Logistics IRP Model for the Supply Chain of Perishable Food

Abstract

The joint management of routing and inventories has become a field of particular importance for the academic community in recent years. However, there were no models identified in the literature review that contemplate the characteristics of the supply chain of fresh food. The authors of this paper propose an IRP (Inventory Routing Problem), multi-objective, multi-product and multi-echelon model of mixed linear programming which considers the shelf life of the food. The model is applied to the supply chain of perishable fruits. The two scenarios evaluated constitute the base for a strategy proposal to reduce costs, maximize the contribution margin and diminish the losses of fruit.

1 Introduction

The Inventory Routing Problem (IRP) in the Supply Chain of Perishable Food (SCPF), has become relevant for the academic community in recent years [1]. The simultaneous study of routing and inventories, although complex, has a positive impact on the overall performance of the Chain [2, 3]. However, inventory management, distribution and routing, produces high losses and logistics costs [4].

Fruits produced in Colombia, compared to other subtropical countries, have a better quality of their organoleptic characteristics: color, flavor, aroma, higher content of soluble solids and degrees Brix [5]. This paper presents a multi-objective mathematical model of the SCPF applied to 5 fruits: mango, blackberry, strawberry, orange and mandarin from a region in Colombia [6].

The SCPF is made up of six echelons: producers, the supply central, wholesalers, hypermarkets, agro-industrial and retailers, the last echelon in turn made up of marketplaces and shopkeepers. The model allows the establishment of the Distribution Plan and inventories for each fruit as well as: the capacities required in each echelon of the chain, the total cost of the chain, the contribution margin and the impact of losses. The model surpasses previous models for considering the reduction from post-harvest losses in more than two fruits.

The article begins with a literature review, methodology and proposed mathematical model, based on the papers [7,8,9,10] and continues with the case study through two scenarios and the strategies formulated in order to reduce the level of losses in the SCPF.

2 Literature on IRP Models for Perishables

The first article that was identified as IRP for perishables included the allocation and distribution of the product from a regional collection center to a defined set of retailers with random demands [11]. There are several methods of solution when the IRP is NP-hard. Chen et al. [12] applied quality time windows for a perishable product with a limited shelf life in order to control the deterioration. Propose a two-stage heuristic to solve the multi-product version of the IRP with multiple clients. Another two-stage heuristic for a linear programming model was proposed by Mirzaei and Seifi [13].

Another IRP for perishable foods assuming uncertain demands, including seasonality factors, was proposed by Sivakumar [14], studied indicators such as gas emissions from transport vehicles, levels of food waste, transport times and the final quality of the distribution chain.

Amorim et al. [15] also propose an IRP for perishable food. They distinguish between two classes: fresh and mature, based on shelf life. The product is discarded at the maximum age in the inventory. They then study the impact of perishability on the IRP models for a perishable product with a fixed shelf life [1]. In both studies they restrict the storage time in the facilities without considering deterioration. Coelho and Laporte [16] include lifecycle monitoring of a perishable product with a fixed shelf life in an IRP. Jia et al. [17] propose an integrated IRP in which a supplier with a limited production capacity distributes a single item to a set of retailers using homogenous vehicles and considering a fixed deterioration of the food. The assumption of an unlimited shelf life of the product in the IRP models does not allow for the consideration of the decay in food quality, an obstacle for the application of the basic models of the IRP on the supply chain of perishable food (SCFP) [18]. These considerations have brought new objectives such as the ability to control quality and reduce waste in the SCPF as well as environmental and social impacts [2].

Al Shamsi et al. [19] propose a model with two chain echelons. Suppliers send products with a fixed shelf life and storage time to retailers. The model includes CO₂ emissions. Soysal et al. [2] show a multi-period IRP model. It includes the level of service to meet an uncertain demand applied on the distribution of fresh tomatoes to supermarkets. Mirzaei and Seifi [13] consider a two-echelon SC. The shelf life is a linear or exponentially decreasing function, the model considers any unit that is still in inventory at the time of the next delivery as lost. Rahimi et al. [20] present a multi-objective IRP model composed of three parts; one associated to economic cost, one to the level of customer satisfaction and one to environmental aspects. Hiassat et al. [21] propose a location, inventory and routing model for perishable foods. The model determines the number and location of required deposits, they develop a genetic algorithm and a heuristic for local search to solve the problem. The work of Azadeh et al. [22] presents an IRP model with transshipment for a single perishable food item. It is solved with a genetic algorithm, calculating the parameters using a Taguchi design.

Future Work in IRP should include the characteristics of SCPF [23], which could include stochastic and dynamic conditions. The solution models and methods should take into account factors such as cold chain, hygiene standards, air pollution, greenhouse gas emissions, waste generation, road occupation and other aspects related to City Logistics [24] and Green Logistics [25].

3 IRP Mathematical Models for Perishable Foods

In this section the IRP mathematical model for perishable foods is formulated.

3.1 Sets

• $\text{V}_{\text{a}}$ = Producers, where a = {1, 2, …, A}

• $\text{V}_{\text{b}}$ = Wholesalers, where b = {1, 2, …, B}

• $\text{V}_{0}$ = Collection center

• $\text{V}_{\text{c}}$ = Hypermarkets, where c = {1, 2, …, C}

• $\text{V}_{\text{d}}$ = Agroindustry, where d = {1, 2, …, D}

• $\text{V}_{\text{e}}$ = Retailers markets, where e = {1, 2, …, E}

• $\text{V}_{\text{f}}$ = Retailers shopkeepers, where f = {1, 2, …, F}

• V = All nodes, where ${\text{V}} = \text{V}_{\text{a}} {\bigcup V}_{\text{b}} {\bigcup V}_{\text{c}} {\bigcup V}_{\text{d}} {\bigcup V}_{\text{e}} {\bigcup V}_{\text{f}} {\bigcup V}_{0}$

• $\text{V}^{{\prime }}$ = All nodes without producers $\text{V}^{{\prime }} = \text{V}_{\text{b}} {\bigcup V}_{\text{c}} {\bigcup V}_{\text{d}} {\bigcup V}_{\text{e}} {\bigcup V}_{\text{f}} {\bigcup V}_{0}$

• $V_{dem}$ = delivery nodes $V_{dem} = \text{V}_{\text{c}} {\bigcup V}_{\text{d}} {\bigcup V}_{\text{e}} {\bigcup V}_{\text{f}}$

• $\text{V}_{int}$ = intermediate delivery and reception nodes $\text{V}_{int} = \text{V}_{\text{b}} {\bigcup } {\text{V}}_{ 0}$

• $A = All arcs {\text{A}}\text{ = }\left\{ {\left( {{\text{i}},{\text{j}}} \right):{\text{i}},{\text{j}} \in {\text{V}},{\text{i}} \ne } \right. \left. {\text{j}} \right\}$

• T = Time periods in weeks, t = {1, 2, …, T}

• P = Food products, p = {1, 2, …, P}

• K = Vehicles, k = {1, 2, …, K}

3.2 Parameters

• ${\mathbf{d}}_{{{\mathbf{p}} ,{\mathbf{t}}}}^{{\mathbf{i}}}$ = Demand by node ${\text{i}} \in {\text{V}}^{{\prime }}$ of food type p, in the time period t, in kg.

• ${\mathbf{m}}_{{\mathbf{p}}}$ = Maximum shelf life of the type of food p, in days.

• $\varvec{ca}_{\varvec{k}}$ = Capacity of a vehicle in entities of 20 kg.

• ${\mathbf{a}}_{{{\mathbf{i}},{\mathbf{j}}}}$ = Distance between the node i.e. j, $\left( {i,j} \right) \in A$, in km.

• ${\mathbf{r}}_{{\mathbf{p}}}$ = Penaltycost for lost food type p, in $/kg.

• ${\mathbf{Co}}_{{{\mathbf{i}},{\mathbf{jk}}}}$ = Variable Transport cost per ${\text{kg*kl}}$, for arc $( {\text{i, j)}} \in \,{\text{A}}$, the type of truck k, in $\$ / {\text{kg*kl}}$.

• ${\mathbf{Cf}}_{{{\mathbf{i}},{\mathbf{jk}}}}$ = fixed Transport cost per kilometer, for arc $( {\text{i, j)}} \in \,{\text{A}}$ the type of truck k, in $/kl.

• ${\mathbf{h}}_{{{\mathbf{i}},{\mathbf{p}}}}$ = Cost of maintaining inventory in the node type ${\text{i}} \in {\text{V}}^{{\prime }}$, of food type p.

• ${\mathbf{mc}}_{{{\mathbf{i}},{\mathbf{p}}}}$ = Contribution margin in the node type ${\text{i}} \in V$, for the food type p, in $/kg.

3.3 Decision Variables

• ${\mathbf{I}}_{{{\mathbf{p}} ,{\mathbf{t}}}}^{{\mathbf{i}}}$ = Quantity of inventory in node type ${\text{i}} \in {\text{V}}^{{\prime }}$, of food type p, at the end of the period ${\text{t}} \in T$. Where ${\text{I}}_{\text{p,0}}^{\text{i}} = 0, \forall {\text{i}} \in {\text{V}}_{\text{e}} \cup {\text{V}}_{\text{f}} \, \forall {\text{p}} \in {\text{P}}.$

• ${\mathbf{B}}_{{{\mathbf{p}} ,{\mathbf{t}}}}^{{{\mathbf{i}} ,{\mathbf{k}}}}$ = Quantity picked up in node type ${\text{i}} \in {\text{V}}^{{\prime }}$, of food ${\text{p}} \in {\text{P}}$, by vehicle ${\text{k }} \in {\text{K}}$, at the beginning of period t, in kg.

• ${\mathbf{Q}}_{{{\mathbf{p}} ,{\mathbf{t}}}}^{{{\mathbf{i}} ,{\mathbf{k}}}}$ = Quantity delivered at node type ${\text{i }} \in {\text{V}}^{{\prime }}$, of food ${\text{p}} \in {\text{P}}$, by vehicle ${\text{k}} \in {\text{K}}$, at the beginning the period t, in kg.

• ${\mathbf{X}}_{{{\mathbf{k}} ,{\mathbf{t}}}}^{{{\mathbf{i}},{\mathbf{j}}}} { = }\left\{ {\begin{array}{*{20}l} {\begin{array}{*{20}c} {1, {\text{if}}\,{\text{the}}\,{\text{flow}}\,{\text{is}}\,{\text{from}}\,{\text{origin}}\,{\text{to}}\,{\text{destination}}\,{\text{i}} \in V\, to\, j \in V, } \\ {{\text{in}}\,{\text{vehicle}}\,{\text{k}} \in {\text{K}}, \,{\text{in}}\,{\text{period}}\,{\text{t}} \in {\text{T}} } \\ \end{array} } \hfill \\ {0,\;{\text{otherwise}}.} \hfill \\ \end{array} } \right.$

• ${\text{F}}_{{{\mathbf{k}} ,{\mathbf{p}} ,{\mathbf{t}}}}^{{{\mathbf{i}} ,{\mathbf{j}}}}$ = Flow between node i and node j, of food p, by vehicle k y, in the period t i, in kg.

• ${\mathbf{W}}_{{{\mathbf{p}} ,{\mathbf{t}}}}^{i}$ = Quantity of loss in node ${\text{i}} \in {\text{V}}^{{\prime }}$, of food ${\text{p}} \in$, in the time period ${\text{t}} \in {\text{T}}$, in kg.

• ${\mathbf{U}}_{{{\mathbf{k}} ,{\mathbf{t}}}}^{{\mathbf{i}}}$ = position in the node path io ${{\text{V}} \mathord{\left/ {\vphantom {{\text{V}} {\left\{ 0 \right\}}}} \right. \kern-0pt} {\left\{ 0 \right\}}}$, in vehicle k ve, in the period ${\text{t}} \in {\text{T}}$.

3.4 Objectives

The mathematical model includes two objectives. The first objective function (1) is composed of inventory, routing, and costs from losses.

$$ \begin{aligned} {{\texttt{Min }}\,{\texttt{F}}}_{ 1} & = \mathop \sum \limits_{{{\text{i}} \in {\text{V}}^{\prime} }} \mathop \sum \limits_{{{\text{p}} \in {\text{P}}}} \mathop \sum \limits_{{{\text{t}} \in {\text{T}}}} {\text{I}}_{{{\text{p}}, {\rm{t}}}}^{\text{i}} {\rm{h}}_{{{\text{i}}, {\rm{p}}}} \\ & \quad + \mathop \sum \limits_{{\left( {{\text{i}}, {\rm{j}}} \right) \in {\text{A}}}} \mathop \sum \limits_{{{\text{k}} \in {\text{K}}}} \mathop \sum \limits_{{{\text{t}} \in {\text{T}}}} \left[ {X_{k,t}^{i,j} Cf_{i,j} a_{ij} + \mathop \sum \limits_{p \in P} Co_{i,j} a_{ij} F_{k,p,t}^{i,j} } \right] + \mathop \sum \limits_{{{\text{i}} \in {\text{V}}^{\prime} }} \mathop \sum \limits_{{{\text{p}} \in {\text{P}}}} \mathop \sum \limits_{{{\text{t}} \in {\text{T }}}} {\text{W}}_{{{\text{p}},{\text{t}}}}^{\text{i}} {\rm{r}}_{\text{p}} . \\ \end{aligned} $$

(1)

The second objective function (2) maximizes the contribution margin of the chain. It differentiates between the quantity delivered and the food losses per echelon, multiplied by the unitary contribution margin in pesos per kilogram (subtraction between the selling price and the cost). This objective maximization function has not been found in the revised literature, it was designed based on the considerations of cost from fruit losses.

$$ {\text{Max F}}_{ 2} = \mathop \sum \limits_{{{\text{i}} \in {\text{V}}^{{\prime }} }} \mathop \sum \limits_{{{\text{p}} \in {\text{P}}}} \mathop \sum \limits_{{{\text{t}} \in {\text{T}}}} \left( {{\text{d}}_{\text{p,t}}^{\text{i}} - {\text{W}}_{\text{p,t}}^{\text{i}} } \right){\text{mc}}_{\text{i, p}} $$

(2)

3.5 Constraints

The constraint number (3) and (4) allows to calculate the inventory levels for each node per period, based on the total cargo delivered, picked and the expected demand, losses are added to the proposal by Soysal et al. [26]. It is assumed that the inventory at the beginning of the planning period is zero.

$$ {\text{I}}_{\text{p,t}}^{\text{i}} = {\text{I}}_{{{\text{p,t}} - 1}}^{\text{i}} + \mathop \sum \limits_{k \in K} {\text{Q}}_{\text{p,t}}^{\text{i,k}} - {\text{d}}_{\text{p,t}}^{\text{i}} - {\text{W}}_{\text{p,t}}^{i} \;\,\forall {\text{i}} \in V_{dem} ,\,{\text{p}} \in {\text{P}},\,{\text{t}} \in {\text{T}} . $$

(3)

$$ {\text{I}}_{\text{p,t}}^{\text{i}} = {\text{I}}_{{{\text{p,t}} - 1}}^{\text{i}} + \mathop \sum \limits_{k \in K} {\text{Q}}_{\text{p,t}}^{\text{i,k}} - \mathop \sum \limits_{k \in K} {\mathbf{B}}_{\text{p,t}}^{\text{i,k}} - {\text{d}}_{\text{p,t}}^{\text{i}} - {\text{W}}_{\text{p,t}}^{i} \forall {\text{i}} \in V_{int} ,\,{\text{p}} \in {\text{P}},\,{\text{t}} \in {\text{T}} . $$

(4)

The constraint number (5) and (6) establishes that the amount of food losses will depend on the inventory of the period, discounting the days of the maximum useful life of the product, also considering the demand and losses of the previous period.

$$ {\text{I}}_{\text{p,t}}^{\text{i}} - \mathop \sum \limits_{a = t}^{{t + m_{p} }} {\text{d}}_{\text{p,a}}^{\text{i}} \le \mathop \sum \limits_{a = t}^{{t + m_{p} }} {\text{W}}_{\text{p,a}}^{i} \forall {\text{i}} \in V_{dem} ,\,{\text{p}} \in {\text{P,}}\,{\text{t}} \in {\text{T }}|\left\{ {t \le \left| T \right| - m_{p} } \right\} $$

(5)

$$ \mathop \sum \limits_{{{\text{i}} \in (V_{int} )}} I_{\text{p,t}}^{\text{i}} - \mathop \sum \limits_{{j \in {\text{V}}^{{\prime }} }} \mathop \sum \limits_{a = t}^{{t + m_{p} }} {\text{d}}_{\text{p,a}}^{\text{j}} \le \mathop \sum \limits_{{{\text{i}} \in (V_{int} )}} \mathop \sum \limits_{a = t}^{{t + m_{p} }} {\text{W}}_{\text{p,a}}^{i} \forall {\text{p}} \in {\text{P,}}\,{\text{t}} \in {\text{T }}|\left\{ {t \le \left| T \right| - m_{p} } \right\} $$

(6)

The constraint number (7) and (8) suggested by [27, 28], guarantees that each vehicle can cover one route at most for a given period of time, and guarantee the continuity of the route.

$$ \mathop \sum \limits_{{{\text{i}} \in {\text{V, i}} \ne {\text{j}}}} {\text{X}}_{\text{k,t}}^{{{\text{i}},{\text{j}}}} = \mathop \sum \limits_{{{\text{i}} \in {\text{V, i}} \ne {\text{j}}}} {\text{X}}_{\text{k,t}}^{{{\text{j}},{\text{i}}}} ,\forall {\text{j}} \in {\text{V}}\backslash \left\{ 0 \right\},\,{\text{k}} \in {\text{K}},\;{\text{t}} \in {\text{T}} . $$

(7)

$$ \mathop \sum \limits_{{{\text{j}} \in {\text{V, i}} \ne {\text{j}}}} {\text{X}}_{\text{k,t}}^{{{\text{i}},{\text{j}}}} \le \text{ } 1,\forall {\text{i}} \in {\text{V}},\;{\text{k}} \in {\text{K}},\;{\text{t}} \in {\text{T}} . $$

(8)

The constraint number (9) and (10) restrict the direct flows from producer to retailers and vice versa

$$ {\text{X}}_{\text{k,t}}^{{{\text{i}},{\text{j}}}} = 0 ,\forall {\text{i}} \in (V_{a} ) ,\;{\text{j}} \in (V_{e} { \cup }V_{j} ) ,\;{\text{k}} \in {\text{K,}}\;{\text{t}} \in {\text{T}} . $$

(9)

$$ {\text{X}}_{\text{k,t}}^{{{\text{j}},{\text{i}}}} = 0 ,\forall {\text{i}} \in (V_{a} ) ,\;{\text{j}} \in (V_{e} { \cup }V_{j} ) ,\;{\text{k}} \in {\text{K,}}\;{\text{t}} \in {\text{T}} . $$

(10)

The constraint number (11) they guarantee the behavior of the flows and their relationship with deliveries and collections.

$$ \begin{aligned} \mathop \sum \limits_{{j \in {\text{V, i}} \ne {\text{j}}}} F_{k,p,t}^{j,i} & - \mathop \sum \limits_{{j \in {\text{V, i}} \ne {\text{j}}}} F_{k,p,t}^{i,j} = {\text{Q}}_{\text{p,t}}^{\text{i,k}} \\ & - {\text{B}}_{\text{p,t}}^{\text{i,k}} \forall {\text{i }} \in {\text{V}}^{{\prime }} ,\,{\text{k}} \in {\text{K,}}\,{\text{p}} \in {\text{P,}}\,{\text{t}} \in {\text{T}} . \\ \end{aligned} $$

(11)

The restriction number (12) guarantees that only flows are given in the arcs that are chosen, and that the flow does not exceed the capacity of the vehicle

$$ {\text{F}}_{\text{k,p,t}}^{\text{i,j}} \le ca_{k} {\text{X}}_{\text{k,t}}^{{{\text{i}},{\text{j}}}} \;\; \forall \left( {i,\,j} \right) \in A,\,{\text{k}} \in {\text{K}},\,{\text{p}} \in {\text{P}},\,{\text{t}} \in {\text{T}} . $$

(12)

The constraint number (13) ensure that vehicles cannot pick up product from a node that does not have that product.

$$ \mathop \sum \limits_{k \in K} {\mathbf{B}}_{\text{p,t}}^{\text{i,k}} \le {\text{I}}_{{{\text{p,t}} - 1}}^{\text{i}} + \mathop \sum \limits_{k \in K} {\text{Q}}_{\text{p,t}}^{\text{i,k}} - {\text{d}}_{\text{p,t}}^{\text{i}} - {\text{W}}_{\text{p,t}}^{i} - {\text{I}}_{\text{p,t}}^{\text{i}} \,\forall {\text{i}} \in V_{int} ,\,{\text{p}} \in {\text{P}},\,{\text{t}} \in {\text{T}} . $$

(13)

The constraint number (14) guarantees the elimination of subtours. It relates the position variable of the node in the route, with respect to the binary variable, which indicates whether the vehicle travels the arc iteratively to complete all the origin destination.

$$ {\text{U}}_{\text{k,t}}^{\text{i}} + 1 \le {\text{U}}_{\text{k,t}}^{\text{j}} + |V|\left( {1 - {\text{X}}_{\text{k,t}}^{{{\text{i}},{\text{j}}}} } \right),\quad \forall \left( {i,j} \right) \in A\left( {V\backslash \left\{ 0 \right\}} \right) ,\,{\text{k}} \in {\text{K}},\,{\text{t}} \in {\text{T}} . $$

(14)

The constraints numbers (15) to (20) are associated with the non-negativity and the conditions of the decision variables.

$$ {\text{X}}_{\text{k,t}}^{{{\text{i}},{\text{j}}}} , \in \left\{ { 0 , 1} \right\} ,\forall \left( {{\text{i,}}\,{\text{j}}} \right) \in {\text{A,}}\,{\text{k}} \in {\text{K,}}\,{\text{t}} \in {\text{T}}. $$

(15)

$$ F_{k,p,t}^{i,j} \ge 0,\forall \left( {{\text{i}},{\text{j}}} \right) \in {\text{A}},{\text{k}} \in {\text{K}},\,{\text{p}} \in {\text{P}},\,{\text{t}} \in T. $$

(16)

$$ {\text{I}}_{\text{p,t}}^{\text{i}} \ge 0 ,\forall {\text{i}} \in {\text{V}}^{{\prime }} ,\,{\text{p}} \in {\text{P,}}\;{\text{t}} \in {\text{T}} . $$

(17)

$$ {\text{W}}_{\text{p,t}}^{i} \ge 0 ,\quad \forall {\text{i}} \in {\text{V}}^{{\prime }} ,{\text{p}} \in {\text{P}},{\text{t}} \in {\text{T}}. $$

(18)

$$ {\text{U}}_{\text{k,t}}^{\text{i}} \ge 0 ,\forall {\text{i}} \in V\backslash \left\{ 0 \right\},\;{\text{k}} \in {\text{K,}}\;{\text{t}} \in {\text{T}} . $$

(19)

$$ {\text{Q}}_{\text{p,t}}^{\text{i,k}} ,\;{\text{B}}_{\text{p,t}}^{\text{i,k}} \ge 0 ,\quad \forall {\text{i}} \in {\text{V}}^{{\prime }} ,\;{\text{k}} \in {\text{K,}}\;{\text{p}} \in ,\,{\text{t}} \in {\text{T}} . $$

(20)

4 Supply Chain of Perishable Fruit Products, Case Study

The multi-echelon and multi-objective model of mixed linear programming is applied to the SC of perishable fruit, having as reference the models [2, 13, 26]. In Fig. 1 a complete directed graph is presented $G = \left( {N,E} \right),$ where N represents the source nodes and E the target nodes.

Fig. 1.

Echelons of the fruit SC identified for the model.

4.1 Input Information

Based on 74 surveys, filled by the agents of the chain and complemented with secondary information [29], the supply was determined for each type of fruit in the following 10 municipalities selected from the department of Cundinamarca - Colombia: La Palma, Topaipí, Tena, Arbeláez, Anolaima, La Mesa, Villapinzón, Suesca, Cogua, Ubalá. Demand was calculated per node for a 4-week planning time frame, in Bogotá D.C, Capital of Colombia with 10 million inhabitants. Distances were determined from the source nodes to the destination nodes. Inventory maintenance costs were determined based on the survey; they are equivalent to 15% of the average selling price of the fruit.

4.2 Assumptions of the Model

Several fruits are offered by the Supply Center, each type with a fixed shelf life. Fruits can be sent to any node of the graph except to producers. Two scenarios were considered. The first one is the direct application of the IRP model to the current fruit SC. In the second scenario the Supply Central is proposed as the node that centralizes the reception of food from the producers and dispatch it to the other nodes. In the literature review it was determined that for IRP models of perishable foods this is a convenient arrangement [30]. The quantity of fruit available at the level of the wholesaler is limited by its shelf life in the time-horizon planning. Each vehicle covers at most one route per day. Each retailer can be serviced by more than one vehicle as the total cargo assigned to each can be divided into two or more vehicles. The demand for fruit is known through the time-horizon planning. For each retailer there is a cost to keep inventory in each period. However, if the fruit is kept in inventory more than indicated, it is calculated as food loss. The first operation that takes place in a day is the delivery, then the consumption and finally, the inventory level is recalculated.

4.3 Results

The proposed multi-objective mathematical model was applied to SC of perishable fruit. It allows calculating the use of transport capacities, costs, analysis of losses as well as routing aspects. The fruit IRP model was programmed in the GAMS (General Algebraic Modeling System).

Analysis of Total Cost and Contribution Margin.

To solve the proposed algorithms, the Epsilon Restriction (ε-constraint) method was used. For this purpose, an objective function that minimizes cost was chosen, and the objective function 2 was included as a constraint. By applying the mathematical model, minimizing the total cost and maximizing the contribution margin for scenarios 1 and 2, the result obtained was that scenario 1 exceeds the contribution margin of scenario 2 by 9.4%. With respect to the total cost scenario 2 had a 4.19% lower cost than scenario 1. As for the total losses, it is clear that in scenario 2 they are 8.24% lower than scenario 1. In Fig. 2 the minimum cost for each level of contribution margin is shown for each scenario.

Fig. 2.

Pareto curve comparison, scenario 1 and 2, between (Total cost) and (Contribution margin).

Inventory Management.

A final average inventory was obtained based on demand, storage capacity and type of fruit. As shown in Fig. 3, scenario 2 has an inventory of 43, 24% less than scenario 1.

Fig. 3.

Level of final average inventory (quantity in kg) per (fruit) for scenario 1 and 2.

Analysis of Losses.

Regarding the fruit it is clear that mango produces the greatest amount of loss for both scenarios. However there is a reduction of 59.12% in scenario 2 for this fruit. But the most significant reduction of losses is for mandarin with 95.64% less losses in scenario 2 as shown in Fig. 4.

Fig. 4.

Level of (total loss per fruit) consolidated from all the nodes (quantity in kg) for both scenarios.

In percentage terms, it is clear that the largest loss in both scenarios takes place at the Central Supply node. The largest difference is in the node of the hypermarket Olímpica. While in scenario 1 the loss is 0.53% it decreases to 0.03% in scenario 2. At the retail level, there is a similar amount of loss in both scenarios as can be seen in Fig. 5.

Fig. 5.

(Percentage of losses per node), (losses in %). Where (CA) is the Central supply, (SS) is Surtifruver in la Sabana, (PC) is the Placita Campesina, (OL) is Olímpica, (EX) is warehouses Éxito, (FR) is Agroindustria Frutisima, (SG) is Agroindustria San Gregorio, (P1) is market 1, (P2) is market 2, and (T1) to (T12) the respective 12 shopkeepers.

Figure 6 shows the percentage of loss for the five fruits and the two scenarios with respect to the quantity demanded. The fruit with the lowest loss is orange and the greatest is blackberry. Although a similar behavior can be observed for each fruit scenario 1 is only better for orange.

Fig. 6.

Percentage of (fruit loss with respect to demand), for each (fruit) scenario 1 and 2.

4.4 Formulation of Strategies

The centralized scenario with the Supply Center reduces the total cost by 4.19% compared with the first scenario. The distribution and inventory plans obtained suggest that it is better to centralize collection and distribution operations [20]. However, this finding can be expanded considering that the city can have such nodes at strategic locations and not exclusively in Supply Centers [21]. The results show that the cost of routing is the most representative as it exceeds 70% of the total costs in both scenarios. Inventory costs are above 20% and the costs of losses do not exceed 2%. Therefore, if the current routing conditions are improved, the total cost may be significantly reduced.

With respect to the losses and according to the results obtained, it can be observed that they increase as a function of the number of nodes and distribution routes at the end of the supply chain, the retailers. Given the central analysis that determines an IRP model and considering the results obtained, the proposed strategies for the fruit chain in Bogotá D.C is as follows:

Strategy: Use a Heterogeneous Fleet of Vehicles.

As the mathematical model shows, the use of vehicle capacity is fundamental for the reduction of costs and losses. Currently the transport in Bogotá DC is carried out in vehicles that are not suitable for cargo and not even transporting fruits exclusively. This increases the risks for mechanical damages and damages from handling the cargo. A heterogeneous fleet, as shown in Table 1, will improve deliveries, as well as the capacity available for the system, which, in conjunction with the inventory plan, would reduce the expected costs.

Table 1. Types of vehicles proposed.

The use of vehicles of different capacity as proposed would allow a reduction in routing costs by 10.3%, and the contribution margin could increase by 4.6%, with a minimum impact on the level of losses.

Strategy: Inventory Control for the Fruit.

According to the study by Sánchez [30] there is no control of the costs of the logistic process for perishable foods in Colombia. This affects the analysis of costs for each fruit.

As for cost discrimination, the second scenario is better, as shown in Table 2. The total cost is higher in scenario 1 given that it does not consider a collection center for fruits. Therefore, the sensitivity analysis of the model shows that the routing costs are inversely proportional to the inventory costs for the fruits analyzed. The strategy proposed is to implement systems that prolong the shelf life of the food in order to increase the quantities that can be delivered.

Table 2. Comparison of scenarios in percentage of total cost.

Strategy: Define a Distribution Plan and Inventory of Fruits According to Demand.

The study presented by [31] shows that most of the production of perishables is not made with a technical analysis of the demand, generating an imbalance between supply and demand. In areas like Bogotá even street vendors hence observe the supply, the importance of applying logistical knowledge to agriculture. The second scenario has the lowest cost and the best margin, with a total loss of 8.20% fruit, derived from transportation (30%), storage (25%), poor demand planning and unsold fruit (15%).

5 Conclusions

The proposed multi-objective, multi-product and multi-echelon IRP mathematical model contributes to the literature of this type of problem, bringing it closer to the real behavior and complexity of the SCPF. The reviewed models have a cost approach, while the presented model allows the analysis and reduction of losses given the inclusion of the shelf life in the model. The objective function relates the losses to the margin of contribution in the studied echelons.

The IRP model validated in the SC of perishable fruits, allows the planning of inventories and routing simultaneously and thus the reduction of post-harvest losses. The experimentation with the mathematical model and analysis of results allowed an evaluation of two scenarios and a proposition of strategies for SC in Cundinamarca - Bogotá D.C, reducing costs, maximizing contribution margin and reducing losses. The analysis of the model determined the minimum cost for the routing and inventory of each fruit and the losses of each agent of the chain, the margin of contribution as well as the losses with respect to demand.

6 Future Work

The multi-objective, multi-echelon, multiproduct mathematical model proposed can be extended to a model that is integrated in Inventory Routing Problem with Production Decisions, in order to establish the most appropriate time and place for sowing fruits. Another model to develop may be Location Inventory Routing, which takes into account the location of collection centers in rural municipalities and distribution centers for urban logistics in the mega-city.

A future model could include heterogeneous vehicle fleets, assessing vehicle speed, fuel consumption and emissions of CO₂ as well as sustainable management aspects for SCPFs. Given the perishability of food over time, the development of a dynamic model it required, with stochastic variables for a probabilistic analysis.