in the construction
of the hubs (CC/WM) generates costs once over several years and the transportation costs
are generated every day. This leads us to conclude that, with more investment on the total
number of hubs (CC/WM) to be located (once over several years), total daily transportation
costs decrease. The managers (decision makers) should take into consideration this trade-off
while designing the fruits and vegetables distribution network.

7 Conclusion

In this work, the aim is to respond to the problem of sizing and optimization of multi-
commodity distribution networks for perishable products such as fruits and vegetables. Through
a location model (MILP), the study of different problems for structuring distribution networks,
encountered and motivated from real-life issues, allows to suggest strategic solutions for decision-
makers. We have demonstrated how certain parameters affect the final decision and lead to a
series of managerial recommendations for achieving the various objectives set.

Designed to improve the performance and viability of fresh food distribution networks,
the model allows to find the optimal number, locations and capacities of CC&WM, according
to agricultural fields and retail outlets, taking into account the specificity of fresh products
(supply availability, products perishability and main roads). The proposed solutions allow better
availability of products at lower cost, improved responses to customer requests and improved
food safety by reducing waste and contamination generated during distribution. These points
constitute one of the important levers for optimizing the overall performance of food distribution
chains, with a significant impact on the development of the sector at the local level as well as
at the international one.

More generally, food marketing logistics chains are often inefficient, impacting producers
incomes and consumers satisfaction level. The implementation of optimization programmes for
logistic infrastructures is therefore a necessity. It helps to solve both the problem of private
and public income growth, but also to improve the growth trajectories of the supply and its
quality at the global level. The study we proposed illustrates through experimental examples,
how strategic and operational needs of logistics chains can be solved by optimization methods
of operational research.
# Acknowledgements

We would like to thank Prof. M. BAÏOU from the LIMOS laboratory, University of Blaise Pascal - Clermont-Ferrand (France) for his helpful comments as well as for allowing us to use the laboratory resources (Cplex software and computing servers). Our respects and our thanks also go to the sttaf of the computer network of the LIMOS laboratory, in particular the person in charge of the HPC platform, H. TOUSSAIT for their assistance and continuous help.

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