Introduction

According to ^{CONADESUCA (2019)}, sugarcane in Mexico is a relevant crop because the agroindustry of this crop has a presence in 15 states and 267 municipalities and generates a significant number of direct and indirect jobs. According to ^{CEMA (2017)} in the 2016-2017 harvest, 184 171 producers participated, of which 71.5% corresponds to ejidatarios (shareholders of common land), 24.6% to smallholders and 3.9% to other types of producers such as tenants and school plots. For the same harvest, ^{CONADESUCA (2019)} reported 68 365 participating cutters. In this way, under the assumption that the producers employ themselves directly in the harvest and that the cutters can effectively be considered as employees, then one will be talking about 252 536 direct jobs.

In addition, if it is also assumed that for every direct job three direct jobs are generated, then one would be talking about the generation of 1 010 144 jobs generated in the field link of the sugarcane agroindustry. However, in the 2018-2019 harvest, 804 060 ha were industrialized with a sugar production of 6.4 million tons. According to the ^{SIAP (2019)}, sugarcane represented in the agricultural year 2019, 6.8% of the total value of agricultural production, ranking third, only behind corn and avocado, which represented 17.7% and 7.7%, respectively.

Sugarcane is the only crop in Mexico that has a regulation that regulates the relations between sugarcane mills and sugarcane farmers. This regulation is the Law of Sustainable Development of Sugarcane published in 2005 (^{SAGARPA, 2005}), which together with the National Program for the Cane Agroindustry (^{SAGARPA, 2014}) are the normative instruments and that give guidelines for the planning, modernization and improvement of competitiveness via crop research.

However, although there is such a specific program for agroindustry, in practice, the operability, profitability and the boost to the manufacturing capacity of agroindustry presents challenges emanating from the almost permanent structural crisis of the sugarcane sector. In this sense, there have been two moments in which the situation of the sugar mills has entered in crisis, which has caused the intervention of the public sector. Between the second decade of the twentieth century and the 80s of the same centuries, the sugar mills were managed by the State.

The characteristic of these periods was the application of subsidies and an erratic business policy. The situation worsened between 1988 and 1990 because the large volumes of sugar imports caused an oversupply, mills were overindebted and with an overdue portfolio, had limited access to credit, high production costs and price fixing was punished up to 15% below the agreed price. In addition to the above, there was stagnation in the productivity of sugarcane fields and mills with loss of profitability, which made it difficult to pay debts, so several mills closed definitively.

It is in this context that in 1988 the government made the decision to sell the mills to a group of businessmen, among whom were partners of the soft drink companies, with the argument of the modernization of agroindustry (^{Molina, 2019}).

The second intervention of the public sector in the sugarcane agroindustry took place in 2001. According to ^{Molina (2019)}, the government decided to nationalize 27 mills with the argument of alleviating the crisis that at that time the mills registered, which dragged fiscal, social security and credit debts and debts with producers and day laborers, which amounted to more than 3 billion dollars. The corollary of such crises has been inefficient productive infrastructure and technology, high production costs and little diversification in the use of co-products and by-products obtained in the sugar mill and distillery.

According to the National Union of Sugarcane Growers, Civil Association, linked to the National Confederation of Rural Owners (^{UNC-CNPR, 1996}), the area of industrialized cane in the 1995-1996 harvest was 577 106 ha with a yield of 69.63 t ha^-1 and net milled cane production of 39 415 468 t produced in 61 mills. However, in the 1996 harvest 61 sugar mills operated while in the 2015 harvest 51 operated. In this way, in a period of 19 years, 10 mills stopped operating in the sugarcane agroindustry; that is, an average of approximately one mill stopped operating or closed every two years.

The analysis of a series of indicators of the productivity of the sugarcane agroindustry shows that it has had a series of inefficiencies and technological backwardness that have had an impact on its profitability and probably influenced the closure of 22% of the existing mills in a period of 19 years. According to the ^{Fideicomisos Instituidos en Relación a la Agricultura (2016)}, sugarcane is an input that could generate products such as biofertilizers, biofuels, carbon credits for the reduction of polluting emissions, etc.

This could generate additional incomes and improve the profitability position of the mills. However, agroindustry is poorly diversified and shows inefficiencies in several technical parameters. Among the main ones are: Industrial plants have only taken advantage of 39% of the total caloric content of one ton of cane (1.2 million kilocalories). The sugarcane agroindustry buys approximately 6.3% of electricity from the Federal Electricity Commission (CFE) to be able to mill sugarcane, which raises its operating costs. That is, the sugarcane agroindustry is not self-sufficient in the generation of electricity for self-consumption, even though the existing legal framework allows it to generate surpluses through cogeneration and sell it to the CFE, which could constitute a source of income that improves its financial balance sheets.

The average lost time per harvest in the period 2005 to 2016 of the total harvest hours per year (209 818 hours) is 19.1%. This 19.1% is distributed as follows: 3.3% in the field, 8.9% in the factory, 0.4% due to holidays, 5.4% due to rains and 1.1% due to aspects attributable to personnel. Most of the total lost time corresponds to mill itself due to the insufficient plant maintenance and replacement of problematic or obsolete equipment. In the period 2005 to 2016, the sugarcane agroindustry has paid on average 118.28 kg of sugar per ton of net milled cane to producers; however, the kilos of sugar actually obtained on average per ton of net milled cane were 117.38 kg.

The mill is not reaching the efficiency indicated in the Law of Sustainable Development of Sugarcane for the payment of cane based on the reference price determined by CONADESUCA, so the mill is paying an extra cost of 0.765 kg t^-1 of sugarcane that in monetary terms has meant an additional expenditure of 3 044 740 million pesos.

A parameter that shows the technological backwardness of the physical equipment of the mills is reflected in the so-called specific consumption of steam of sugarcane; a totally inefficient use of cane bagasse and oil, due to the use of boilers and turbogenerators that operate at very low pressures and low steam temperatures.

As mentioned, in the period 1995-1996, 61 sugar mills operated while in the 2015-2016 harvest only 51 did, in a period of 19 years 10 mills closed, which represents 19.6% of these units, so it is of interest the analysis of the 51 sugar mills that operated in the 2015-2016 harvest.

The data envelopment analysis methodology has been used to measure the efficiency in the use of resources in various economic activities. ^{Becerril et al. (2011)} applied it to measure the technical efficiency of the Mexican agricultural sector. ^{Fontalvo and De la Hoz (2016)} used it to measure the efficiency of hospitals in a Colombian city; ^{Quintanilha et al. (2012)} used this methodology to measure the efficiency of Brazilian airlines, it has also been used to measure efficiency in credit unions (^{Belmonte and Plaza, 2008}) and in the measurement of crops such as coffee in Colombia (^{Perdomo and Mendieta, 2007}). ^{Jaforullah and Whiteman (1999)} used the data envelopment analysis methodology to measure the scale efficiency of New Zealand’s dairy industry and examine the relationship between farm size and technical efficiency.

^{Tetteh et al. (2016)} assess the technical efficiency and scale efficiency of small-scale rice farmers in northern Ghana as well as the effect of farm size on efficiency, using data envelopment analysis to measure efficiency. In this context, the present study aims to evaluate the efficiency performance of Mexico’s sugar mills for the 2010-2011 harvest by estimating the scale efficiency and nature of the returns to scale at which they operated in that harvest and to determine which mills could stop operating given their inefficiencies.

Materials and methods

For the determination of the efficiency and nature of the returns to scale, the data envelopment analysis was used in the research. The concept of technical efficiency, in the context of envelopment data analysis, was initiated by ^{Charnes et al. (1978)}, whose challenge was originally to estimate such a concept of technical efficiency considering multiple inputs, multiple products and without the usual price information. The result of that work was the formulation of the Charnes-Cooper-Rhodes (CCR) model of the DEA in the form of a quotient or fractional programming.

In contrast to parametric approaches, which aim to optimize a single plane of regression through the data, the DEA optimizes a ‘model’ for each individual observation with the aim of calculating a discrete piecewise frontier determined by the set of Pareto-efficient decision-making units (DMUs). Both approaches, the parametric and the non-parametric use all the information contained in the data. In parametric analysis, the optimized regression line is assumed to apply to each of DMU.

DEA analysis optimizes the measure, in contrast, of the performance of each DMU. This results in a revealed understanding about each DMU rather than the typical average DMU. In other words, the DEA’s focus is on individual observations; that is, observation by observation, in contrast to the approach of averages and estimation of parameters that are associated with the statistical approaches of a single optimization (^{Charnes et al., 1994}).

The CCR assumes, as mentioned, constant returns to scale (CRS). This model is appropriate when all the firms analyzed operate on the optimal scale. ^{Charnes et al. (1978)} define efficiency as the maximum of a quotient of weighted output between weighted inputs subject to the fact that similar quotients for each DMU are less than or equal to the unit. According to ^{Huguenin (2012)}, in mathematical form these ideas can be expressed as follows: TEK=∑r=1suryrk∑i=1mvixik. Where: TEk is the relative technical efficiency of the firm k that uses k inputs to produce s products and also; y_rk= quantity of product r produced by firm k; x_ik= amount of input i consumed by firm k; u_r= weight of product r; v_i= weight of input i; s= number of products; and m= number of inputs.

The technical efficiency of company k is maximized under two constraints. First, the weights applied to the outputs and inputs of firm k cannot generate an efficiency score greater than 1 when applied to each company in the dataset. Second, the weights of outputs and inputs are strictly positive. In this way, for each firm, the following linear programming problem must be solved: maximizar ∑r=1suryrk∑i=1mvixik subject to ∑r=1suryrj∑i=1mvixij≤ 1 j= 1,..., n ≤ 1 j= 1,..., n j= 1,..., n ur, vi> 0 , vi> 0 ∀r= 1,..., s; i= 1,..., m .

The notation of this fraction al programming problem is the same as described in the last two paragraphs. This linear programming problem can be dealt with by following two different approaches. In the first case, the weighted sums of inputs are minimized holding outputs constant (product-oriented model). In the second case, the weighted sums of inputs are maximized holding inputs constant (oriented-input model).

As is known, the solution can be obtained by processing the primal or the dual of the linear programming problem. The primal equations for each model, expressed in its multiplicative form, are as follows and also the notation is the one described for the expression of the relative technical efficiency of the unit k (TE_k).

Product-oriented CRS model

Min∑i=1mvixik Sujeto a ∑imvixij-∑r=1suryrj≥ 0 j= 1,..., n ∑r=1suryrk= 1 ur,vi> 0 ∀r= 1,..s; i= 1,..., m

Input-oriented CRS model

Max∑r=1suryrk Sujeto a ∑i=1mvixij-∑r=1suryrj≥ 0 j= 1,.., n ∑i=1mvixik= 1 ur,vi> 0 ∀r= 1,...,s; i= 1,..., m

The materials from which the information was obtained to feed the mathematical programming models, on which the data envelopment analysis method is based, was the ‘statistical report of the agro-industrial sector of sugarcane in Mexico harvests 2006-2007-2015-2016’ (^{CONADESUCA, 2017}). This document contains abundant official information on sugarcane both in the field and in the factory.

The official information was contrasted with the information published by the ^{UNC-CNPR (2017)}, which is one of the two organizations that supply raw material for sugar mills and represents the smallholders who grow sugarcane. The selected period, the 2010-2011 harvest, is because the information of all the mills is complete and does not contain lost data. It should be noted that cross-sectional data were used in this study; that is, for a single period considering all decision-making units (DMUs) or sugar mills.

The software used to run the models of data envelopment analysis was the Data Envelopment Analysis Program 2.1 (DEAP 2.1) of ^{Coelli (1996)}. ^{Tetteh et al. (2016)} point out that the DEA approach generates a data envelopment surface by linking the points in the input-output space in such a way that it no longer allows the production of more product using the same input level or the production of the same product using fewer inputs. The data envelopment surface serves as a reference point for measuring the relative efficiency of the rest of the companies below the envelope surface.

Empirically this means that all efficient units are linked by a continuum to form an efficient frontier in which the efficiency score for each decision-making unit is measured by how far it deviates from the efficient frontier.

The frontier for constant returns to scale is linear, while for variable returns to scale it is a convex piecewise envelope. It is possible to break down the efficiency analysis into three dimensions to understand the possible sources of inefficiency. The first dimension relates to overall technical efficiency, the second relates to pure technical efficiency, while the third relates to scale efficiency.

Estimating overall technical efficiency involves measuring the ratio between the distance between inefficient points and the efficient frontier of constant returns to scale (CRS), while estimating pure technical efficiency involves measuring the ratio between the distance between inefficient points and the efficient frontier under variable returns to scale (VRS). Scale efficiency (SE) is the ratio between overall technical efficiency (OTE) and pure technical efficiency (PTE). The study by ^{Tetteh et al. (2016)} uses input orientation in estimation because producers have more control over inputs than over production.

Results and discussion

According to ^{CONADESUCA (2018)}, in the 2015-2016 harvest, the sugarcane agroindustry in Mexico operated with 51 sugarcane mills. Of these, there are 18 in Veracruz (35.3%), 6 in Jalisco (11.8%), 4 in San Luis Potosí (7.8%), 3 in Michoacán (5.9%) and 3 in Oaxaca (5.9%). These five states account for 66.7% of the sugarcane agroindustry in Mexico.

The variables used to construct the DEA model based on linear programming, calculate the technical efficiency of decision-making units under constant returns to scale, variable returns to scale and scale efficiency are shown in Table 1, while descriptive statistics of such variables are shown in Table 2. The units of the variables are those shown in the second column of Table 1.

Once an exploratory analysis of the data was performed and analyzing the descriptive statistics to detect observations that could distort the analysis, the DEA model was processed for the calculation of technical efficiency under the assumption of constant returns to scale and the DEA model under the assumption of variable returns to scale. As mentioned above, scale efficiency is obtained as the quotient of technical efficiency (under the assumption of constant returns to scale) and technical efficiency (under the assumption of variable returns to scale). Before proceeding with the analysis, it is necessary to mention that the efficiency obtained under constant returns to scale is called total efficiency, the efficiency obtained under variable returns to scale is called pure efficiency (^{Coelli et al., 1998}; ^{Coelli and Prasada et al., 2005}; ^{Zhu, 2013}).

Conclusions

It was found that 23 mills operated at their optimal scale and presented constant returns to scale, which implies that the management of the resources they have is efficient. A second group of mills that presented increasing returns to scale was found, which implies that they can improve their efficiency because if they increase the use of inputs in a certain proportion, the product obtained will increase more than proportionally. In this group were 29 sugar mills. Finally, a third group that presented diminishing returns to scale; that is, if resources, such as harvested area, machinery, among others, increase, their cane production will decrease. In this case is the mill Pánuco. The overall conclusion of the study is that the presence of a higher number of mills with increasing returns to scale (29) implies that there is an opportunity to improve the efficiency of the national sugarcane agroindustry. This study is the first to determine the scale efficiency and nature of returns to scale using the efficient frontier methodology in the sugarcane agroindustry in Mexico