An Improved Grey Wolf Optimizer for a Supplier Selection and Order Quantity Allocation Problem

: Supplier selection and order quantity allocation have a strong inﬂuence on a company’s proﬁtability and the total cost of ﬁnished products. From an optimization perspective, the processes of selecting the right suppliers and allocating orders are modeled through a cost function that considers di ﬀ erent elements, such as the price of raw materials, ordering costs, and holding costs. Obtaining the optimal solution for these models represents a complex problem due to their discontinuity, non-linearity, and high multi-modality. Under such conditions, it is not possible to use classical optimization methods. On the other hand, metaheuristic schemes have been extensively employed as alternative optimization techniques to solve di ﬃ cult problems. Among the metaheuristic computation algorithms, the Grey Wolf Optimization (GWO) algorithm corresponds to a relatively new technique based on the hunting behavior of wolves. Even though GWO allows obtaining satisfying results, its limited exploration reduces its performance signiﬁcantly when it faces high multi-modal and discontinuous cost functions. In this paper, a modiﬁed version of the GWO scheme is introduced to solve the complex optimization problems of supplier selection and order quantity allocation. The improved GWO method called iGWO includes weighted factors and a displacement vector to promote the exploration of the search strategy, avoiding the use of unfeasible solutions. In order to evaluate its performance, the proposed algorithm has been tested on a number of instances of a di ﬃ cult problem found in the literature. The results show that the proposed algorithm not only obtains the optimal cost solutions, but also maintains a better search strategy, ﬁnding feasible solutions in all instances. A.M., and E.O.-B.; methodology, E.C.; software, A.A.-R. and A.R.; and formal and and resources,


Introduction
The purchase of raw materials for industrial manufacturing is an important task. Materials must be purchased at the right times and quantities since a shortage (an interruption of the production due to the lack of raw materials) causes large monetary losses. In these activities, one of the main challenges is determining the optimal purchasing parameters, the supplier, or the suppliers to order the raw material from, and how many items must be ordered from each supplier. This also involves the average inventory (and then, the size of the storage facility) and the monthly demand of items. A cost is calculated for each aspect of the purchasing, such as the setup cost, holding cost, and the cost of the items.
On the other hand, metaheuristic methods are optimization schemes inspired by our scientific understanding of biological or social systems, which at some abstraction level can be considered as search strategies [17]. Some examples of popular metaheuristic methods include Particle Swarm Optimization (PSO) [18], Genetic Algorithms [19], the Artificial Bee Colony (ABC) algorithm [20], the Differential Evolution (DE) method [21], the Harmony Search (HS) strategy [22], the Gravitational Search Algorithm (GSA) [23], and the Flower Pollination Algorithm (FPA) [24]. Metaheuristic schemes do not need convexity, continuity differentiability, or certain initial conditions, which corresponds to an important advantage considering classical techniques.
Alternatively to linear programming techniques, the problems of purchasing have also been conducted through metaheuristic schemes. In the literature, metaheuristic methods have demonstrated to obtain a better performance than those based on classical techniques in terms of accuracy and robustness. As a result, some approaches have been proposed considering different metaheuristic schemes. Some examples include techniques such as Genetic Algorithm (GA) [25][26][27][28][29] and PSO [30][31][32][33]. Although these schemes present interesting results, they have a critical problem-their low premature convergence. This fact generates that such methods frequently obtain sub-optimal solutions, mainly in multi-modal objective functions.
The GWO algorithm [34] is a recent metaheuristic technique based on the hunting behavior of grey wolves. It mimics the leadership, hierarchy, and hunting mechanism of grey wolves. They considered four types of wolves (alpha, beta, delta, and omega) for simulating the leadership hierarchy. Furthermore, they implemented the four main steps of hunting (searching for prey, hunting, encircling prey, and attacking the prey). Its interesting characteristics have motivated its use in several engineering problems, such as sustainable manufacturing [35] and supply chain [36]. In spite of its interesting results, the limited exploration of GWO presents great difficulties in its search strategy when it solves highly multi-modal optimization problems.
In this paper, an improved version of the GWO scheme is introduced to solve the highly multi-modal problem of purchasing. In the enhanced method, two additional elements have been included: (I) weighted factors and (II) a displacement vector. With such inclusions, the new method maintains its important characteristics, increasing its explorative properties so that the algorithm can converge to difficult high multi-modal optima. Different from linear programming techniques, the proposed method can solve supplier selection and purchasing problems under very complex and realistic scenarios, since it does not assume linearity and unimodality in its operation. On the other hand, in comparison to the original GWO and other metaheuristic schemes, our approach is capable of obtaining global optimal solutions due to the improved capacity to explore the search space extensively.
With the purpose of testing our approach, a representative model popular in the literature have been selected. The model [37] considers multiple suppliers with limited capacity. It assumes that suppliers do not have 100% non-defective parts. The model considers a known demand over a finite planning horizon. Additionally, the maximum storage space for the buyer is considered to maximize the total profit. The decision variables are the order quantity for each product, selected suppliers, and purchasing order cycle; the formulation models a problem of supplier selection and lot-sizing inventory. The results show that the proposed algorithm does not just obtain the optimal cost solutions, but also maintains a better search strategy in all instances of the problem, finding feasible solutions in all instances.
The remainder of this article is organized as follows. In Section 2, the problem description and model formulation are presented. Section 3 describes the GWO algorithm. Section 4 describes the proposed modifications to the algorithm. Section 5 presents an illustrative example, along with numerical results and a statistical analysis. Finally, some important conclusions are summarized in Section 6.

Problem Description and Model Formulation
This section introduces the problem under study [37]. It consists of solving the supplier selection and order quantity allocation problem incorporating the total income, which considers the income not only of perfect items but also of imperfect items. The model considers several costs, such as the purchasing, ordering, screening, and holding costs. The model under study has been selected for two main reasons: (i) it provides a complex formulation considering several costs in the optimization, constraints, and decision variables; (ii) this model uses several parameters than can be changed in the design of experiments for comparison purposes.
The model characterizes the management of a supply chain where multiple products and multiple suppliers are considered. All the suppliers have a limited capacity. The model implements the scenario of receiving items that may not meet the requirements for the percentage of non-defective parts-a percentage of parts are not of perfect quality. The non-perfect items are sold as a single batch, prior to receiving the next shipment. These items are sold at a lower cost than the non-defective items. The demand is known along the finite planning horizon. The items can be purchased from potential suppliers. A holding cost applies to each item that must be stored. Maximum storage space is considered. With the aim to maximize the total profit, the company needs to determine who are the best suppliers for assigning an order to and how much order quantity must be placed for each product and in which period.

1.
The ordering cost O j for each supplier j (if an order is assigned) does not depend on the variety and order quantity of the items involved.

2.
The holding cost h i of the product i represents the cost of maintaining an item in stock.

3.
Demand d it represents the amount of the product i that is required in period t, and it is known along the planning horizon.

4.
It is possible that suppliers do not offer perfect quality; the purchased items can contain a percentage P ij of defective products; the percentage of perfect products would be (1−P ij ). 5.
The purchased imperfect items are stored apart and sold prior to the next purchasing period as a single batch. 6.
The purchasing price (of item i) from supplier j is defined as b ij . The perfect quality items are sold at a price S gi per unit, and the defective items are sold as a single batch at a lower cost S di . 7.
The 100% of the screening process of the order is made, which is defined with a unit screening cost v i of item i.

8.
Each supplier has a limited capacity for providing items per period. 9.
The requirements of the items must be fulfilled in each period. Shortage or back-ordering is not allowed. 10. Each product requires a storage space w i , and it considers the total available storage space W. Table 1 summarizes the description of the parameters that will be used along with the model.

Objective Function
The objective function is composed of two elements which will be described in this subsection. The first element is the total income of the company (R). It is computed through the transactions of good quality items plus the income of selling the imperfect quality items.
where X ijt symbolizes the ordered quantity (in units) for item i from supplier j in period t.
The processes of generating an order and purchasing the materials have an impact on several costs, such as the purchasing cost, ordering cost, screening cost, and holding cost. The sum of these costs represents the total expenditure of the company (E), which represents the second element. E is calculated as follows: where the first term represents the purchasing cost, which is calculated by the total items of certain types of products ordered at each supplier in any period, multiplied by the price of the item from the supplier. The second term determines the transaction cost for the suppliers, which does not depend on the variety of the ordered items nor on the order quantity. Ordering cost is calculated for each period in which an order is assigned at a supplier. The third term represents the total screening cost, which is calculated as the product of the total ordered items of each type of product and the respective screening cost per type of item. The last term represents the holding cost of maintaining each item that should be stored.
Therefore, the objective function corresponds to the total profit (Z) of the company, represented by the total income minus the total expenses.
As mentioned before, the objective is to find the ordered quantity for the product i from supplier j in period t, so as to maximize the total profit function. The formulation is summarized below: Subject to, 0 ≤ X ijt ≤ c ij , ∀i = 1, . . . , n, ∀j = 1, . . . , r, ∀k = 1, . . . , t.
The first constraint, represented by Equation (5), ensures that the demand for each type of item in each period is covered with the purchased items. The second constraint in Equation (6) ensures that all orders are accompanied by a transaction cost; if an order is assigned to supplier j in period t, then Y jt is equal to 1; otherwise, it is equal to 0. The third constraint, Equation (7), determines that the total storage space is limited by W. Finally, the constraint represented by Equation (8) ensures that the order quantity per supplier does not exceed their capacity per period c ij .
Deterministic methods usually find a global solution when the complexity of the problem is low. The complexity of this model can be determined by the number of constraints, as follows: (n·t) + (n·r·t) + 1 + 2(n·r·t), (9) where n is the total number of different products, r determines the number of available suppliers, and t represents the number of periods. When the size of the problem is large, it is extremely difficult to obtain a global solution in a reasonable time, and other strategies such as metaheuristics can be used to solve this type of problem. Table 2 shows how the number of constraints grows considerably when the type of items, the available suppliers, and the number of periods increase. The size of the problem (dimension) is also determined by the number of decision variables. In this problem, the total number of decision variables is equal to: If we consider the use of metaheuristic algorithms, this number of variables can be reduced. Therefore, the model is simplified because there is a dependence between the variable Y it (if an order was assigned at supplier j in the period t) and X ijt . If X ijt > 0, then Y it = 1; otherwise, Y it = 0. The total number of variables using this simplification is as follows: (n·r·t). (11) Obtaining a global solution by commercial software, based on classical techniques, can take too long. For this reason, it is necessary to explore other strategies such as metaheuristics for solving this type of problem. Some metaheuristic methods, such as PSO and GA, have been used to obtain a good solution in a lower computational time [38]. However, a disadvantage of these methods is that they present a premature convergence, producing frequently suboptimal solutions.

Original Grey Wolf Optimizer
The Grey Wolf Optimizer (GWO) [34] algorithm is a new metaheuristic method inspired by the hunting behavior of the grey wolf in nature. Generally, they live in groups of 5-12 grey wolves and form a pack. The algorithm is based on the social hierarchy behavior of the wolves and their mechanism of obtaining prey (hunting). The wolf pack has several hierarchical levels: the alpha wolf (α) is responsible for making decisions about sleeping or hunting. They lead the herd, and the members follow the decisions of alpha wolves. The beta wolf (β) helps the alpha wolf, coordinating and collaborating with the management of the herd. They are subordinate to the alpha wolves. They represent the second level within a hierarchy. The other hierarchical level is fulfilled by delta wolves (δ). They complement the alpha and beta wolves in managing the herd. The omega wolves (Ω) are the lowest level of the hierarchy. They must obey the alpha, beta, and delta wolves.
GWO algorithm emulates the position of the prey as the optimal solution to an optimization problem. Then, using operators based on the wolves hunting process, the algorithm tries to obtain the position of the prey. The algorithm considers four stages in their structure: Attacking prey, • Searching for prey.

Encircling Prey
The grey wolves begin the hunting process by encircling (surrounding) the prey. This action is determined using the following formulations (12), (13) to update the position of the wolves in the encircling action: A coefficients are calculated as follows: where → a is linearly decreased from 2 to 0 along the course of iterations, and → r 1 and → r 2 are random values in the range [0, 1].

Hunting
In the real process of hunting, the alpha wolf determines the position of the prey, and the beta and delta wolves follow the alpha wolf and participate in the hunting. The positions of alpha (best candidate solution), beta, and delta have a better understanding of the potential location of prey. The method saves the first three best solutions obtained so far and forces the other search agents (including omegas) to update their positions according to the position of the best search agents.

Attacking Prey
Wolves capture the prey when it stops moving. This action is modeled decreasing the value of → a over the course of iterations from 2 to 0, then If random values → A are in [−1, 1], the next position of a search agent may be in any position between the position of the prey and its position, when |A| < 1, then the grey wolves are forced to attack the prey. With the use of these operators, the algorithm allows its search agents to update their position based on the position of the alpha, beta, and delta. Only using these operators, the algorithm is susceptible to stay in local solutions; for this reason, more operators are needed.

Search for Prey
The search is done according to the position of the wolves (alpha, beta, delta). The wolves diverge from each other with the purpose of searching for prey and converge to attack it. The divergence is reached using random values → A > 1 or → A < 1 to force the search agent to diverge from the prey. This process helps in exploration and allows finding a global solution.

Improved Grey Wolf Optimizer
The problem of supplier selection is discrete and can become extremely complex when the number of suppliers and items increases. These conditions and their numerous constraints produce objective functions with a high multi-modality. In spite of its interesting results, the limited exploration of GWO presents great difficulties in the search strategy when it solves highly multi-modal optimization problems. Likewise, the GWO has been designed to operate in continuous spaces. For this reason, it experiences inconsistencies when it is used in problems of a discrete nature. Under such conditions, an improved version of GWO is necessary in order to overcome this issue. In this work, an improved version of the GWO method, called iGWO, has been introduced to solve the problem under study. The enhanced version incorporates two new elements: (1) weighted factors and (2) a displacement vector. With such inclusions, the new method increases and improves the explorative properties so that the algorithm can converge to difficult high multi-modal optima.

Weighted Factors
In the original GWO, particles are updated by considering the average combination of the alpha, beta, and delta wolves (Equation (18)). This mechanism guides individuals in the same proportion towards the best elements. However, it has been proved that this is not the best strategy [39], since that mechanism produces a limited exploration of the search space. Therefore, in the improved version of GWO, particles are updated using the following formulation: where w 1 , w 2 , and w 3 are the weighted factors that determine the contribution of each alpha, beta, and delta wolf. These weights are used to guide the search process towards the best elements but considering different proportions according to the hierarchy of grey wolves.

Displacement Vector
In the new iGWO, a displacement vector (19)) has been included in order to increase the exploration and prevent the consideration of unfeasible solutions. Here, → r 3 is a random value in the interval [−1, 1] that controls the direction of the search. The element → b is included to promote exploration and prevent stagnation in local optima. This element is considered a tuning parameter that must be set with an initial value. To ensure convergence, → b is non-linearly decreased throughout iterations. The definition of → b is given by the following formulation: where t max is the maximum number of iterations. Under this update mechanism, occasionally random steps are permitted to jump into a feasible area in case the global best is stuck in an unfeasible solution. In the beginning, larger steps are allowed. However, the displacement vector is non-linearly decreased over time to balance the exploration-exploitation rate. Besides, since the supplier selection problem requires an integer solution, the updated positions given by Equation (19) are rounded to the nearest integer toward negative infinity.

Experimental Results
A representative formulation introduced in [37] has been considered as an illustrative problem to test the performance of the proposed method. It has been selected in order to maintain compatibility with other studies reported in the literature. The problem consists of three different products, three suppliers, and four-time periods. Assuming Equation (10) as a basis, we have 48 decision variables. They can be reduced to 36 decision variables (Equation (11)). The parameters for this problem are described in Tables 3-7.  The capacity c ij of product i from supplier j per period is 1000 units for all suppliers. The total available space W is limited to 200.
The popular software LINGO and the proposed Improved Grey Wolf Optimizer (iGWO) have been used for solving the model. The experiments have been implemented using MATLAB R2019a, in a computer with an intel(R) Core (TM)i7-8550u cpu@1.80 GHz 1.99 GHz processor.
The results are shown in Tables 8 and 9. Observe that iGWO presents a higher profit than the classical optimization tools. The algorithm obtains a result that is 60% better than the result obtained by LINGO.

Weighted Factors
An experiment was performed with the purpose of analyzing the accuracy and consistency of the proposed algorithm (iGWO). In the experiment, several parameters of the model were changed to confirm the robustness of the algorithm. These parameters are the demand d it , the total available space W, and the capacity of the supplier for each item c ij . For each parameter, three levels were analyzed. The demand (d it ) of the problem instance presented in Table 3 was changed at 75% and 125% of the actual demand. Case 1 (for demand) corresponds to the original demand presented in Table 3; case 2 and case 3 correspond to the new demand considering 75% and 125%, respectively, of the original demand. The total available space (W) was considered for case 1, case 2, and case 3 at 200, 400, and 600, respectively. The capacity of suppliers (c ij ) was changed. Case 1 considers the original demand at 1000 units per item and per period; for case 2 and case 3, the demand is presented in Table 10.  1  600  600  600  1  450  450  450  2  580  580  580  2  435  435  435  3  620  500  480  3  465  375  360 When modifying the parameters, 27 different scenarios were generated. All the scenarios have been solved considering the proposed iGWO method. The results have been compared with those produced by other methods such as LINGO, original Grey Wolf Optimizer (GWO) [34], Modified Grey Wolf Optimizer (mGWO) [39], Proportional-based Grey Wolf Optimizer (PGWO) [40], Tournament-based Grey Wolf Optimizer (TGWO) [40], Particle Swarm Optimization (PSO) [30], Differential Evolution (DE) [21], and Success-History based Adaptive DE with Linear population size reduction (L-SHADE) [41]. In the comparisons, the parameters of these methods have been configured according to the reported values provided by their own references. All these settings are summarized in Table 11. Table 11. Parameter configurations of metaheuristic algorithms.

Settings Configuration
a linearly decrease from 2 to 0 mGWO a linearly decrease from 2 to 0 PGWO a linearly decrease from 2 to 0 TGWO a non-linearly decrease from 2 to 0 The 27 scenarios are identified as follows: instance (1,2,3) indicates that it considers case 1 of demand, case 2 of total available space, and case 3 for supplier capacity. The original instance is defined as (1,1,1). Since metaheuristic algorithms are stochastic methods, the optimization process is repeated in 10 independent executions for every metaheuristic algorithm (with 1000 iterations) to verify the consistency of the results. The population for the algorithms was 100 individuals, and the size dimension is 36. For each algorithm, 10 results are obtained, which represent the best-found solutions. With this information, the performance of the algorithms are statistically compared considering the following indicators: the average profit Z a , the median of the results Z m , the best profit Z b , the worst profit Z w , and the standard deviation S. Indicators Z b , Z w , Z a , and Z m evaluate the accuracy of the algorithms, and S evaluates the consistency of the solutions and, therefore, the robustness of the metaheuristic algorithms. First, the performance of the algorithms in the instances where only one parameter is changed is analyzed. These instances are: (1,1,1), (2,1,1), (3,1,1), (1,2,1), (1,3,1), (1,1,2), (1,1,3). Table 12 presents the statistical indicators of these instances for the 10 executions per method.
From all instances in Table 12, only the iGWO algorithm found a feasible solution in all the 10 executions of the seven instances. In the instances (1,1,1) and (2,1,1), the best result was presented by iGWO at $18,433.30 and $18,008.18, respectively. GWO and mGWO found only one solution. PGWO, TGWO, PSO, DE, and L-SHADE did not find a feasible solution. For the instance (3,1,1), the best result was presented by DE with $24,041.09; therefore, the algorithm only managed to find three solutions out of 10 feasible solutions. The profit of iGWO is only 7% lower than the best solution; also, the average profit and median of the profit of iGWO are better than those of DE. GWO and mGWO found seven and nine solutions out of 10, respectively; PGWO, TGWO, PSO, and L-SHADE did not find a feasible solution. For the instance (1,2,1), the best result was presented by iGWO with $33,842.24. iGWO, mGWO, and DE found a feasible solution for each execution. PSO found two feasible solutions out of 10. PGWO, TGWO, and L-SHADE did not find a feasible solution. For instance (1,3,1), the best result was presented by mGWO with $44,099.66; therefore, the average profit and median of the profit of iGWO is better than all algorithms. GWO, mGWO, and DE found a feasible solution for each execution. PGWO, TGWO, PSO, and L-SHADE found two, one, eight, and seven solutions out of 10, respectively.
For the instance (1,1,2), the best result was presented by iGWO, at $22,432.70. GWO and mGWO found three and two solutions out of 10, respectively. PGWO, TGWO, PSO, DE, and L-SHADE did not find a feasible solution. For the instance (1,1,3), the best result was presented by iGWO, at $22,432.70. GWO, mGWO, and PGWO found one, two, and one solution out of 10, respectively. TGWO, PSO, DE, and L-SHADE did not find a feasible solution. Figure 1 shows that the profit of the found the best solution by LINGO and the iGWO, GWO, mGWO algorithms. These metaheuristic algorithms were selected because they managed to find more feasible solutions than the others.   found three and two solutions out of 10, respectively. PGWO, TGWO, PSO, DE, and L-SHADE did not find a feasible solution. For the instance (1,1,3), the best result was presented by iGWO, at $22,432.70. GWO, mGWO, and PGWO found one, two, and one solution out of 10, respectively. TGWO, PSO, DE, and L-SHADE did not find a feasible solution. Figure 1 shows that the profit of the found the best solution by LINGO and the iGWO, GWO, mGWO algorithms. These metaheuristic algorithms were selected because they managed to find more feasible solutions than the others.   Table 13 summarizes the results of the best solution with profit Z b for the seven instances presented previously.   Observe in Table 13 the values for the decision variable X ijt ; the total profit for each solution; and the behavior of the purchasing, ordering, screening, and holding cost.
As a second analysis, the best profit found for each instance (27 instances) is presented. See Table 14, and observe that iGWO achieved 21 best results out of the 27 instances (77%). There are three instances ((3,3,3), (3,2,3), (3,1,3)) in which only the LINGO and iGWO algorithms found a result, therefore the best results for these instances were generated by iGWO.  Table 15 shows both the best results and the processing time for each instance using iGWO.  Figure 2 shows the main effects of the best solutions for the 27 instances considering the iGWO algorithm. The best results are presented considering case 3 of demand, case 3 of the total available space, and case 3 of supplier capacity. There is a large difference in the profit when the total available space is increased.
(2,2,2) $30,960.60 39.55 Figure 2 shows the main effects of the best solutions for the 27 instances considering the iGWO algorithm. The best results are presented considering case 3 of demand, case 3 of the total available space, and case 3 of supplier capacity. There is a large difference in the profit when the total available space is increased.     From the numerical results, it can be stated that, different from the linear programming techniques, the proposed method is able to solve the supplier selection and purchasing problems under very complex and realistic scenarios, since it does not assume linearity and unimodality in its operation. On the other hand, in comparison to the original GWO and other metaheuristic schemes, our approach is capable of obtaining optimal solutions due to the improved capacity to avoid suboptimal search locations. Despite its interesting performance properties, the proposed scheme maintains two disadvantages of very high computational cost and difficulty in implementation, as it is not incorporated within the suite of commercial software.
The instances were executed using LINGO and the metaheuristic algorithms, each algorithm for 10 independent times. Then, the non-parametric statistical technique, the Kruskal-Wallis test, was used to test for significance. Recall that this statistical test compares the medians among the nine methods used. Table 16 shows the p-values, which present evidence of a significant difference between the medians of the methods (LINGO, iGWO, mGWO, PGWO, TGWO, PSO, DE, and L-SHADE) around the total profit; also, it is possible to observe that the iGWO algorithm presents the best median in five out of seven instances.  From the numerical results, it can be stated that, different from the linear programming techniques, the proposed method is able to solve the supplier selection and purchasing problems under very complex and realistic scenarios, since it does not assume linearity and unimodality in its operation. On the other hand, in comparison to the original GWO and other metaheuristic schemes, our approach is capable of obtaining optimal solutions due to the improved capacity to avoid sub-optimal search locations. Despite its interesting performance properties, the proposed scheme maintains two disadvantages of very high computational cost and difficulty in implementation, as it is not incorporated within the suite of commercial software.
The instances were executed using LINGO and the metaheuristic algorithms, each algorithm for 10 independent times. Then, the non-parametric statistical technique, the Kruskal-Wallis test, was used to test for significance. Recall that this statistical test compares the medians among the nine methods used. Table 16 shows the p-values, which present evidence of a significant difference between the medians of the methods (LINGO, iGWO, mGWO, PGWO, TGWO, PSO, DE, and L-SHADE) around the total profit; also, it is possible to observe that the iGWO algorithm presents the best median in five out of seven instances. approach initially promotes exploration. However, as the generations progress, the exploitation should be intensified to improve existing solutions. Schemes based on metaheuristic principles involve a set of solutions to exploit and explore the search space in order to obtain the optimal solutions for an optimization task. In their operation, the best quality solutions attract other agents conducting the search process towards their locations. As a result of this effect, the distance among individuals decreases while the results of the exploitation increase. Conversely, if the distance among solutions increases, the consequences of the exploration in the metaheuristic scheme are reinforced.
To evaluate the distance among search agents, a diversity index called the dimension-wise diversity assessment [43] is assumed. Under this index, the diversity is computed as follows: where median x j corresponds to the median value of the j-th dimension from the complete population.
x j symbolizes the j-th dimension corresponding to the i-th search agent. n represents the total number of individuals in the population, whereas m corresponds to the number of variables that involve the optimization formulation to be solved. Under this procedure, the evaluation of the diversity in every dimension Div j is formulated as the mean distance between the j-th dimension of each individual and the median value from that dimension. Therefore, the diversity of the complete population Div is evaluated by calculating the averaged value of Div j for each dimension. Div is computed in each iteration during the complete evolution process.
Once computed the value of Div, the exploration-exploitation balance can be computed as the percentage of the time that the processes of exploring or exploiting invest in terms of its diversity. Such values can be evaluated at every iteration by using the following models: XPL% = Div Div max × 100,XPT% = |Div − Div max | Div max × 100, where Div max corresponds to the maximum Div obtained in the complete optimization process. XPL% represents the percentage of exploration, which corresponds to the level of exploration. It relates the diversity in each iteration with the maximal reached diversity. On the other hand, XPT% represents the percentage of exploitation that expresses the level of exploitation. It is computed as the complementary percentage of XPL%, since the difference between the maximum diversity and the current diversity from a particular iteration is generated as a result of the attraction of search agents. Therefore, both indexes XPL% and XPT% are mutually complementary. Figure 4 shows the evolution of the balance between exploration and exploitation obtained by the original GWO (Figure 4a) scheme and the improved GWO (Figure 4b) method, considering as an optimization problem the instance (1,2,2). This instance corresponds to a representative optimization task that reflects the complexity of the purchasing problems from an optimization perspective. In the simulation, a total number of 100 iterations have been considered.
In order to compare their performance, the point in which both process exploration and exploitation maintain the same proportion (XPL% = 50, XPT% = 50) is evaluated. This point represents the location at which the algorithm changes its behavior from the exploration (where the value of XPL% > XPT%) into exploitation (XPL% < XPT%).  In order to compare their performance, the point in which both process exploration and exploitation maintain the same proportion ( % = 50, % = 50) is evaluated. This point represents the location at which the algorithm changes its behavior from the exploration (where the value of % > %) into exploitation ( % < %).
As can be seen from Figure 4, the improved GWO maintains a higher level of exploration, since the balance point (B) is reached in 500 generations. On the other hand, the original GWO method presents a lower exploration level, considering that its balance point (A) is located around the 200 generations. This fact demonstrated that the improved version of GWO is able to explore the search space extensively in order to obtain globally optimal solutions to the complex purchasing problems. This remarkable result is provoked by the inclusion of (I) weighted factors and (II) a displacement vector. These elements avoid the excessive concentration of the search agents in locations, allowing a better distribution within the search space.

Conclusions
Supply chain management requires that processes and models may be able to provide solutions in a fast and efficient manner. This paper addresses the supplier selection and order quantity allocation problem. This problem is characterized by its discontinuity, non-linearity, and high multimodality. In this paper, a modified version of the GWO scheme is introduced to solve this type of complex optimization problem. The improved GWO method called iGWO includes weighted factors and a displacement vector to promote the exploration of the search strategy, avoiding the use of unfeasible solutions.
A representative difficult problem of the literature was selected with the purpose of testing the behavior of the proposed algorithm. Solutions were obtained using LINGO and the proposed iGWO scheme. After exhaustive experimentation, the results demonstrate that the proposed algorithm does not just lead to lower total cost solutions, but also performs a better search strategy in all the compared scenarios.  As can be seen from Figure 4, the improved GWO maintains a higher level of exploration, since the balance point (B) is reached in 500 generations. On the other hand, the original GWO method presents a lower exploration level, considering that its balance point (A) is located around the 200 generations. This fact demonstrated that the improved version of GWO is able to explore the search space extensively in order to obtain globally optimal solutions to the complex purchasing problems. This remarkable result is provoked by the inclusion of (I) weighted factors and (II) a displacement vector. These elements avoid the excessive concentration of the search agents in locations, allowing a better distribution within the search space.

Conclusions
Supply chain management requires that processes and models may be able to provide solutions in a fast and efficient manner. This paper addresses the supplier selection and order quantity allocation problem. This problem is characterized by its discontinuity, non-linearity, and high multi-modality. In this paper, a modified version of the GWO scheme is introduced to solve this type of complex optimization problem. The improved GWO method called iGWO includes weighted factors and a displacement vector to promote the exploration of the search strategy, avoiding the use of unfeasible solutions.
A representative difficult problem of the literature was selected with the purpose of testing the behavior of the proposed algorithm. Solutions were obtained using LINGO and the proposed iGWO scheme. After exhaustive experimentation, the results demonstrate that the proposed algorithm does not just lead to lower total cost solutions, but also performs a better search strategy in all the compared scenarios.

Conflicts of Interest:
The authors declare no conflict of interest.