The design of distribution networks of agricultural products implies the analysis of different conditions related to the own nature of the products. In this sense, the operations research (OR) has contributed through the application of mathematical techniques that allow optimizing resources in agriculture [9].

Through the OR hybrid approaches have been suggested based on simulation, whose objective has been to integrate uncertainty within the optimization model [10], with the advantage of providing, with this models, a better understanding of a supplying chain analyzing an optimal solution under stochastic conditions [11,12].

Hybrid models are also an alternative to reduce complexity in a problem with a lot of variables, generally not linear variables [3,12].

Authors like [13] suggest a model to design a grain transportation network considering the reduction of post-harvest losses, applying a series of sensitivity analyses under different scenarios in a distribution network in the United States.

[14] suggest a MILP to evaluate the feasibility of opening new grain processing plants with the purpose of increasing the production capacities in Turkey, this model looks for determining the location of new plants while minimizing fixed costs of installation and costs of transportation.

[7] identify several strategies to enhance the distribution cost through the implementation of a hybrid model of optimization and simulation in a supplying chain of wheat in Canada. In Holland, [1] assumed uncertainty in supplying, transportation and processing operations evaluating a hybrid model of optimization and simulation to design a network in flower sector.

Other approaches include suggestions like the one of [12] which combine optimization and stochastic environments with the objective of reducing the costs of production, transportation and inventory in a supplying chain integrated by different suppliers, producers and distributors who are connected through different means of transportation. [8] present a model based on optimization and simulation techniques to estimate an optimum global solution to combinatorial problems under uncertainty conditions.

According to [15] harvest risk constitutes one of the main factors in the production environment of agriculture, in this sense, the effects on harvest yield and production risks are suggested by [16] within a model of simulation and optimization that considers different scenarios to plan production strategies.

[17] evaluate different potential locations to install malt processing enterprises and later propose a MILP model to distribute beer and barley.

In sum, the operational aspects in production and distribution costs, and the uncertainty regarding harvest and quality are considered to suggest models that support decision making to design a network. However, there were no research works that take into account, from a CF point of view, the interaction between the buyers’ capacities and production uncertainty. Therefore, our approach includes an optimal solution considering stochastic variables in harvest yields, loss risk and penalties/incentives under different scenarios in CF.

The suggested mixed-integer linear programming model (MILP) was formulated considering the reduction of distribution cost of the harvest based on the malt producers’ demand. The results of this model show in a deterministic way the best allocation between farmers -hubs, farmers-malt producers and hubs -malt producers indicating the tons of grains that should be sent in each one of these nodes.

Indices and sets:

i ∈ I: set of farmers

j ∈ J: set of hubs

k∈ K: set of malt producers

n∈ N: set of candidate solution

Decision variable:

Xij: tonnes to transport from farmer i to the hub j.

X ik: tonnes to transport from the farmer i to the malt producer k.

X jk: tonnes to transport from the hub j to the malt producer k.

Y j : hub j is open or not.

Zn: binary variable for incorporating the cost of uncertainty for candidate solution n.

Parameters:

cij :freight cost (in dollars) from the farmer i to the hub j.

cik: freight cost (in dollars) from the farmer i to the malt producer k.

cjk: freight cost (in dollars) from the hub j to the malt producer k.

sumi: available offer (tonnes of harvest) from the farmer i.

capj: storage capacity (demand in tonnes) of the hub j.

demk:demand (in tonnes) of the malt producer k.

capmk: capacity to receive directly from the malt producer k.

fj : opening cost of the hub j.

Sn: uncertainty found by simulation for candidate solution n.

Tnj: binary parameter (1, if binary variable Yj in candidate solution n is 1; 0, otherwise).

Qmin: mínimum total expected cost with uncertainty obtained so far from simulations.

M: large number.

Minimize

Subject to

The objective function (1) is to minimize the transportation costs from the farmers to the hubs, from the hubs to the malt producers, from the farmers to the malt producers, the opening costs of hubs and the impact on uncertainty obtained in the simulation of the candidate solution n. A candidate solution is defined as an optimal deterministic scenario previously obtained under certain parameters. Equations (2) and (4) are restrictions indicating compliance in the demand and supply, restriction (3) represents a balance between ins and outs of the hub to avoid accumulating stock at the end of the planning period, this means, the amount of tons of barley sent from the farmer to the hub must be equal to the amount taken from the hub to the malt producer. Restriction (5) shows that the amount sent from the different farmers to the hubs must not be higher than the available capacity of storage; restriction (6) based on the CF points out that the amount sent from the farmers to the malt producers must be lower than the capacity of receiving, meaning that the malt producers only allow receiving directly (without going through a hub) certain amount of grain tons. Restriction (7) indicates that the amount sent to malt producer through hubs must be lower or equal to their available capacity. Restrictions (8) and (9) are similar to the ones suggested by ^[8] to include uncertainty of the candidate solution. Restriction (10) is included in the model to establish a ceiling limit obtained in the simulation of the previous candidate solution; finally restrictions (11) - (15) define the non-negative and binary variables.

Based on the frame of reference suggested by authors like ^[18] the simulation has as objective to evaluate under uncertainty conditions the feasibility of the solution suggested by the MILP model, stochastically integrating possible variations in the levels of supply, storage capacity and demand.

The harvest yield, loss risk, penalties/incentives, as well as hubs’ capacities and malt producers’ demands were determined as stochastic entries of the simulation model, while locations of hubs and malt producers, transportation costs, opening costs of hubs and production capacities of farmers are considered as deterministic entries.

For this simulation, data from historical harvest periods since 2003 to 2015 were collected by using information from ^20] which were analyzed through distribution adjustments with the software ExpertFit selecting the top ranked distributions. Harvest yields for each farmer were adjusted to Beta, Weibull and Pearson type VI distributions depending on the historical data (2004-2016) of each one of them, in the case of loss risks, Beta, Weibull and Johnson distributions were obtained as best adjustments.

To determine penalties/incentives that originate in the grain delivery, data were adjusted to a uniform distribution based on the quality levels suggested in the standard NMX-FF-043-SCFI-2003 ^4]. Demands in malt producers through adjustments of distribution were generated randomly between 1,000 and 13,000 tons of grain based on a uniform distribution. The capacities of hubs according to the adjustments of obtained historical data are considered normally distributed with variations in the standard deviation of 15%, 25% and 35% compared to the average, in the case of production capacities they are considered within a range from 20 to 80 available hectares for cultivation per farmer.

The costs of transportation were calculated based on the distance between the farmer, the hubs and the malt producer. The opening cost of the hubs was determined between a range of 3,000 and 4,500 dollars.

In this simulation it is also assumed that:

Barley production (expressed in tons) obtained with a certain yield is considered an entity which is totally consumed by malt producers, being them the only clients.

Fig. 2 shows the layout used for this simulation, in which 243 farmers generate certain levels of tons of barley (entities) based on the suggested distributions and then, according to the MILP model, it is sent to one of the 9 available hubs which will send these entities to the malt producers.

Figure 2 Source: The authors.

The way out of this model are the total transportation and opening costs between the different nodes, as well as the distribution cost for all the suggested supplying chain considering uncertainty in harvest yields, capacities, penalties/incentives and loss risk.

The hybrid approach in this proposal uses the deterministic solution of a MILP model to later, through simulation, estimate the uncertainty impact (S _n ) in the objective function, the difference between the cost in the MILP model and the simulation is incorporated again to formulate the optimization with this parameter. This process is performed in an iterative way until finding the network design that minimizes the total cost of distribution and the impact in the uncertainty under the particular conditions of this supplying chain.

The steps for the solution of this approach are shown in Fig. 3, this procedure starts by obtaining a possible design for the distribution network with the MILP model that indicates the hubs (Y _J) that must be opened and the flows in the amount to be sent ( x _ij , x _ik, x _jk ), then, the network is evaluated through a simulation to calculate the impact of uncertainty(S _n ) also obtaining a value Q _min updating if necessary, the values Z _n and T _nj that are incorporated to evaluate a possible solution in particular. The obtained information from the previous steps serves as feedback on the optimization model and the network design, the procedure continue to evaluate other candidate solutions until finding the one with the best design under conditions of uncertainty.

Figure 3 Source: The authors.

This model was solved through coding in Matlab R2015a, Microsoft Excel 2016 and the simulator ProModel v4. First, the MILP model is executed in Matlab using the intlinprog algorithm which applies three methods (Branch-and-Bound, Cutting Plane and Heuristics) to find a solution and later, through a Microsoft Excel interface, the obtained results are included as entries to the simulation model in ProModel.

Each one of the random problems formulated within the different scenarios were solved through a MILP model with the purpose of finding the best combinations to design the network, subsequently, the deterministic solutions were simulated under conditions of uncertainty considering ten replicas suggested by ^[18] to enhance the precision of the simulation results. Afterwards, the hybrid model determined the best solution under conditions of uncertainty minimizing the difference between the deterministic model and the stochastic model.

For example, in the case of the scenario two where the variation of a malt producer capacity is between 41% and 70% and there is a variation of 25% in the capacity of hubs, the results indicate four possible arrangements to design the network considering the opening of four to five hubs, these arrangements were analyzed in an iterative way under conditions of uncertainty resulting as optimal solution a design considering just four hubs with a total cost of 259,759 USD. Fig. 5 shows the procedure, rejecting in this case those solutions above Q _min in each iteration.

Figure 5 Source: The authors.

The suggested solution in this scenario indicates a network design like the one showed in Fig. 6.

Fig. 7 shows the obtained results for the set of scenarios, initially for a level one (assignment of grain 10% - 40%) with a variation in the capacity of hubs of 15% the best cost was 272,723 USD considering the opening of five hubs, with variations of 25% and 35% were obtained for this level with a minimal cost of 259,759 and 268,261 USD respectively suggesting the opening of four hubs in both cases.

Figure 6 Source:The authors.

In the level two (assignment of grain 41% - 70%) the cost was 260,821; 265,436 and 262,238 USD for variations of 15%, 25% and 35% respectively considering the opening of four hubs in all cases. In the last level (assignment of grain 71 % - 90%) the obtained costs were 235,245; 232,463 and 242,571 USD for the variations of 15%, 25% and 35% respectively considering in the first case the opening of five hubs and four hubs in the other cases. In Fig. 7 it can be observed the improvement of costs as the capacities of malt producers vary, in this case, the higher the assignment, the better the distribution cost.

Figure 7 Source: The authors.

Finally, the previous results were applied in an Analysis of Variance (ANOVA) with the purpose of determining the effects on the suggested variations (table 2).

Table 2.

Source: The authors.

It was found that there is a dependent relationship between the variation of capacities of grain assignment of malt producers and the distribution cost in this supplying chain, meaning that the different levels in the capacities have an influence on the cost of the network design.

The level of significance associated to malt producers in this study was of 0.001 which demonstrates that the capacities of grain assignment planned by malt producers through CF explain a significant part of the observed variation in the distribution cost for this chain. The level of significance of the hubs’ capacity was of 0.801 and for the interaction hub capacity ( malt producer capacity was of 0.852 indicating that these factors are not significant.

The value R-squared obtained in the ANOVA was of 0.817 proving that, in general, factors of capacity of hubs and of malt producers explain the 81.7% of variance of the cost-dependent variable.