Introduction

Cocoa (Theobroma cacao) is a culture native to Mexico, of cultural importance for the country, so there is a general statement of protection of the appellation of origin of the ‘Cacao Grijalva’ that protects green or roasted/ground cocoa of the species in mention that takes place in the Grijalva Region of Tabasco, this one is integrated by three productive subregions: Chontalpa, Mountain range and Center and they group 11 municipalities of Tabasco (^{DOF, 2016}). The tropical region of America presents the optimal conditions for the cultivation of cocoa, so in these areas it has been cultivated for about three thousand years by the Olmec culture; however, the Maya are attributed the spreading of the grain when they use it, even as a currency in their barter system (^{Nisao, 2007}).

The peoples of Mesoamerica gave a great deal of sentimental value to cocoa since they considered it a gift from the gods; in fact, Theobroma in Greek means ‘food of the gods’, the fruit symbolized the human heart and chocolate contained blood; currently, cocoa and its derivatives have a prominent role in international markets, especially in agro-industry (^{Salas and Hernández, 2015}). In 2017, according to the Agri-Food and Fisheries Information Service (SIAP), in Mexico, 27 287.25 t of cocoa were produced, the producing states of this grain were: Tabasco with 17 430 21 t harvested, Chiapas with 9 611 63 t and Guerrero with 245 41 t (^{SIAP, 2018}).

It should be noted that the production of cocoa in Mexico is in the hands of 37 thousand producers, approximately, Tabasco includes 68% of these, Chiapas has 31% of them and Guerrero has about 1% remaining (^{Orozco, 2018}).

“The 80% of world production is concentrated in cocoas: outsiders or ordinary, trinitario and creole and Mexico has all three types and promotes the increase of creole cocoas, since it is the country of origin” (^{Jaramillo, 2017}). However, in terms of cocoa production, “the socioeconomic and parasitological factors that limit production in a precise and precise manner have not been identified in Mexico and everything points to the fact that diseases contribute significantly to the disappearance of this important crop” (Hernández et al., 2015), given that the import of this grain is essential to meet domestic demand. In 2017, 41 321 97 t of cocoa were imported from around the world, the main suppliers were: Ecuador with 27 012 64 t, Ivory Coast with 9 076 7 t, Colombia with 3 022 33 t, Peru with 1 350 98 t and Republic Dominican Republic with 840.63 t (^{SIAVI, 2018}).

The objective of the work was to state a transport model that optimizes the distribution of cocoa in Mexico, minimizing the cost of transportation. The hypothesis was that the national production of cocoa could only supply the demand of the southern states of the country, given the geographical proximity they have with respect to the producing states, while the states of the center-north of the country would depend on the imports of cocoa to see your demand satisfied. Imports have been gaining more weight in the consumption of national cocoa, mainly due to the growing demand of large chocolate companies that fail to meet their requirements in the domestic market due to the scarce national production of grain (^{Andrade, 2017}). Therefore, in a complementary manner, a minimum cost transport model for an open economy was developed.

Materials and methods

Within the applications of linear programming highlights the problem of transport whose purpose is to determine the optimal way to move goods, since “transportation generally represents the single most important element in logistics costs for most companies. It has been observed that cargo movement absorbs between one and two thirds of total logistics costs” (^{Quintero et al., 2016}). It should be noted that “the general problem of transport -refers literally or figuratively- to the distribution of any goods from any group of supply centers, called origins, to any group of reception centers, called destinations, in such a way that the total distribution costs are minimized” (^{Hillier and Lieberman, 2010}).

In general, the origin O_i(i=1, 2, 3,…, n) has Xi units to offer and the destination D_j (j=1, 2, 3,…, n) has a demand of X_j units. In the case of Mexican cocoa, there are two origins in a closed economy and 30 destinations. In some applications, the quantities of supply or resources and of demand have integer values, and when working with the model it will be required that the quantities distributed take integer values; it should be noted that the entire programming is a problem of linear programming in which it is required that some or all of the variables adopt non-negative integer values (^{Winston, 2005}).

The property of whole solutions establishes that for transport problems in which X_i and X_j have an integer value, all the basic variables (assignments), in all the feasible basic solution (including the optimal one), also have integer values. For its part, ownership of feasible solutions establishes that a necessary and sufficient condition for a transportation problem to have feasible solutions is that (^{Hillier and Lieberman, 2010}).

This property can be verified by observing that the restrictions require that.

The condition that the total resources must equal the total demand requires that the system be balanced, if the problem has some physical meaning and this condition is not met, it would imply that X_i, or X_j, represent a dimension and not an exact requirement. In this case, an imaginary or fictitious origin or destination can be introduced to capture the slack, in order to convert the inequalities into equalities and satisfy the feasibility condition (^{Hillier and Lieberman, 2010}). In this sense, given that cocoa production in Mexico does not satisfy domestic demand, it is necessary to import this product, so, for the open economy model, two fictitious origins were added, Jalisco and Yucatán.

To determine which states of Mexico are deficit or surplus in the consumption of cocoa, the apparent national consumption was calculated, the result was divided among the national population to obtain the apparent consumption per capita. The data obtained was multiplied by the population of each state and its consumption was estimated. The production minus the demand for this grain determined whether the state is a supplying or demanding cocoa. The formulas used were the following (^{Miranda, 2005}).

Apparent national consumption= production + import - export

Where: national apparent consumption= is the quantity demanded of cocoa in tons by the Mexican market; production= it refers to the amount of cocoa harvested in Mexico in tons, the data was consulted in the Agri-Food and Fisheries Information Service (SIAP), the year of study was 2015; import= it is the imported quantity of cocoa expressed in tons, tariff item 18010001 “Cocoa beans, whole or split, raw or roasted” was used. The data was consulted in the Internet Tariff Information System (SIAVI) for the year 2015; export= it is the quantity exported of cocoa expressed in tons; tariff item 18010001 “cocoa beans, whole or split, raw or roasted” was used. The data was consulted in the SIAVI for the year 2015.

Where: apparent consumption per capita= it is the quantity demanded of cocoa for each person in Mexico; National population= express the number of inhabitants in Mexico in 2015, the data was consulted in the National Institute of Statistics and Geography (INEGI).

State consumption = apparent consumption per capita *state population

State population= is the number of inhabitants in each of the states of Mexico in 2015, the data was consulted in the INEGI.

The transportation costs were estimated from the trial version of the ^{GlobalMap Software “roads of road transportation of Mexico 2018”}. They were calculated for a type T3-C2 transport that, according to NOM-012-SCT-2-2017, has a capacity of 30 t (^{DOF, 2017}). Therefore, the deficit or surplus quantities used in the model are presented in truck units.

The origin (O_i) states were those whose production was greater with respect to their consumption; that is, those that had a surplus, while the destination states (D_j) were those that presented a deficit. Origins were taken as the larger cities near the area of production of each state, while destinations for state capitals were used, considering that the bulk of economic activity, mostly concentrated in them.

The variable X11 corresponds to the origin ‘Chiapas’ and destination ‘Aguascalientes’, the variable X12 belongs to the origin ‘Chiapas’ and destination ‘Baja California’, ..., the variable X130 belongs to the origin ‘Chiapas’ and destination ‘Zacatecas’. For its part, the variable representing the origin ‘Tabasco’ and destination ‘Aguascalientes’ is X21, ..., and the origin ‘Tabasco’ and destination ‘Zacatecas’ is X230.

For the open economy, two fictitious origins were used: Jalisco and Yucatán, initiating their variables with X3 and X4, respectively.

Model for a closed economy

Objective function

MIN 22282X11+49509X12+51168X13+11251X14+34585X15+15348X16+24842X17+25499X18+26931X19+20120X110+18263X111+15362X112+22842X113+16381X114+19493X115+15934X116+26495X117+25739X118+6757X119+13456X120+18042X121+10104X122+20197X123+33333X124+41904X125+18344X126+13872X127+11987X128+12903X129+23480X130+17745X21+44647X22+46554X23+4691X24+30061X25+10486X26+19980X27+20965X28+22068X29+15586X210+13726X211+10502X212+18303X213+11518X214+14923X215+11071X216+21633X217+20880X218+6648X219+8591X220+13500X221+5783X222+15660X223+28796X224+37046X225+13809X226+9010X227+7129X228+6346X229+18946X230.

Offer restrictions

X11+X12+X13+X14+X15+X16+X17+X18+X19+X110+X111+X112+X113+X114+X115+X116+X117+X118+X119+X120+X121+X122+X123+X124+X125+X126+X127+X128+X129+X130=270

X21+X22+X23+X24+X25+X26+X27+X28+X29+X210+X211+X212+X213+X214+X215+X216+X217+X218+X219+X220+X221+X222+X223+X224+X225+X226+X227+X228+X229+X230=578.

Demand constraints

X11+X21<=18

X12+X22<=47

X13+X23<=10

X14+X24<=12

X15+X25<=10

X16+X26<=74

X17+X27<=50

X18+X28<=128

X19+X29<=25

X110+X210<=83

X111+X211<=43

X112+X212<=40

X113+X213<=112

X114+X214<=231

X115+X215<=65

X116+X216<=27

X117+X217<=16

X118+X218<=73

X119+X219<=56

X120+X220<=88

X121+X221<=29

X122+X222<=21

X123+X223<=38

X124+X224<=42

X125+X225<=41

X126+X226<=49

X127+X227<=18

X128+X228<=116

X129+X229<=30

X130+X230<=22.

Model for an open economy

Objective Function

MIN 22282X11+49509X12+51168X13+11251X14+34585X15+15348X16+24842X17+25499X18+26931X19+20120X110+18263X111+15362X112+22842X113+16381X114+19493X115+15934X116+26495X117+25739X118+6757X119+13456X120+18042X121+10104X122+20197X123+33333X124+41904X125+18344X126+13872X127+11987X128+12903X129+23480X130+17745X21+44647X22+46554X23+4691X24+30061X25+10486X26+19980X27+20965X28+22068X29+15586X210+13726X211+10502X212+18303X213+11518X214+14923X215+11071X216+21633X217+20880X218+6648X219+8591X220+13500X221+5783X222+15660X223+28796X224+37046X225+13809X226+9010X227+7129X228+6346X229+18946X230+9530X31+31437X32+33336X33+24273X34+21846X35+9478X36+14564X37+3244X38+14153X39+7366X310+5352X311+9190X312+6766X313+8086X314+4436X315+9472X316+8424X317+15466X318+15561X319+11196X320+7824X321+25551X322+10491X323+15590X324+24161X325+13522X326+10828X327+12483X328+26412X329+10731X330+24749X41+51651X42+53553X43+2149X44+37065X45+17491X46+26984X47+27966X48+29073X49+22260X410+20405X411+17504X412+25307X413+18520X414+21602X415+18076X416+28637X417+27886X418+13652X419+15596X420+20179X421+4123X422+22664X423+35475X424+44046X425+20813X426+16017X427+14130X428+534X429+25947X430.

Offer restrictions

X11+X12+X13+X14+X15+X16+X17+X18+X19+X110+X111+X112+X113+X114+X115+X116+X117+X118+X119+X120+X121+X122+X123+X124+X125+X126+X127+X128+X129+X130=270

X21+X22+X23+X24+X25+X26+X27+X28+X29+X210+X211+X212+X213+X214+X215+X216+X217+X218+X219+X220+X221+X222+X223+X224+X225+X226+X227+X228+X229+X230=578

X31+X41+X32+X42+X33+X43+X34+X44+X35+X45+X36+X46+X37+X47+X38+X48+X39+X49+X310+X410+X311+X411+X312+X412+X313+X413+X314+X414+X315+X415+X316+X416+X317+X417+X318+X418+X319+X419+X320+X420+X321+X421+X322+X422+X323+X423+X324+X424+X325+X425+X326+X426+X327+X427+X328+X428+X329+X429+X330+X430=766.

Demand constraints

X11+X21+X31+X41=18

X12+X22+X32+X42=47

X13+X23+X33+X43=10

X14+X24+X34+X44=12

X15+X25+X35+X45=10

X16+X26+X36+X46=74

X17+X27+X37+X47=50

X18+X28+X38+X48=128

X19+X29+X39+X49=25

X110+X210+X310+X410=83

X111+X211+X311+X411=43

X112+X212+X312+X412=40

X113+X213+X313+X413=112

X114+X214+X314+X414=231

X115+X215+X315+X415=65

X116+X216+X316+X416=27

X117+X217+X317+X417=16

X118+X218+X318+X418=73

X119+X219+X319+X419=56

X120+X220+X320+X420=88

X121+X221+X321+X421=29

X122+X222+X322+X422=21

X123+X223+X323+X423=38

X124+X224+X324+X424=42

X125+X225+X325+X425=41

X126+X226+X326+X426=49

X127+X227+X327+X427=18

X128+X228+X328+X428=116

X129+X229+X329+X429=30

X130+X230+X330+X430=22.

Once the objective function and the respective supply and demand restrictions for each of the models were proposed, they were resolved using the Linear, Interactive, and Discrete Optimizer package (Lindo).

Results and discussion

In 2015, the national production of cocoa in Mexico was 28 006.59 t (^{SIAP, 2018}), exports were of 133.83 t and imports totaled 23 521.37 t (^{SIAVI, 2018}), so the national apparent consumption was 51 394.13 t. The total population of Mexico, in that year, was 119 938 472 inhabitants (^{INEGI, 2018}), therefore, the apparent per capita consumption of cocoa in the country was 0.43 kg, this figure coincides with that presented in the Agroalimentary Atlas 2016 of the SIAP (^{SAGARPA, 2016}).

Given that cocoa, in 2015, was only produced in the states of Tabasco, Chiapas and Guerrero, and in the latter the production was not enough to supply the state consumption of this crop, they are considered as national supplying states only to Tabasco and Chiapas, with a volume of 17 363.57 and 7 143.91 t respectively, once their domestic demand has been met. Table 1 shows the calculation of the deficit or surplus of cocoa of each state in tons of product, also has a column that was called equivalence that refers to the number of trucks that supply or demand each state. The total amount of trucks in surplus was 848, while in deficit were 1 614 trucks, so it was necessary to import 766 trucks to fulfill the feasible solution property in programming.

Elaboration with data from the SIAP and INEGI.

Under conditions of a closed economy, production and distribution must be efficient for the country, but this does not guarantee that the entire product is consumed, in that case there is a excess of production that can be subjected to an agro-industrial process to obtain by-products; or, export it through international trade (^{Ayllón et al., 2015}). In the case of cocoa, production does not supply domestic demand, therefore, international trade is also used, but in the sense of acquiring abroad the quantity of product necessary to cover domestic demand.

In the Table 2 shows the transportation costs of the two national origins, considering the place of production and two international origins, considering two possible ports of entry for international cocoa.

Elaboration with data from GlobalMap.

The total number of trucks that must be distributed nationwide to cover the apparent demand of all states is 1 614; however, in the case of the analysis of a closed economy, the available number of trucks in Mexico in 2015 was 848, so there is a deficit of 766 trucks (Table 3). According to the results of the programming carried out to reduce the transportation costs of cocoa, considering only the national production, the value of the objective function is $9 500 068.00.

Elaboration based on the results of the Lindo program.

According to the results of the programming carried out to reduce the transportation costs of cocoa, considering an open economy, the value of the objective function is $18 123 640. Table 4 shows what the optimal distribution would be by reducing the costs of transport.

Elaboration based on the results of the Lindo program.

The reduced costs show all the routes that were not selected by the model in the optimal solution, whose value is different from zero. The interpretation of these values indicated that by introducing these routes the value of the objective function would increase. For example, transporting a truck with cocoa from Chiapas to Aguascalientes (X11) would raise the value of the objective function by $2 789.00 pesos, in the same way, forcing the model to transport a cocoa truck from Chiapas to Baja California (X12) would raise the value of the objective function in $30 016.00 pesos.

Conclusions

The national production of cocoa in Mexico in 2015 was insufficient to satisfy domestic consumption, so imports of this product were made, the national apparent consumption was 51.3 thousand tons. Under the system of closed economy, it will only be able to distribute 848 trucks, so minimizing transport costs only the states of central-south of the country could fully meet their demand for this grain, while Michoacán can get 14 of the 65 truck that demand. The optimal cost of transportation with a closed economy was $9.5 million pesos.

In order to cover the national demand for cocoa, it is necessary to resort to foreign markets, which is why the port of Lazaro Cardenas, Jalisco, turned out to be a viable option, while Puerto Progreso, Yucatán was only feasible to supply the demand of its state; therefore, it is not a viable option for the importation of cocoa. The optimum cost of the market with an open economy was $18.1 million pesos.

Finally, it is recommended to carry out the transport model considering the agroindustrial demand of this crop in each state since, being a crop that is part of the chocolate production chain, the industry has great weight in the demand for it. In addition, a new model can be run considering other ports in the Gulf of Mexico.