ORIGINAL ARTICLE

Rational Organization of the Sugarcane Harvest-Transport Complex with the Integration of Mathematical Models

Dr.C. Yanara Rodríguez-López^{1  *} 

Dr.C. Yanoy Morejón-Mesa¹ 

Ing. Claudia Cruz-Arredondo¹ 

Ing. Orlando Martínez-Bao¹ 

¹Universidad Agraria de La Habana, Facultad de Ciencias Técnicas, San José de Las Lajas, Mayabeque, Cuba.

ABSTRACT

The present investigation was carried out in the Base Business Unit “Héctor Molina Riaño”, during the sugarcane harvest corresponding to the periods 2014-2015 to 2016-2017, to determine the rational organization of the harvest-transport brigade, through the integration of mathematical models, which guarantee the stability of the system flow. The rational conformation of the harvest-transport brigade was determined using methods such as Linear Programming and Queuing Theory for a single service station and for cascade stations. When analyzing the stability of the compositions with the Markov Chain Model, the most stable variant is when the queue theory is used in cascades while working in fields of 75 t/ha, with two CASE IH 8800 Harvesters and four Tractors BELARUS 1523 + four VTX 10000 self-balancing trailers, four HOWO SINOTRUCK aggregates + two trailers and three reception centers. All that with a 53,79% probability that the cycle will not be interrupted and a system-stop cost of 33,05 peso/h , being possible to reduce the costs for stops by more than 30% and observing a marked influence when increasing the number of collection centers.

Keywords: optimization; mathematical models; probability; stability; costs for stops

INTRODUCTION

As it is known, the biggest problems that arise during the process of obtaining sugar are the loss of time in the harvest-transport system. In the work of the harvest-transport-reception links, waits are originated. Due to these losses of time during the wait in the queue, considerable material means, productive capacities and human energy are lost (^{Matos and Iglesias, 2012}). The harvest-transport process of sugarcane can be stopped due to the lack of substantiation of the rational composition of the harvest-transport brigade. This results in the low flow stability of the technological process and its cost, which is why its determination is required based on scientific criteria (^{Giraldo, 1995}).

In Cuba, from the organization of the transportation of sugarcane, losses arise that are not solved immediately (^{Martínez et al., 2012}). In the specific case of the Base Business Unit (UEB) Attention to Sugarcane Producers "Héctor Molina Riaño", not only industrial problems have arisen, but also problems of an organizational and productive nature have occurred.

The analysis of the relation between the harvest and transport links requires the study of a waiting system in which it is necessary to balance that relationship, so that the loss due to these causes is minimal. In the work of the sugarcane combined, it happens that it waits for the means of transport or the transport waits for it to receive the load, in addition to waiting in the reception center for the unloading process (^{Kleinrock, 1975}, ^{Prawda, 1988} and ^{Wolff, 1989}). This type of phenomenon is characteristic and is associated with the development of the productive forces. A rational way to bring down queues is to study the laws of queue formation, to learn calculating the necessary number of service units and on this basis, to organize the work of service systems (^{Suárez, 2006}).

For this, it is essential to apply mathematical modeling in scientifically based decision-making, alien to all kinds of improvisations, that allow justifying, for example, the levels of production, use of technical means and material resources whose combination produces the maximum efficiency (^{Buffa, 1968} and ^{Cooper, 1981}). Among the mathematical methods and models used in research to determine the rational organization of the harvest-transport process, the following can be mentioned: Markov Chains, Linear Programming, and the Theory of Mass or Queue Service (^{Escudero, 1972}, ^{Fonollosa et al., 2002} and ^{Morejón et al., 2012}).

MATERIALS AND METHODS

The rationalization of the harvest-transport-reception process should be based on the technological, economic and operational evaluation of the technical means involved in the harvest and transport, as well as the observation of the components of shift time in the reception center. In the present case study, it was taken into account that the harvest was carried out in fields with different agricultural yields; in transport, the road conditions were considered, as well as the type and capacity of the means of transport. It was also taken into account that the distance from the harvest field to the reception center will vary. In the case of costs, direct and specific operating costs were determined in each subsystem, as well as costs for shutdowns.

For the rational composition of the harvest-transport brigade from Linear Programming and Queuing Theory for single service stations and cascades, it was followed the methodology set forth in the article “Mathematical modeling of the sugar cane harvest-transport complex” (^{Rodriguez et al., 2016}).

After having determined the composition of the brigade by the methods described above, it was determined which of the resulting compositions offers more stability in the operation of the crop in flow and for this the Markov Chain Method was used.

A Markov chain has stationary transition probabilities, if for any pair of states E_i and E_j there is a transition probability pij such that:

$\text{P}\left({\text{X}}_{\text{n+1}}\text{=}{\text{E}}_{\text{j}}|{\text{X}}_{\text{n}}\text{=}{\text{E}}_{\text{i}}\right)\text{=}{\text{p}}_{\text{ij}}\text{para n=1, 2, ….}$ (1)

For the best analysis of the Markov chain, the transition matrix is established, within which the following can be stated:

Stochastic matrix:

It is a square matrix whose elements are non-negative and such that the sum of the elements in each row is equal to 1.

Transition matrix in one step:

Given a Markov chain with k possible states E₁,. . ., E_k and stationary transition probabilities.

$\text{Si} {\text{p}}_{\text{ij}}\text{=P}\left({\text{X}}_{\text{n+1}}\text{=}{\text{E}}_{\text{j}}|{\text{X}}_{\text{n}}\text{=}{\text{E}}_{\text{i}}\right)\text{→P=}\left(\begin{array}{ccc}{\text{p}}_{\text{11}}& \text{…}& {\text{p}}_{\text{1k}}\\ {\text{p}}_{\text{12}}& \text{…}& {\text{p}}_{\text{2k}}\\ \begin{array}{c}\text{⋮}\\ {\text{p}}_{\text{k1}}\end{array}& \begin{array}{c}\\ \text{…}\end{array}& \begin{array}{c}\text{⋮}\\ {\text{p}}_{\text{kk}}\end{array}\end{array}\right)$ (2)

The transition matrix P of any finite Markov chain with stationary transition probabilities is a stochastic matrix.

Foundations and applications of Markov Model can be used in the formation of the sugarcane harvest-transport-reception brigade, to determine the stability of the system, based on the probability of passing from one link to another, that is, it is used as an additional conformation method, which allows with various possibilities of structures of the harvest-transport-reception complex of sugarcane, to determine which is the most stable. In the particular case of this research, to establish the model, authors started from the definition of the states that make up the process, which are: sugarcane in harvest E_c, sugarcane in transport E_t and sugarcane in the reception center E_r , defining sugarcane as the element that transits, since it goes from being in the field to being cut, transported out of the field and towards the mill, where it is unloaded, classified and processed. After specifying the states, the number of necessary means of transport are defined and that is combined with the transition probability criteria from the elaborated matrix (^{Kijima, 1997}, ²⁰¹³; ^{Hermanns, 2002}; ^{Ching y Ng, 2006}; ^{Ibe, 2013}).

In order to form the transition matrix, the probability of transition or non-transition from one state to another must be determined through the Poisson tables according to ^{Yesin and Sevostyanov (2014)}, as shown in Expression (3)

${\text{Pt}}_{\text{(x)}}\text{=}\frac{{\text{e}}^{\text{-}\text{λ}}\text{*}{\text{λ}}^{\text{n}}}{\text{x!}}$ (3)

It will be defined the probability of transition from the sugarcane state in harvest to that of sugarcane in transport and from it, to be unloaded in the reception center, considering the fact that there are the necessary inputs for their correct exploitation, due to the reliability of the technical means involved in the process.

For harvesting and transportation, it is determined from the technical availability coefficient of the technical means involved, because what defines the transition from the sugarcane state at harvest to the transport means and from it to the reception center, is that the technical means work.

${\text{λ}}_{\mathrm{m}\mathrm{t}}\text{=}{\text{n}}_{\text{mt}}\text{*}{\text{k}}_{{\mathrm{d}}_{\mathrm{m}\mathrm{t}}}$ (4)

where:

nmt	- number of technical means;
${\text{k}}_{d_{mt}}$	- availability coefficient of the technical means.

From the calculation of the mathematical expectation of the operation of the technical means in the sugarcane at harvest and sugarcane in transport (λ_c and λ_t) states determined by Expression 4, the transition probability is established using the Poisson Tables where x is the value of number of technical means in each state.

To determine the mathematical expectation of the operation of technical means in the sugarcane state at reception λ_cr, 15 observations were made, which were averaged, determining the number of trucks waiting to deliver the product (n_cer) and the total number of trucks in the reception center. (n_tr);

${\text{P}}_{\text{nt}}\text{=}\frac{{\text{n}}_{\text{cer}}}{{\text{n}}_{\text{tr}}}$ (5)

Taking into account the probabilities of non-transition in each state, the transition probabilities can be obtained:

${\text{P}}_{\text{t=}}\text{1-}{\text{P}}_{\text{nt}}$ (6)

Then the transition matrix shown in Expression 7 is constructed. The product of the transition probabilities indicates the probability that the flow harvest works, which allows determining the solutions proposed in each model, which will make the system work with fewer stops, that is, more stable.

$\begin{array}{c}{\text{E}}_{\text{c}}\\ {\text{E}}_{\text{t}}\\ {\text{E}}_{\text{r}}\end{array}\begin{array}{c}\begin{array}{ccc}{\text{E}}_{\text{c}}& {\text{E}}_{\text{t}}& {\text{E}}_{\text{r}}\end{array}\\ \left(\begin{array}{ccc}{\text{P}}_{\text{nt}}& {\text{P}}_{\text{t}}& \text{0}\\ \text{0}& {\text{P}}_{\text{nt}}& {\text{P}}_{\text{t}}\\ {\text{P}}_{\text{t}}& \text{0}& {\text{P}}_{\text{nt}}\end{array}\right)\end{array}$ (7)

Based on the analysis previously carried out, an estimate of the economic impact due to the break of the C_pet cycle can be obtained from the determination of the costs for shutdowns in each element of the cycle and the probability that it will not transit from one state to another.

${\text{C}}_{\text{pet=}}\left({\text{C}}_{\text{pc}}\text{*}{\text{P}}_{\text{ntc}}\right)\text{+}\left({\text{C}}_{\text{pt}}\text{*}{\text{P}}_{{\text{nt}}_{\text{t}}}\right)\text{+}\left({\text{C}}_{\text{pcr}}\text{*}{\text{P}}_{{\text{nt}}_{\text{r}}}\right)\text{;peso/h}$ (8)

where:

Cpc, pt and cr

- cost per stop at harvest, transportation and central reception center, respectively; peso/h

Regarding the costs for stops in the reception center, the wages of the workers linked to the process, the fuels and lubricants consumed in the process, as well as the maintenance and the electrical energy consumed as shown in Expression 9 are taken into account.

${\text{C}}_{\text{pcr}}\text{=}{\text{C}}_{\text{s}}\text{±}{\text{C}}_{\text{c}}\text{±}{\text{C}}_{\text{mr}}\text{±}{\text{C}}_{\text{e}},\mathrm{ }\mathrm{p}\mathrm{e}\mathrm{s}\mathrm{o}/\mathrm{h}$ (9)

RESULTS AND DISCUSSION

For the rational composition of the complex harvest-transport of sugarcane, combining mathematical methods, three variants are analyzed, which are listed below:

• Variant I: The estimated agricultural yield of the field is 65 t/ha, two CASE AUSTOFT IH 8800 harvesters are used, for intermediate transport the BELARUS 1523+ VTX 10000 transport aggregates are used (10 t capacity) and for external transport HOWO SINOTRUK + 2 TRAILER trucks (total capacity 60 t, 20 t on each trailer and 20 t on the truck). The distance to transport is 18 km, 6 km of them are on a paved road and 12 km are an embankment.

• Variant II: The estimated agricultural yield of the field is 70 t/ha, two CASE AUSTOFT IH 8800 harvesters are used, for intermediate transport the BELARUS 1523+ VTX 10000 transport aggregates are used (10 t capacity) and for external transport KAMAZ trucks + 1 TRAILER (total capacity 20 t, 10 t on the trailer and 10 t on the truck). The distance to transport is 20 km, of them 7 km are on a paved road and 13 km are an embankment.

• Variant III: The estimated agricultural yield of the field is 75 t/ha, two CASE AUSTOFT IH 8800 harvesters are used, for intermediate transport the BELARUS 1523+ VTX 10000 transport aggregates are used (capacity 10 t) and for external transport HOWO SINOTRUK + 2 TRAILER trucks (total capacity 60 t, 20 t on each trailer and 20 t on the truck). The distance to transport is 20 km, of them 7 km are on a paved road and 13 km are an embankment.

Determination of the Rational Composition of the Sugarcane Harvest-Transport Complex Using Linear Programming

The POM-QW FOR WINDOWS version 3 software was used to determine the rational composition of the media involved in the sugarcane harvest-transport process using Linear Programming. The quantity variables were taken as decision variables of harvesters necessary in the process (X₁) and the number of external means of transport (trucks) necessary in the process (X₂). Table 1 shows the restrictions 1 and 2 as well as the objective function used in the formation of the models according to the agricultural performance of the field (^{Sadleir, 1970}).

TABLE 1 Objective function and restrictions 1 and 2 according to the agricultural performance of the field 

Ra, t/ha	Objective function	R 1	R 2	R-3	R-4
65	Z=22,18*X1+9,52*X2	446,71*X1=391,92	54,96*X2=138,79	X1≥2	X2 ≥ 1
70	Z=29,46*X1+6,94*X2	459,82*X1=675	18*X2=75
75	Z=22,11*X1+9,52*X2	807,99*X1=748	55,26*X2=134,83

For the given production volume and taking into account the agricultural yields of the fields, it was obtained with the method used that the necessary number of harvesters is in all cases 2 (Table 2), while the number of trucks varies depending on the agricultural yield being that, in fields with 65 t/ha, two HOWO SINOTRUCK trucks must be used with a minimum cost of 66.47 peso/h, needing three when the field has 75 t/ha with a minimum cost of 64,04 peso/h. In fields of 70 t/ha, four KAMAZ trucks are needed to obtain a minimum cost of 77.25 peso/h.

TABLE 2 Result of rational conformation using linear programming 

	65	70	75
Z, peso/h	66,47	77,25	64,04
Number of combines	2	2	2
Number of trucks	2	4	3

As it can be observed, as the agricultural yield of the field increases and the distance to transport the necessary number of means of transport increases, this occurs due to the high values reached by the cycle time, which is affected by the conditions of the fields.

Determination of the Rational Composition of the Sugarcane Harvest-Transport Complex Using the Queue Theory for Consecutive Stations or Cascades

^{Moskowitz and Wright (1991)} and ^{Medhi (2002)}, using the Queue Theory, the waiting model with two stations in cascades with a limited number of clients and in a system where the arrivals of the means of transport to the field behaved according to the Poisson distribution, obtained the following results (Table 3) for each subsystem (Harvest, Transport and Reception).

In Variant I, in the Harvest Subsystem, there were four self-balancing trailers (servers) to service two harvesters (customers), where the filling time of the self-balancing trailer was 0,27 h. To satisfy the demand of the two combines, four self-balancing trailers are needed, which will be able to satisfy 1,17 combines in 1 hour, with a channel capacity of 3,11 and a system request of 0,36. The probability that there are no units in the subsystem is 25%, the mean queue length is 2,25 and the length of stay 1,93 h. In the Transport Subsystem, it was found that, with two means of transport, in one channel hour, 0,39 requests can be satisfied. With a channel capacity of 1,42, the density of requests for the system is 0,27. The average number of units in the queue is 0,37, the queue length is 0,10, the length of stay is 0,27 h and the probability that there are no units in the subsystem is 73%. In the Reception Subsystem, two Reception Centers are needed to satisfy 1,34 means of transport with a channel capacity of 2,32 and a system request density of 0,58. The average queue length is 1,37, with a stay time of 0,58 h.

In Variant I, working with two harvesters, four intermediate means of transport, two external means of transport (HOWO SINOTRUCK) and three reception centers, the queue probability is 28%, with a minimum cost for stops of 63,15 pesos/h.

In Variant II in the Harvest Subsystem, there were four self-balancing trailers (servers) to service two harvesters (customers), where the filling time of the self-balancing trailer is 0,3 h. To satisfy the demand of the two combines, two self-balancing trailers are needed, which can satisfy 1,79 combines in 1 hour, with a channel capacity of 3,58 and the system request is 0,50. The probability that there are no units in the subsystem is 50%, with the mean queue length being 1 and the residence time being 0,28 h. In the Transport Subsystem, it was found that, with four means of transport, 0,90 requests can be satisfied in one channel hour, with a channel capacity of 2,25, the system request density being 0,40. The average number of units in the queue is 0,66, the queue length being 0,264, the stay time is 0,29 h and the probability that there are no units in the subsystem is 60%.

In the Reception Subsystem, three Reception Centers are needed to satisfy 2,30 means of transport with a channel capacity of 2,99 and a system request density of 0,51. The average length of the queue is 1,05, the stay time being 0.24 h.

In Variant II, working with two combines, two intermediate means of transport, four external means of transport (KAMAZ) and three Reception Centers, the queue probability is 21%, with a minimum cost per stop of 74,26 peso/h.

In Variant III, in the Harvest Subsystem, there were four self-balancing trailers (servers) to service two combines, where the filling time of the self-balancing trailer is 0,26 h. When applying the model, it was obtained that four self-balancing trailers (servers) must be used in the Harvesting System; in one hour the canal can satisfy 1,13 combines. The capacity of the service channel in one hour is 1,50. The system request density was 0,75 with a probability of no units in the subsystem of 96%. The average number of units in the queue is 0,0045, the queue length being 0,002, which translates into a time spent in the system of 0,6 h.

When analyzing the Transport Subsystem, it was found that, with four means of transport, 0,75 requests can be satisfied in one channel hour, with a channel capacity of 2,84, the system request density being 0,26. The average number of units in the queue is 0,15, with the queue length being 0,02 and the stay time is 0,5 h. The probability that there are no units in the subsystem is 87%.

To satisfy the demands of the Transport Subsystem and, consequently, those of the Harvest Subsystem, it is necessary that the Reception Subsystem be made up of three centers satisfying 1,34 transport aggregates in 1h of channel, with the service capacity at that time being 3,09 and the system request density of 0,58. The average number of means of transport in the queue is 1,37, with the stay time of one unit in the Reception Center being 0,59 h and the probability that there are no units in the subsystem of 42%.

From the above, it is obtained that in Variant III, working with two harvesters, four intermediate means of transport, four external means of transport (HOWO SINOTRUCK) and three reception centers, the probability of queuing is 8%, with a minimum cost of stops of 55,04 peso/h.

TABLE 3 Results of the waiting model with two stations in cascades using the Queue Theory, for each subsystem 

Ra, t/ha Indicator		65	70	75
Ab		4	2	4
n		2	4	4
Cr		3	3	3
λ	SC	1,17	1,79	1,13
	ST	0,39	0,90	0,75
	SR	1,34	2,30	1,34
µ	SC	3,11	3,58	1,50
	ST	1,42	2,25	2,84
	SR	2,32	2,99	3,09
φ	SC	0,36	0,50	0,75
	ST	0,27	0,40	0,26
	SR	0,58	0,51	0,58
Po, %	SC	25	50	96
	ST	73	60	87
	SR	42	49	42
Mc	SC	2,25	0,5	0,002
	ST	0,10	0,264	0,02
	SR	0,78	0,54	0,79
Ms	SC	3	1	0,045
	ST	0,37	0,66	0,15
	SR	1,37	1,05	1,37
T est , h	SC	1,93	0,28	0,6
	ST	0,27	0,29	0,5
	SR	0,77	0,24	0,59
P cola , %		28	21	8
S, peso/h		63,15	74,26	55,04

Determination of the Stability of the Technological Flow Harvest-Transport-Reception Using the Markov Model

In order to determine which mathematical model of the previous ones used is the most stable and economically feasible in determining the rational composition of the media involved in the process, the Markov Model was used, in which three (3) states were defined for resolution; which were the cane in harvest E_c (Harvest Subsystem), the cane in transport E_t (Transport Subsystem) and the cane in reception E_r (Reception Subsystem). For the application of this model, the mathematical expectations (λ_m) for the state of cane at harvest, in transport and reception (only for the Queue Theory Model with consecutive stations or in cascades) necessary to obtain the values of no transition using Poisson Tables. Starting from it, the probability of their transition is determined. The coefficients of technical availability of the combines, the means of transport and the Reception Center are also determined.

From the elaborated transition matrices, it can be verified that as the number of technical means (harvesters, self-balancing trailers, trucks and reception centers) increases, as the coefficients of technical availability do not vary much, the probability of transition increases, due to which the possibility of interrupting the process flow is reduced. In the state of cane at harvest and cane at reception, the results obtained were analyzed together using Linear Programming and Queuing Theory for a single service station because the values coincide in the case of Harvest, and, in the case of the Center of Reception, the criterion used to analyze the probability of transition and non-transition is the same. In all the matrices the value of the probability of the cane passing from the harvest state to the reception state is null, since the cane has to be transported, likewise, the probability of the cane passing to be transported to the Harvest status is nil, because once the cane is cut it becomes transported. A similar situation occurs in the cane state when it is received, because once the cane reaches the mill it is not returned.

When Linear Programming is used in all the variants analyzed, the probability of entering the reception state is 77%, and there is a probability of not doing so of 23%, which may be due to breakage of the combine harvester or lack of means of transport. The same occurs with the rest of the models in the other variants except for Variants I and III, when analyzing the conformation of the brigade with the Queuing Theory Model in cascades, which in both cases has a probability of transition of 80,5%, these being the most stable.

In the case of the transport state, the probability of transition to the Reception Center is between 75 and 80% in the case of Variants I and III when the Queue Theory Method is used in cascades, and in Variant I in the rest of the models used. In the rest of the models and variants it is between 80 and 83%.

In the reception state in the Linear Programming and Queuing Theory Models for a single service station, the transition probability is between 76 and 78%, being 82% when using the cascade queuing theory method in the Variants from I to III.

After having defined the transition matrices, the economic impact due to instability or technological failure of the different cycles that represent the different conformations of the sugarcane harvest-transport-reception complex, was determined, which can be seen in Figure 1.

In Variant I, the method that shows the best results in terms of stability and minimum economic losses is the Queuing Theory in cascades (two CASE AUSTOFT IH 8800 combines, four BELARUS 1523+ VTX 10000, two HOWO SINOTRUK + two TOWING (each) and three reception centers) with a loss due to system stops of 32,74 peso/h (6,48 and 3,94 peso/h less than if using Linear Programming and Queue Theory of a single service station) . With this conformation, the probability of the transition of sugarcane from harvest to transport is 80,5%, from transport to reception 73%, of being processed in the Reception Center of 82% and that the harvest is completed in 48,16% flow.

Likewise, in Variant II, the best results are obtained with the mathematical method proposed, being the optimal conformation of two CASE AUSTOFT IH 8800 combines, two BELARUS 1523+ VTX 10000, four KAMAZ + one TRAILER (each) and three reception centers with a loss for system stops of 38,33 peso/h (3,35 and 3,05 peso/h less than if Linear Programming and Queue Theory of single service station are used). With this conformation, the probability of transition of the system is 48,39%, being that of the cane from harvest to transport of 73%, from transport to reception of 80,9% and of being processed in the Reception Center of 82 %.

In Variant III, the same value of losses due to system stops (25,54 peso/h) is obtained when analyzed with Linear Programming and Queue Theory of single service station, reducing losses to 9,53 and 8,98 peso/h when the conformation variant resulting from applying the Queue Theory in cascades is used, which is two CASE AUSTOFT IH 8800 combines, four BELARUS 1523+ VTX 10000, four HOWO SINOTRUK + two TRAILERS (each) and three centers of reception. With this conformation, the probability of transitioning the cane from harvest to transport is 80,5%, from transport to reception 81,6%, and of being processed at the Reception Center, 82%, with 53,78 % probability that the process will occur in flow.

FIGURE 1 Costs for stops and probability of transition of the system. 

As observed in Figure 1, in all cases, when the cascade Queue Theory Method is used, the highest probability of transition is obtained in the entire system, that is, the cycle is more likely to be maintained, being The most stable conformations are those offered by this method for fields of 75 and 90 t/ha (53,78 and 51,29%, respectively) and in which the losses due to system shutdowns are lower.

When analyzing the conformation of the harvest-transport-reception brigade that was used in the real process, with which each mathematical model throws (Figure 2), it can be seen that in the fields of 65 t/ha the difference lies in that with the Queue Theory Method for single stations aims to increase the number of external means of transport to 3 while using the Queue Theory Method in cascades it is proposed to increase the number of internal means of transport from two to four. In the fields of 65 t/ha of estimated agricultural yield, it is proposed with all methods to double the number of aggregates of external transport, the same happens in the fields of 75 t/ha using the Queue Theory when using the Linear Programming Methods and Queue Theory for single service stations. In these fields with the last two related agricultural yields, it is proposed to double the number of aggregates of internal transports when used to form the brigade, the Queue Theory Method in cascades.

FIGURE 2 Formation of the real brigade and that obtained by mathematical methods 

As it can be seen in Figure 2, when working in all fields, using the cascading Queue Theory Method, it is proposed to increase the reception centers from one to three, which would decrease transport cycle times and times waiting by external means of transport in the field, helping to reduce costs for system stops.

Likewise, when comparing the stop costs that were obtained experimentally, with those obtained by integrating the mathematical methods Linear Programming, single-station Queuing Theory and cascade Queuing Theory with the Markov Chain Method, it can be seen that the difference oscillates between 17,34 and 44,86 peso/h, decreasing the costs for stops calculated between 27,89 and 68,31% with respect to those obtained experimentally

In order to know what percent, the costs for shutdowns decrease, if the conformations of the proposed brigades are established, the Figure 3 must be observed. In fields of 65 t/ha the decrease would be 46,48% (37,69 peso/h). In the fields of 70 and 75 t/ha, it decreases by 42,19 and 44,86 peso/h, which represents 47,60 and 36,19% of the costs for shutdowns of the system obtained experimentally.

FIGURE 3 Comparison of stop costs. 

CONCLUSIONS

• The rational conformation of the transport harvest brigade was determined using the methods: Linear Programming, Queue Theory for a single service station and for cascade stations.

• When analyzing the stability of the compositions with the Markov Chain Model, it is found that the most stable variant is the one resulting from conforming the brigade with the cascade Queuing Theory Method when working in fields of 75 t/ha, with two CASE IH 8800 Combine Harvesters and four BELARUS 1523 Tractors + four VTX 10000 self-balancing trailers, four HOWO SINOTRUCK aggregates + two trailers and three reception centers with a probability of 53,79% that the cycle is not interrupted and a cost per stops of 33,05 peso/h system.

• From the rationalization criteria, based on the integration of the mathematical models, it is possible to reduce the costs for shutdowns by more than 30%, observing a marked influence as the number of reception centers increases.

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⁶The mention of trademarks of specific equipment, instruments or materials is for identification purposes, there being no promotional commitment in relation to them, neither by the authors nor by the publisher.

Received: December 19, 2019; Accepted: May 14, 2020

^*Author for correspondence: Yanara Rodríguez López, e-mail: [e-mail]

Yanara Rodríguez-López, Prof. Auxiliar, Universidad Agraria de La Habana, Facultad de Ciencias Técnicas, San José de Las Lajas, Mayabeque, Cuba, e-mail: [e-mail]

Yanoy Morejón-Mesa, Prof. Titular, Universidad Agraria de La Habana, Facultad de Ciencias Técnicas, San José de Las Lajas, Mayabeque, Cuba, e-mail: [e-mail]

Claudia Cruz-Arredondo, Profesora, Universidad Agraria de La Habana, Facultad de Ciencias Técnicas, San José de Las Lajas, Mayabeque, Cuba, e-mail: [e-mail]

Orlando Martínez-Bao, Prof. Universidad Agraria de La Habana, Facultad de Ciencias Técnicas, San José de Las Lajas, Mayabeque, Cuba, e-mail: [e-mail]

The authors of this work declare no conflict of interests.