# ABSTRACT

The current pricing process of raw milk in Brazil discourages producers from improving milk composition, which affects both yield and quality of dairy products. Furthermore, small and medium-sized dairies face great difficulties when it comes to planning production. Thus, a linear programming model was developed to price the raw milk and determine the optimal mix (combination of quantities) of dairy products that maximizes total contribution margin (TCM) under daily scenarios of high (January) and low (July) raw milk supplies (summer and winter, respectively) by comparing optimal solutions with actual results. The TCM of optimal and actual mixes were higher in January due to the greater availability of raw material. Packaging was a limiting factor in the production of cheese in optimal mixes. The relationship between unit contribution margin (UCM) and the required amount of raw materials per product unit and resource availability is crucial to defining the mix of dairy products and TCM of the dairy. Casein and raw milk volume showed shadow prices. Under both scenarios, the calculated prices of raw milk were higher than the prices charged by the dairy and were higher in January. The proposed model remunerates the producers based on the quantity and quality of raw milk. The dairy can maximize its TCM by better planning its mix of products with the use of linear programming.

Keywords:
contribution margin; dairy products; linear programming; milk components; milk pricing

# Introduction

Brazilian dairies face two very usual problems: one is related to the quality of its main raw material, milk, and the other to production planning of the mix (combination of quantities) of dairy products, such as cheeses, milk, cream, etc. for sale.

Raw milk quality affects industrial yield, milk processing, dairy product quality and profitability of producers and dairies (Blowey and Edmondson, 2010Blowey, R.W.; Edmondson, P. 2010. Mastitis Control in Dairy Herds. 2ed. CABI, Wallingford, UK.). A number of Brazilian dairies currently have in place compensation systems to producers with awards and penalties assigned to the liter price (L) of raw milk based on quality criteria, such as Total Bacterial Count (TBC), Somatic Cell Count (SCC), protein and fat contents, as an incentive for milk quality improvement (Roma et al., 2009Roma, L.C.; Montoya, J.F.G.; Martins, T.T.; Cassoli, L.D.; Machado, P. F. 2009. Seasonability of protein and other milk components related with quality payment program. Arquivo Brasileiro de Medicina Veterinaria e Zootecnia 61: 1411–1418 (in Portuguese, with abstract in English).). However, these programs have not increased the protein and fat content of raw milk supplied to dairies (Botaro et al., 2013Botaro, B.G.; Gameiro, A.H.; Santos, M.V. 2013. Quality based payment program and milk quality in dairy cooperatives of southern Brazil: an econometric analysis. Scientia Agricola 70: 21–26.).

Programming the mix of dairy products that maximizes the total contribution margin (TCM), i.e. the difference between revenue and direct costs, is a major challenge faced by dairies, mainly small and medium-sized ones (Kerrigan and Norback, 1986Kerrigan, G.L.; Norback, J.P. 1986. Linear-programming in the allocation of milk resources for cheese making. Journal of Dairy Science 69: 1432–1440.). This challenge is attributed to milk production seasonality, milk quality variability, low qualification of staff in charge of dairy production planning, failures in production planning, lack of awareness of decision-support tools, the absence or adoption of rudimentary practices of accounting controls and the lack of familiarity with technical coefficients and indicators of industrial efficiency related to the use of equipment, structure, material and human resources of the dairy processing plant (Banaszewska et al., 2013Banaszewska, A.; Cruijssen, F.; Vorst, J. van der; Claassen, G.; Kampman, J. 2013. A comprehensive dairy valorization model. Journal of Dairy Science 96: 761–779.).

Defining a more profitable mix becomes very difficult without support from mathematical and computational tools for decision making when planning the production of dairy products, since this task requires calculations that involve many technical and economic factors (Geary et al., 2010aGeary, U.; Shalloo, L.; Lopez, N. 2010a. Development of a dairy processing sector model for the Irish dairy industry. Advances in Animal Biosciences 1: 335–335.). However, computer programs based on linear programming (LP) are available and can resolve the issues regarding the definition of the optimal mix to maximize the TCM in a simplified form which respects operational and market constraints. LP is a mathematical technique for solving systems of linear equations or inequations by inverting matrices, being a mathematical model used for more efficient allocation of resources (Dantzig, 1998Dantzig, G.B. 1998. Linear Programming and Extensions. Princeton University Press, Princeton, NJ, USA.; Sharma, 2006Sharma, S. 2006. Introductory Operation Research. Discovery Publishing House, Grand Rapids, MI, USA.).

A model that supports strategic decisions and milk pricing strategies would be helpful in determining the mix of milk products and the characteristics of the raw milk supplied (Geary et al., 2010bGeary, U.; Lopez-Villalobos, N.; Garrick, D.; Shalloo, L. 2010b. Development and application of a processing model for the Irish dairy industry. Journal of Dairy Science 93: 5091–5100.). To fulfill these needs, we developed a mathematical model for small-sized dairies to price raw milk in terms of its quality and to maximize TCM through the definition of the optimal mix of products using LP.

# Materials and methods

A mathematical model was developed to maximize total contribution margin (TCM) of dairies and to determine the maximum price per liter (L) of raw milk supplied by dairy farmers. The model was developed and tested using technical and economic data from a small-sized dairy in the municipality of Piracicaba, in the state of São Paulo, Brazil. Technical and accounting coefficients were calculated as below.

The total contribution margin and maximum price per volume (L) of raw milk were calculated by definitions of the optimal mix (ideal combination of quantities) of dairy products ‘i’ and of the shadow price of the limiting components of raw milk under constraints in two daily scenarios, respectively. The shadow price can be interpreted as the marginal profit that would be obtained if another unit of the limiting resource were processed. Therefore, one can understand that the shadow price of the limiting resource is the maximum price that can be paid for such resource without making a loss.

The model was processed by means of the GAMS 24.0 computational program (General Algebraic Modeling System 24.0) using the Cplex solver version 12.5.0.0 (McCarl et al., 2013McCarl, B.A.; Meeraus, A.; Eijk, P. van der; Bussieck, M.; Dirkse, S.; Steacy, P.; Nelissen, F. 2013. McCarl GAMS User Guide. GAMS Development Corporation, Washington, DC, USA.) under both scenarios (Figure 1).

Figure 1
Flowchart of summarized processes adopted in the research methodology.

## Scenarios

We defined two daily scenarios we named ‘j’ corresponding to the months of 2011 with high (January) and low (July) raw milk supplies (in liters) with different average compositions (%), simulating the peak (summer) and off-season (winter) periods of raw milk production, respectively (Table 1).

Table 1
Average volumes and contents of the components of two types of raw milk obtained by the dairy under daily scenarios of high (January) and low (July) raw milk supplies.

## Objective function

The objective function of the model is to maximize the TCM of the dairy in each daily scenario ‘j’ by adding the products of multiplications of decision variables (dairy products ‘i’) to the respective contribution margins per unit (UCM) (Equation 1).

where: TCMj = total contribution margin (US$of ‘j’ scenario, Xij = quantity (kg) or volume (L) of the dairy product ‘i’ in the ‘j’ scenario and UCMij = contribution margin per unit (US$) of the dairy product ‘i’ in the ‘j’ scenario.

The contribution margin per unit is the diffierence between the selling price and the direct cost per unit of each dairy product. Raw milk costs were not included as direct costs in order to determine the shadow prices per volume (L) of raw milk or per mass (kg) of its components. Interest, taxes, depreciation and amortization were also not considered. The selling prices of dairy products were fixed under both scenarios in order to evaluate the effect of differences in the quantity and quality of raw milk only on TCMs and shadow prices of the limiting resources of the dairy under both scenarios.

## Decision variables

The 10 dairy products produced by the dairy denominated as ‘i’ consisted of two brands of milk (brand X cooled standardized pasteurized milk with 3 % fat (XM) and brand Y – a cooled standardized pasteurized milk with 3 % fat (YM)), milk cream with 40 % fat (C40) from XM and YM standardization and seven kinds of cheese (Minas Standard (MS), Mozzarella Cheese (MC), Mozzarella Ball (MB), Minas Fresh (MF), Provolone Ball (PB), Spicy Provolone (SP) and Ricotta Cheese (RC)). A single producer provided the raw milk (RM1) for XM production daily while approximately 50 producers supplied raw milk (RM50) for processing the other dairy products. The milk cream was sold to other food mills while the other nine dairy products were sold to retail customers from neighboring counties (bakeries, grocery stores and small markets).

## Restrictions and equations

The maximization of the dairy's TCM was subject to technical, market and resources constraints in both scenarios. We calculated daily average quantities (kg) available in fat, protein and casein according to daily average volumes (L) and contents (%) of daily components of RM1 and RM50 types of raw milk (Table 2), according to the following equations:

where PRMj = quantity (kg) of raw milk protein in the ‘j’ scenario, VRMj = average volume (L) of raw milk collected in scenario ‘j’, % PRMj = average protein content (%) of raw milk in the ‘j’ scenario, 1.03 = average density of raw milk (kg L−1), CRMj = quantity (kg) of raw milk casein in the ‘j’ scenario, % CRMj = average casein content (%) of raw milk in the ‘j’ scenario, FRMj = quantity (kg) of raw milk fat in the ‘j’ scenario and % FRMj = average fat content (%) of raw milk in the ‘j’ scenario.

Table 2
Average constraints of volume and components of two types of raw milk obtained by the dairy in daily scenarios of high (January) and low (July) raw milk supplies.

The whey volume generated in production of each type of cheese ‘i’ was determined under both scenarios ‘j’ by the following equation:

where: VWHQij = whey volume (L) generated per quantity (kg) of cheese ‘i’ produced in the ‘j’ scenario, 1.03 = average density (kg L−1) of raw milk, VRM50Qij = volume (L) of RM50 required per kg of cheese ‘i’ (L kg−1) in the ‘j’ scenario, Qij = quantity (kg) of cheese ‘i’ produced in the ‘j’ scenario and 1.025 = average density (kg L−1) of whey. The sum of VWHQij corresponds to the total volume (L) of whey from cheeses ‘i’ available for ricotta production under the ‘j’ scenario.

The quantities (kg) available of fat and whey protein were calculated according to volumes (L) and average compositions (%) of whey (Teixeira and Fonseca, 2008Teixeira, L.V.; Fonseca, L.M. 2008. Physical-chemical profile of whey from Mozzarella and Minas-Standard cheesemaking from various regions of the state of Minas Gerais. Arquivo Brasileiro de Medicina Veterinária e Zootecnia 60: 8 (in Portuguese, with abstract in English).), according to the following equations:

where: KPSj = mass (kg) of whey protein under the ‘j’ scenario, ΣVWHQij = sum of whey volumes (L) in cheese production ‘i’ in the ‘j’ scenario, 0.82 = average content (%) of whey protein, 1.025 = average density (kg L−1) of whey, KFSj = fat quantity (kg) in whey under the ‘j’ scenario and 0.725 = average fat content (%) in whey.

Beyond the restrictions of raw materials in Table 2 (milk components), the optimal solution of the mix of dairy products to maximize the TCM should respect the restrictions of daily demand from retail customers for dairy products (Table 3), equipment capacities and available working hours of the labor force.

Table 3
Maximum and minimum demands of each dairy product produced by the dairy in daily scenarios of high (January) and low (July) raw milk supplies.

The dairy sold its products to other food mills and retail customers from neighboring counties (bakeries, grocery stores and small markets). In both scenarios, the dairy had to meet the minimum demands but operate within the maximum sales limit of their products. Minimum demand (DMin) for dairy XM, YM and MC corresponded to orders and contractual requirements of customers, while the minimum demands of other dairy products were due to the strategy of positioning and maintenance of the company's brand in the market. Maximum demand (DMax) for dairy products corresponded to the maximum sales recorded in both January and July. The demand for each product was represented by an interval based on a quantity representative of the dairy plant's sales. This option to represent demand brings flexibility to the model proposed for calculating the best production mix designed to maximize the TCM of the dairy given its likely actual demand reality. Every dairy product produced was considered to have been sold as there were no records of returns of products not sold by retailers to end of chain consumers in either January or July. The restriction imposed on demand is represented by the following mathematical expression:

where: DMinXij = minimum demand for dairy product ‘i’ under the ‘j’ scenario, MaxXij = maximum demand for dairy product ‘i’ under the ‘j’ scenario. or volume (L) of the dairy product ‘i’ under the ‘j’ scenario and D

The dairy had six employees working directly on the production line. Each one worked 6.29 h daily (44 h per week) totaling 37.71 daily man-hours. The time available for daily use of each equipment corresponded to a workday of 6.29 hours.

# Results and discussion

## Dairy product mixes and total contribution margins

The optimal solutions, as well as the actual mixes, were obtained for models processed in accordance with the constraints of both scenarios (Table 6). Under the scenario of high raw milk supply (January), great differences were seen between the optimal and actual mixes. In the actual mix, the raw milk was used for processing eight dairy products (XM, YM, MC, MF, PB, SP, RC and C40), while in the optimal mix, all the 10 dairy products would be processed to meet minimum demand. The differences between the shares of these products in the optimal and actual mixes ranged from 0.43 (RC) to 3228.15 kg (YM) in absolute values, and from 8 % (RC) to 100 % (MS and MB) in relative values. The cooled pasteurized milk had a higher share in both mixes. As expected, the TCM was higher in the optimal solution than in the actual one by US$2021.10 (34 %). Table 6 Combinations and contribution margins of optimal and actual mixes of dairy products produced in daily scenarios of high (January) and low (July) raw milk supplies. Under the scenario of low raw milk supply (July), despite all dairy products participating in both mixes, there were also differences between the optimal and actual mixes. The differences in the shares of XM (4 %), YM (12 %), PB (1 %), SP (0 %) and RC (6 %) between the optimal and actual mixes were very small. The greatest differences in share between the two mixes were found in MS (97 %), MC (33 %), MB (68 %) and MF (69 %). The cooled pasteurized milk had a higher share in both mixes, as well as under the January scenario. As expected, TCM was higher in the optimal solution than in the actual one by US$ 224.08 (5 %). The TCM in both mixes were higher in January due to the greater availability of components from two raw milk types for processing dairy products during this period, demonstrating the importance of raw milk volume and quality to the TCM of dairies.

Compared to XM and YM products, the processed volumes indicated in the optimal solution were higher in January due to the greater availability of RM1 and RM50 in this period, respectively. Because of this difference in availability of RM50 for processing YM and cheeses, the relative YM participation in the optimal mix of July was lower than that of January. On the other hand, the XM participation in the optimal mix of July was higher than that of January, because there was little difference in RM1 supplied to produce XM between both scenarios. Therefore, XM production decreased slightly compared to that of YM.

Although the optimal mix of January showed greater XM and YM production, a higher volume of C40 was produced in July due to higher fat content in both raw milk types in this scenario. Interestingly, C40 generation in the actual mix for both months was identical. Consequently, the relative C40 share in both mixes rose in July.

The four products (SP, PB, MB and MS) with higher UCM, plus the RC and C40 products, showed the smallest shares in the mixes under both scenarios. This is explained by the result of dividing UCM by the required amount of raw material (raw milk and its components) for processing one L or kg of each product (Table 7).

Table 7
Relationship between the UCM and the required quantities of raw materials (raw milk volume and its components) of dairy products in daily scenarios of high (January) and low (July) raw milk supplies.

Although the UCM of three (XM, YM and MF) of the four highest participating products in the optimal mixes of both scenarios are within the six lowest for the dairy product line, these products, except for C40, presented the highest UCM per unit of raw material (US$L−1 or US$ kg−1) under both scenarios (Table 7). Thus, the manufacturing and marketing of products with greater UCM by limiting factor must be prioritized to optimize the use of critical raw materials as limiting factors. This means that producing more products with lower UCM can be more profitable than producing a product with higher UCM in smaller quantities. Therefore, the relationship between UCM and amounts of required raw materials per unit of product was decisive in defining the optimal mixes. The third largest share of the MC product in optimal mixes of both scenarios was due to the need to meet the minimum demands of this product because its UCM per unit of raw material was the lowest.

The UCM per unit of raw materials was higher for the cheese line in July due to a lower amount of raw material required for processing a single unit of milk product. However, for XM and YM milks, it occurred otherwise, because both types of raw milk had higher levels of solid components, including fat, in July than in January. Therefore, the raw milk loss in the standardization of fat XM and YM milks in July was larger.

The XM milk would be included in the optimal mixes regardless of the relationship between UCM and the requirement for raw material, because it does not the raw milk from a single supplier, is exclusively used for its processing. On the other hand, the other dairy products are processed with raw milk from the other 50 milk suppliers. Therefore, these products compete for raw materials to participate in the mixes.

In the optimal mixes of both scenarios, the YM milk participates only because of its restriction of MF cheese production by the packager, since this product was given preference for participation in the mixes due to its higher UCM per the required quantity of raw material.

Under both scenarios, the quantities of dairy products MS, MC, MB, PB, SP and RC determined by the optimal solution, were equal to the quantities defined by the minimum demand. These products were included in the production mix just to meet the minimum demand constraints. If the minimum demand constraints do not exist, these products would not be included in the production mix as they have UCM per unit of raw material lower than those of the other participants in the dairy production mix (Table 7). As all types of cheese competed for packaging, this was the limiting factor in MF cheese production in the optimal mixes (Table 8).

Table 8
Balances and shadow prices of resources of the dairy in daily scenarios of high (January) and low (July) raw milk supplies.

The C40 creams were included in the mixes because they are intermediate products obtained from the standardization of XM and YM pasteurized milks. In dairies, whose production processes a wide variety of products that may generate intermediate products, such as cream and whey, which are raw materials for other products, optimization techniques have already been efficiently deployed (Banaszewska et al., 2013Banaszewska, A.; Cruijssen, F.; Vorst, J. van der; Claassen, G.; Kampman, J. 2013. A comprehensive dairy valorization model. Journal of Dairy Science 96: 761–779.; Geary et al., 2010aGeary, U.; Shalloo, L.; Lopez, N. 2010a. Development of a dairy processing sector model for the Irish dairy industry. Advances in Animal Biosciences 1: 335–335.; Geary et al., 2010bGeary, U.; Lopez-Villalobos, N.; Garrick, D.; Shalloo, L. 2010b. Development and application of a processing model for the Irish dairy industry. Journal of Dairy Science 93: 5091–5100.; Geary et al., 2012Geary, U.; Lopez-Villalobos, N.;Garrick, D.; Shalloo, L. 2012. An analysis of the implications of a change to the seasonal milk supply profile in the Irish dairy industry utilizing a seasonal processing sector model. Journal of Agricultural Science 150: 389–407.; Guan and Philpott, 2011Guan, Z.; Philpott, A. 2011. A multistage stochastic programming model for the New Zealand dairy industry. International Journal of Production Economics 134: 289–299.; Milad and Faezeh, 2012Milad, G.A.; Faezeh, M. 2012. Considering different types of constraints for production planning for a dairy product: a case study. The Journal of Commerce 4: 4.). This optimization relies heavily on the production process, together with marketing and management of raw materials – raw milk which is considered a constraint, because of its irregular availability throughout the year given the seasonality of its production (Banaszewska et al., 2013Banaszewska, A.; Cruijssen, F.; Vorst, J. van der; Claassen, G.; Kampman, J. 2013. A comprehensive dairy valorization model. Journal of Dairy Science 96: 761–779.; Brockington et al., 1992Brockington, N.R.; Veil, J.M.; Zoccal, R. 1992. Herd dynamics and management strategies for small-scale milk production systems in southeast Brazil. Agricultural Systems 39: 201–225.; Geary et al., 2012Geary, U.; Lopez-Villalobos, N.;Garrick, D.; Shalloo, L. 2012. An analysis of the implications of a change to the seasonal milk supply profile in the Irish dairy industry utilizing a seasonal processing sector model. Journal of Agricultural Science 150: 389–407.;Roma et al., 2009Roma, L.C.; Montoya, J.F.G.; Martins, T.T.; Cassoli, L.D.; Machado, P. F. 2009. Seasonability of protein and other milk components related with quality payment program. Arquivo Brasileiro de Medicina Veterinaria e Zootecnia 61: 1411–1418 (in Portuguese, with abstract in English).).

All dairy products produced were deemed to be sold because there were no records of product returns in either January or July. This fact is due to increasing urban demand for dairy products in Brazil where 80 % of the fluid milk, condensed milk and cream are sold in supermarkets. In the state of São Paulo the retail sector is even more important, reaching 85 % of sales of these products (Novo et al., 2010Novo, A.; Jansen, K.; Slingerland, M.; Giller, K. 2010. Biofuel, dairy production and beef in Brazil: competing claims on land use in São Paulo state. The Journal of Peasant Studies 37: 769–792.). Conditional cash transfer programs implemented by the Brazilian Federal Government have contributed to this increasing demand for dairy products. More than 54 % of households included in the program increased consumption of dairy products such as cheese, yogurt, curd and chocolates made with milk (Lignani et al., 2010Lignani, J.B.; Sichieri, R.; Burlandy, L.; Salles-Costa, R. 2010. Changes in food consumption among the Programa Bolsa Família participant families in Brazil. Public Health Nutrition 14: 785–792.).

## Shadow prices of limiting resources

In January, casein in RM1 and RM50 raw milks and packaging were the limiting operating resources, because their supply was fully deployed in the processing of dairy products. Thus, they were also the only resources to show shadow prices. In July, limiting operating resources that presented shadow prices were casein of RM1 raw milk, the liter of RM50 raw milk and packaging (Table 8).

As the fat from both raw milk types was not fully used, it did not show a shadow price, because it did not limit the processing of dairy products. However, if the dairy had fat-based products, such as butter, cream cheese or milk cream in its line of dairy products, fat from milk RM50 may have a shadow value. Other studies (Banks, 2004Banks, J.M. 2004. The technology of low-fat cheese manufacture. International Journal of Dairy Technology 57: 199–207.; Cash et al., 2005Cash, S.B.; Wang, C.; Goddard, E.W. 2005. Dairy products and consumer demand for health foods. Advances in Dairy Technology 17: 67–80.; Davis et al., 2012Davis, C.G.; Dong, D.; Blayney, D.; Yen, S.T.; Stillman, R. 2012. U.S. fluid milk demand: a disaggregated approach. International Food and Agribusiness Management Review 15: 25–50.; FAO, 2013Food and Agriculture Organization [FAO]. 2013. Milk and Dairy Products in Human Nutrition. FAO, Rome, Italy.; Gruebele, 1978Gruebele, J.W. 1978. Effects of removing the dairy price-support program. Illinois Agricultural Economics 18: 30–38.; Hill et al., 2002Hill, D.S.; Knox, J.; Hamilton, J.; Parr, H.; Stringer, M. 2002. Reduced-fat foods: the shopper´s viewpoint. International Journal of Consumer Studies 26: 44–57.; Sandrou and Arvanitoyannis, 2000Sandrou, D.K.; Arvanitoyannis, I.S. 2000. Low-fat/calorie foods: current state and perspectives. Critical Reviews in Food Science and Nutrition 40: 427–447.; Zandstra et al., 2001Zandstra, E.H.; Graaf, C.; Staveren, W.A. van. 2001. Influence of health and taste attitudes on comsumption of low- and high-fat foods. Food Quality and Preference 12: 75–82.) also demonstrated fat devaluation relative to other raw milk components due to changes in consumers’ eating habits. The value of milk fat may decrease as a result of lower demand for higher-fat foods, such as butter, to the detriment of increased consumer preference for lower-fat foods, such as “dairy spreads” and other low-fat dairy products, rich in protein (Banks, 2004Banks, J.M. 2004. The technology of low-fat cheese manufacture. International Journal of Dairy Technology 57: 199–207.;Cash et al., 2005Cash, S.B.; Wang, C.; Goddard, E.W. 2005. Dairy products and consumer demand for health foods. Advances in Dairy Technology 17: 67–80.; Davis et al., 2012Davis, C.G.; Dong, D.; Blayney, D.; Yen, S.T.; Stillman, R. 2012. U.S. fluid milk demand: a disaggregated approach. International Food and Agribusiness Management Review 15: 25–50.; FAO, 2013Food and Agriculture Organization [FAO]. 2013. Milk and Dairy Products in Human Nutrition. FAO, Rome, Italy.; Gruebele, 1978Gruebele, J.W. 1978. Effects of removing the dairy price-support program. Illinois Agricultural Economics 18: 30–38.; Hill et al., 2002Hill, D.S.; Knox, J.; Hamilton, J.; Parr, H.; Stringer, M. 2002. Reduced-fat foods: the shopper´s viewpoint. International Journal of Consumer Studies 26: 44–57.; Sandrou and Arvanitoyannis, 2000Sandrou, D.K.; Arvanitoyannis, I.S. 2000. Low-fat/calorie foods: current state and perspectives. Critical Reviews in Food Science and Nutrition 40: 427–447.; Zandstra et al., 2001Zandstra, E.H.; Graaf, C.; Staveren, W.A. van. 2001. Influence of health and taste attitudes on comsumption of low- and high-fat foods. Food Quality and Preference 12: 75–82.).

A study on consumer demand for dairy products in the region of Boston, USA, indicates that there is a preference for fat-containing conventional milk, especially in households with children under 15 years of age, and the preference increases with an increased number of children in that age group. On the other hand, the higher the household income, the greater the preference for special types of milk, such as organic milk, lactose-free milk and, above all, low-fat milk (Lopez and Lopez, 2009Lopez, E.; Lopez, R.A. 2009. Demand for differentiated milk products: implications for price competition. Agribusiness 25: 453–465.).

In California, for example, the change of consumers’ preference from whole milk to skimmed milk was one of the reasons to establish the payment of milk components. This trend in consumers’ eating habits leads the dairy industry to invest in increased production of protein-rich dairy products, causing appreciation of this nutrient in relation to other solid milk components, and the payment system should reflect this valuation (Gruebele, 1978Gruebele, J.W. 1978. Effects of removing the dairy price-support program. Illinois Agricultural Economics 18: 30–38., 1979Gruebele, J.W. 1979. The California experience in component pricing. Journal of Dairy Science 62: 1368–1373.).

## Raw milk prices and EBTIDA

The shadow prices of components of both raw milk types, the TCM and fixed and indirect costs of an L of raw milk allowed for the calculation of prices for a L of both raw milk types as well as optimal and actual EBITDA of the dairy (Table 9).

Table 9
Total contribution margins, average weighted indirect costs per liter of raw milk, maximum, actual and intermediates prices of the two types of raw milk and EBITDA from the dairy under the daily scenarios of high (January) and low (July) raw milk supplies.

The TCM was higher in January due to the greater volume of milk available for the production of dairy products. This result was expected because January and July are the peak and off-season periods, respectively, in southeastern Brazil. Therefore, the indirect and fixed costs for each L of raw milk processed were lower in January, because the higher the volume of raw milk available for processing by the dairy, the greater the dilution of fixed and indirect costs for the volume of processed milk. Under both scenarios, as expected, the optimal TCM was higher than the actual one due to a better allocation of resources for products with higher UCM per unit of active limiting feature.

Applying the average indirect costs of a litter of raw milk and shadow prices of components from the two types of raw milk to Equation 16, formulated for calculating the maximum price of 1 L of raw milk, we obtained maximum raw milk prices higher than those actually realized by the dairy. The maximum price of RM1 raw milk was higher than the RM50 raw milk maximum price under both scenarios. Prices calculated for both raw milk types were higher in January, contrary to reality where the highest prices are realized during the off-season (July scenario). This occurred as the UCM per quantity of limiting resource required per dairy product was higher in the January scenario.

Each dairy determines its own criteria for bonuses or penalties depending on the quality of the milk (TBC and SCC, besides solid milk compounds). Law No. 12669 has been in effect since 19 June 2012, which obliges dairies to inform the price of a L of raw milk to suppliers by the 25thday of the month preceding milk purchase (Brazil, 2012). Every month, Conseleite (the Brazilian Joint Council of Producers and Milk Processors) calculates the reference price for standard raw milk depending on the mix of dairy products, yield of industrial products, production costs and raw milk quality (Canziani and Guimarães, 2003Canziani, J.R.; Guimarães, V.D.A. 2003. Instruction Manual Conseleite – Paraná = Manual de Instruções do Conseleite -Paraná. Centro de Documentação, Informações Técnicas e Biblioteca do SENAR, Curitiba, PR, Brazil (in Portuguese).). It is more advantageous for the dairy to remunerate the raw milk supplier for a raw material that improves production efficiency and yield (Geary et al., 2010bGeary, U.; Lopez-Villalobos, N.; Garrick, D.; Shalloo, L. 2010b. Development and application of a processing model for the Irish dairy industry. Journal of Dairy Science 93: 5091–5100.). However, there is no Brazilian program for pricing a kg of solid milk components, as proposed in this study.

We obtained the optimal and actual EBITDA through the application of the values from Table 9 to Equations 17 and 18. The actual daily EBITDA was higher than the optimal daily EBITDA under both scenarios because actual prices were lower than the maximum prices calculated for a L of both raw milk types. Still, the results of the optimal daily EBITDA in both scenarios demonstrate that it is possible to make a profit even when paying more for raw materials.

However, the dairy can get the same actual EBITDAs from January and July by producing optimal mixes for both scenarios without the need for extra purchase cost of raw material. For this, it is necessary to calculate the intermediate average price of one L of raw milk that the dairy could pay its suppliers in January and July. Equating equation 18 of optimal EBITDA to the values obtained in the actual EBITDAs for January (US$1316.43) and July (US$ 971.37), respectively, and isolating the response variables (intermediate average price of one L of milk raw of both scenarios) of the equations, it was possible to determine their values. The price of raw milk would amount to US$0.60 L−1 and US$ 0.52 L−1 in January and July, respectively, to get the same actual EBITDAs with optimal mixes without additional acquisition costs of raw materials.

We calculated maximum prices of RM50 from two suppliers of the dairy producing raw milk in different quantities and qualities under both scenarios (Table 10). The maximum price per liter of raw milk to be paid by dairy to both producers was higher in January. Although Producer 2 supplied higher raw milk volume to the dairy than Producer 1, the latter produced raw milk with higher solid contents, including casein, whose shadow price was US$33.44 kg−1, consequently, its raw milk had a higher price under both scenarios. In July, the maximum price per liter of raw milk was the same for both producers because its value was determined by the volume shadow price of RM50 (US$ 0.88 L−1), less the indirect cost (US$0.14 L−1 raw milk) under this scenario. Table 10 Volumes, compositions, maximum prices and total revenue of two raw milk suppliers of the dairy under daily scenarios of high (January) and low (July) raw milk supplies. # Conclusions Linear programming is a useful tool for dairies for production planning, for maximizing TCM which should be incorporated into raw milk pricing and consider shadow prices of milk limiting components. The relationship between UCM and the required amount of raw material per product unit and resource availability are crucial to defining the mix of dairy products and the TCM of the dairy. Raw milk pricing in the proposed model remunerates the dairy farmer based on the raw milk components produced. The dairy can increase its EBITDA and remunerate better its suppliers based on raw milk quality with better planning of its product mix. Although the data used are from a Brazilian dairy that produces regional cheeses, the optimization model developed in linear programming can be applied to any dairy with appropriate adaptations throughout the world. In addition, the portfolio of dairy products herein is not restricted to regional cheeses; it also includes Mozzarella, Provolone and Ricotta, which are produced worldwide. # Acknowledgments The authors wish to thank the CAPES agency (Coordination for the Improvement of Higher Level Personnel) for granting a PhD scholarship to the first author and the Milk Clinic Lab (Laboratory of the Animal Science Department of the College of Agriculture “Luiz de Queiroz” of University of São Paulo accredited by Brazilian Milk Quality Analysis Laboratory Network) for providing data on raw milk composition purchased by the dairy. # References • Anderson, D.L.; Mistry, V.V.; Brandsma, R.L.; Baldwin, K.A. 1993. Reduced fat cheddar cheese from condensed milk. 1. Manufacture, composition, and ripening. Journal of Dairy Science 76: 2832–2844. • Andreatta, E.; Santos, M.V.; Mussarelli C.; Marques, M.C.; Gigante, M.L. 2009. Quality of minas frescal cheese prepared from milk with different somatic cell counts. Pesquisa Agropecuária Brasileira 44: 320–326. • Applied Economic Research Institute [IPEA]. 2013. Brazil Central Bank. Balance of Payments Bulletin. Commercial exchange rate for sale: real (R$) / US dollar (US$) – 2011 annual average = Banco Central do Brasil. Boletim de Balanço de Pagamentos. Taxa de câmbio comercial para venda: real (R$) / dólar americano (US\$) – média anual de 2011. Available at: http://www.ipeadata.gov.br/ [Accessed Mar 26, 2013] (in Portuguese).
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# Publication Dates

• Publication in this collection
Jan-Feb 2016