A linear programming optimization model applied to the decision-making process of a Brazilian e-commerce company Um modelo de otimização de programação linear aplicado ao processo de tomada de decisão de uma empresa brasileira de comércio eletrônico

Professor at the Institute of Industrial Engineering and Management of the Federal University of Itajubá (UNIFEI), PhD in Mechanical Engineering at the São Paulo State University (UNESP), Master in Production Engineering at the UNIFEI and Bachelor in Mechanical Engineering at the UNIFEI. Itajubá, Minas Gerais, Brazil. pinho@unifei.edu.br Abstract The decision-making process is not always simple and requires a more careful analysis to maximize the company's revenue. This paper proposes a linear programming model applied to the decision-making of the section of quality monitoring and packaging of a Brazilian company of e-commerce, in which the simplex method was used to maximize the company's revenue from historical time data of the activities for each type of product. From the results, it was verified which products should be prioritized, providing a revenue of US$ 74,681.50. In addition, a simulation was applied to include two employees in the process, which would provide a 32.76% increase in the company's profitability and a new revenue of US$ 99,145.00.


Introduction
The scarcity of resources and high competitiveness cause companies to seek improvement of their processes. However, focusing only on the production process cannot bring satisfactory results (Almeida et al., (2018a)).
Prioritizing other sectors, such as inspection and assembly, favours the outcome of the process by maximizing the company's efficiency. Therefore, many companies seek to make decisions through process planning in which continuous improvement programs are often applied.
Decision-making is characterized by being critical for organizations (Freitas et al., 1997) and can be applied to situations of risk or uncertainty.
The decision-making process is often not intuitive, requiring detailed analyses to the best choice. Proposing analytical solutions can bring satisfactory results, since the decision theory has a mathematical foundation. Thus, modelling the problem through linear programming (LP) can bring satisfactory results, since, according to Cooper, Edgett, and Kleinschmidt (2000), quantitative models are widely used by organizations.
The Linear Programming (LP) is a technique used to find an optimal point for several variables from an established function, satisfying a set of constraints. LP stands out as one of the most efficient techniques for management tools, in which it is applied in several sectors as in Hall (2010), Nash (2000, Yang and Lin (2000). Among the algorithms used in linear programming, the Simplex algorithm is the most used for LP problem solving.
It is possible to find in the literature several studies involving mathematical modeling applied for decision making, such as Meng et al. (2014), who developed a study in the area of B2B e-business, proposing a stochastic programming model in which the objective function is to minimize the total cost. This work is similar to the study proposed by our paper. Meng et al. (2014) Therefore, and the reduced cost vector . ,..., 1 , If not, calculate: be the variable that comes out.
In view of this, it must define As well, De Cosmis and De Leone (2012) states that when a non-degenerate step is performed, the value of the objective function decreases strictly. In this way, the Simplex method will be completed after a finite number of iterations Exacta, 17 (3) if all BFS are non-degenerate. If there is degeneracy, the value of the objective function must remain constant and the algorithm will go into infinite loop.
In conclusion, specific rules must be implemented to this algorithm to avoid this situation. According to the company's sales history, it is known that the product P1 has a daily demand from 3 to 12 units. The product P2 has a minimum demand of 2 daily units and a maximum demand of 12 daily units. For products P3 and P4, the maximum daily demand is, respectively, 7 and 8 units per day.
Finally, product P5 has a minimum demand of 2 units and a maximum demand of 11.
Before being dispatched, each product goes through the processes of quality and packaging. An overview of the process without any kind of interference was done before collecting the data and designing a mathematical model that represents the reality of the company. After obtaining this wide view of the process, the data collection was performed randomly for seven days from 10 a.m. to 3 p.m. using filming. The data are available in Table   1. Using a statistical counter, the mean time spent in this process was estimated in minutes; this is illustrated in Figure 1. Therefore, one has:  ΦP1 -10 minutes;  ΦP2 -9.5 minutes;  ΦP3 -7 minutes;  ΦP4 -8.5 minutes;  ΦP5 -6 minutes.

Results
Through the application of linear programming by the Simplex method, it was reached an optimal solution, generating the values of P1 to P5 aiming to maximize the daily revenue and respecting the restrictions imposed. According to the results found by Solver®, it is possible to verify that all available time (5:00 hour or 300 min) has been used to check and pack the products. Thus, only P3 and P4 products meet every daily demand.
Considering the restrictions, it is necessary to prepare and pack the quantity demanded of these products (7 and 8 units, respectively).
Products P1, P2 and P5 meet the minimum restriction of their units. However, the result shows that these products do not meet part of the daily demand in quantities of 5, 2 and 8 units, respectively. Therefore, the model shows the quantity required to be checked and packaged for each product, in which seven units will be made for P1, ten for product P2, seven for product P3, eight for product P4 and three units for product P5, thus reaching an optimal solution for the model, maximizing revenue by US$ 74,681.50.

Simulated scenario
From the result found, it is verified that the time restriction was totally used, i.e., it is a scarce resource where any change in it will cause a change in the optimal solution and consequently in the revenue. There are also scarce restrictions on P3 and P4 products, in which every quantity demanded is attended by the company.
The products P1, P2 and P5 in the solution shows that the company no longer meets the market with these three products in 5, 2 and 8 units respectively, even when meeting the minimum demand. For this problem, the company has some solutions such as: application of resources (since the demand for these products is bigger that the company can meet, it would be up to it to use part of the investments in marketing these products for another activity); increase of the market value of these products; increase of the number of employees for this function. Exacta, 17(3 In view of this new analysis, it would be up for the company to assess whether it is feasible to add it to such activity, given the availability and value of the labour of another operator and equipment, since a significant increase in revenue occurred.

Conclusion
This paper uses a linear programming model to optimize a decision-making process of a Brazilian company of e-commerce using the Simplex method.
The data of the times of each product of the process was collected to represent the reality of the company.
From the mathematical abstraction of the problem, it was possible to apply the optimization method to find the optimal solution of the problem.