An approach for a sustainable decision-making in product portfolio design of dairy supply chain in terms of environmental, economic and social criteria

The production of dairy products is related to water and energy costs and generation of large amounts of emissions of pollutants. Full sustainability of these systems can be achieved by optimizing all activities in the supply chain (SC) taking into account not only the environmental and economic aspects, but also the social ones. This study proposes a multi-objective modeling approach for optimal design of three-echelon SC for production of dairy products according to different recipes while satisfying environmental, economic and social criteria defined in terms of costs. The environmental costs are associated with the generated wastewater from dairy production and CO2 emissions due to energy consumed and transport of raw materials and products. The social ones are related to the employees hired for implementation of the SC activities. It was implemented on a real case study from Bulgaria. Four mix integer nonlinear programming optimization models were defined–one without and three with social impact consideration. They were solved at different values of the environmental and social constraints. The obtain results showed that stricter environmental constraints lead to higher economic costs and lower profit. Conversely, higher environmental constraints result in higher profit and lower economic costs. The greatest share in the environmental impact has the wastewater generated, followed by CO2 emissions related with energy consumed and CO2 emissions due to transport. The obtained solutions can be used in the decision-making process in terms of seeking a trade-off between profit, environmental and social impact.


Latin symbols AQ s
Average quantities of raw materials that employees can process per day in suppliers (kg) AQ p Average quantities of products that employees can process per day in dairies (kg) AQ m Average quantities of products that employees can process per day in markets (kg) BOD 5 Biochemical Oxygen Demand during 5 days (kg O 2 ) BOD Cu BOD load related to losses of curd (kg O 2 /kg curd) BOD M BOD load related to losses of skimmed or standardized whole milk (kg O 2 /kg milk) BOD Pa BOD load related to deposits on pasteurizers walls during skimmed milk pasteurization (kg O 2 /kg pasteurized milk) BOD Wh BOD load related to spills of whey produced as by-product during discharging of curd vats (kg O 2 /kg whey) CF Cream fat content of standardized whole milk (%) CMI Costs for medical insurances of employees (BGN) CS Costs for salaries of employees (BGN) CSB Costs for social benefits of employees (BGN) CWCl Costs for working clothes of employees (BGN) EC Energy for cooling skimmed milk in pasteurization process (kWh/kg skimmed milk) ECO 2 Mass of CO 2 emissions associated with kWh energy (kg CO 2 /kWh) EH Energy for heating skimmed milk during pasteurization process (kWh/ kg skimmed milk) EIMCO 2 Environmental impact of CO 2 emissions associated with the energy consumed in the pasteurization process (kg CO 2 /kg curd) F BOD_Cost Total BOD costs paid for treatment of wastewater generated during the production of products (BGN) FC Fuel consumption of the engines of the vehicles used for transportation of raw materials and products (L/100 km) FCO 2 Quantities of CO 2 emissions generated during fuel combustion (kg CO 2 / kWh) F CO2_E_Cost Total CO 2 emissions costs associated with energy consumed by dairy production for realization of the pasteurization process (BGN) F CO2_T_Cost Total costs associated with the CO 2 emissions of pollutants due to transportation of raw materials and products (BGN) FDM Fat in Dry Matter, quality indicator for products (%); FE Energy of fuel used for transportation (kWh/L) F M_Cost Total costs incurred by the dairy complex for purchasing the necessary quantities of milk from suppliers for production of the products (BGN) FP Fuel price (BGN/L) F P_Cost Total production costs for dairy complex (BGN) F Profit Profit of dairy complex obtained after accounting for economic, environmental and social costs (BGN) F R Revenue from the products sale at markets (BGN) F Social_Costs_Suppliers Social costs related with the employees who should be hired by the suppliers (BGN) F Social_Costs_Dairies Social costs related with the employees who should be hired by the diaries (BGN) F Social_Costs_Markets Social costs related with the employees who should be hired by the markets (BGN) F T_Cost Total costs for transportation of raw materials and products between suppliers, dairies and markets (BGN) I Number of dairies L Number of production tasks LS Losses of raw materials, by-products and products (%) LS2 Cu Losses of curd during realization of production task acidification (production task 2, production recipe 1) (%) LS3 Cu Losses of curd during realization of production task draining (production task 3, production recipe 1) (%) LS4 Cu Losses of curd during realization of production draining (production task 4, production recipe 2) (%) LS SM Losses of skimmed milk during realization of milk pasteurization (%) LS Wh Losses of whey during realization of acidification and draining of produced curd (%) LS WM Losses of whole standardized milk related with the pre-processing of milk ( Continuous variable accounting for the used milk for production of 1 kg curd-target product according production recipe 2 (kg) XX Quantities of products produced in dairies and sold at markets (kg) YP Yield of target products (kg) YP1 Yield of curd-raw product (containing residual whey), which is produced according production recipe 2 for production of 1 kg curd-target product

Introduction
The increase in dairy production as a result of population growth leads to increased water and energy costs as well as the generation of GHG emissions and wastewater associated with the products' life cycle in the context of so-called dairy supply chains. This negatively affects energy and water security and results in climate change, competition for natural resources; water, soil and air contamination; increased food costs which arise social concerns (Beck and Walker 2013). All this is a prerequisite for application of approach of an analysis of food-energy-water (FEW) systems nexus and developing supporting mechanisms and methodologies to identify and manage these nexus (Bergendahl et al. 2018) in order to improve the sustainability of the considered supply chains. Their successful implementation can provide a template for other pressing global challenges, including security and health care. Thiyagarajan (2015) has identified a connection between the prices of consumed electricity and water in implementation of activities in the dairy supply chain and the total costs, products demand and the amount of generated GHG emissions. The author has designed a distribution network where distribution centers of milk products are strategically positioned and the corresponding shipping costs and GHG emissions are calculated. The influence of electricity prices and water prices on the electricity and water consumption is observed, and it is seen that there is a noticeable decrease in the demand level as price increases. Santos Junior et al. (2017) have performed a sensitivity analysis including evaluations of different fuels for generating thermal energy, strategies for cleaning of dairy plant and utensils, variations in the way of cheese production based on the fat content and production percentage changes. The results showed that the skimmed milk and thermal energy productions, electricity usage and water consumptions were the main elementary flows. The pallet residues showed the best to be used as fuel for thermal energy. Detergent combinations did not influence the impact categories. Toorn et al. (2020) have investigated the environmental impact of the protein use on material and energy flows in current meat and dairy supply chains in terms of GHG emissions. Furthermore, the environmental impact of adopting low meat and dairy diets has been assessed. The authors have declared that the by-products that were consumed by livestock could substitute other goods leading to further GHG reductions. In contrast, replacing meat and dairy by-products would lead to more GHG emissions. Stanchev et al. (2020) have investigated the potential of anaerobic digestion (AD) of dairy processing effluents close the water, energy and nutrient circular loops together with the relevant environmental costs and benefits at different levels of the dairy supply chain. The developed methodology was based on Material Flow Analysis (MFA) and Life Cycle Assessment (LCA) applied at three different system levels: the AD plant, the dairy processing facility and the entire dairy supply chain. In addition, the wastewater from dairy production can be used for algae cultivation which can be used as a food and/ or energy resource (Zhu et al. 2013).
On the other hand, the loss of raw materials and products along the dairy supply chain as a result of their permanence and poor storage can also cause the loss of abiotic resources such as energy, water, land and unnecessary emissions of pollutants into the air, water and soil. Brancoli et al. (2017) have performed an analysis of the environmental impact of food waste in retailers identifying the waste fractions with the largest environmental impact. The results provide information to support development of strategies and actions to reduce of the supermarket's environmental footprint. Kazancoglu et al. (2018) have proposed an optimization collection center model based on grey method for prediction of potential milk losses across dairy supply chain for improving its environmental, social and economic sustainability.
The approaches represented above apply the principles of Life Cycle Analysis (LCA) to identifying environmental hotspots along the supply chain which provide valuable information with the potential to reduce GHG emissions.
The great part of developed approaches represents deterministic or stochastic models for optimization of all activities in the network aiming minimizing only of GHG emissions associated with the material and energy flows, transportation and the use of cleaning products as well as certain types of packaging materials (Ferreira et al. 2020). Other methods seek for some level of trade-off between environmental and economic aspects in distribution networks from milk suppliers with optimized overall costs, energy consumption and transport-related GHG emissions. Validi et al. (2015) have proposed an MOGA-II optimization method for determination of the routes for transportation in milk distribution supply chain network which result in minimum total costs and total GHG emissions. As a result, a set of non-dominated solutions distributed along the Pareto frontier have been obtained. Rohmer et al. (2019) have provided a model for maintain a sufficient dietary intake level while minimizing different environmental and economic objectives in the global food system. Taking into account several echelons and interlinkages between different food supply chains, the authors broaden the scope of the considered network and incorporates sourcing, processing and transportation decisions within a common framework.
A few approaches extend the optimization framework to include social aspects, along with environmental ones in sustainable design of considered dairy supply chains. Mota et al. (2015) have proposed a ε-constraint multiobjective mixed integer linear programming model with two different environmental and one social metrics.
Some approaches consider the three aspects of sustainability as the environmental impact is assessed only in terms of GHG emissions as a result of transport, storage (contemplating the activities inventory management and material handling), packaging (Satolo et al. 2020). In this case, the economic and social impact is related with reduction in energy consumption, and creating opportunities for additional training of employers, as a result of which losses in the production process are reduced. The problem is formulated based on linear programming, and further analysis is carried out by applying the ε-constraint method and compromise programming. Djekic et al. (2018) have represented an optimization model of dairy supply chain compromising four criteria-resource depletion, climate impact, economy and society with a total of 13 indicators into one transportation sustainability index. The authors have identified the economic and social factors applicable to local and big dairy companies.
Some of developed mathematical approaches have taken into account the impact of uncertainties on the life (durability) of products as well as transport traffic on the operation of the supply chain (SC). Jouzdani and Govindan (2021) have developed a multi-objective stochastic mathematical programming model to optimize the cost, energy consumption and the traffic congestion associated with dairy supply chain operations. The authors have shown that road congestion and the uncertain perishability of the products are critical factors that can, although differently, affect the operation and the design of the supply chain. Yavari and Geraeli (2019) have proposed an approach for optimal design of multi-period and multi-product supply chains that comprises suppliers, manufacturers, warehouses, retailers and collection centers. As uncertain parameters, the demands, the rate of return and the quality of returned products have been considered. A mixedinteger linear programming (MILP) model has been projected to minimize the cost and environmental pollutant, simultaneously. The results indicate that the cost objective and environmental objective function have different behaviors associated with the changing values of the lifetime. Moghaddam et al. (2019) have proposed a mathematical model for the reverse supply chain of perishable goods (taking into account both strategic and tactical issues simultaneously) under uncertain conditions. They have considered four objective functions to maximize profitability and the level of satisfaction with the use of technology, minimize costs and measure environmental impacts. The results of the implementation of the proposed model have shown that objective functions are sensitive to demand, so the change in demand changes the objective functions, in particular the profitability function. Tavana et al. (2017) have provided an integrated multi-criteria decision-making approach to sustainable dairy supplier selection problems based on analytic network process and quality function deployment. The model has identified a clear hierarchical structure for all the relevant sustainable factors and subfactors while weighting the decision criteria based on the importance given to customer requirements.
The presented overview shows that in recent years research on the sustainability of food, in particular dairy supply chains, is developing at a very rapid pace. A large number of approaches have been developed that aim at managing the food-water-energy nexus in the considered milk supply chains in order to achieve either only environmental or in combination with social and economic sustainability. There are many approaches in which all three aspects of sustainability are considered simultaneously. Most of the developed approaches are based on the use of mathematical models (deterministic and stochastic) to optimize the material, energy and waste flows. The formulated problems include as optimization criteria-maximizing profitability, maximizing satisfaction of used technology, minimizing overall costs, reduction in energy consumption, reduction in GHG emissions due to transport, storage and packaging, creating opportunities for additional training of employers. Some of developed approaches have taken into account the impact of uncertainties on the demand, transportation, production and holding costs, facilities' capacity, the life (durability) of products, transport traffic on the operation of the SC, etc.
There are no approaches in the literature that address the three aspects of sustainability in designing optimal dairy supply chains that produce different products according to different recipes, which provide a broader environmental framework that includes not only the impact of CO 2 emissions but also and industrial wastewater.
The present study proposes an extended version of the already developed multi-objective approach for optimal design of SC for the production of different types of dairy products (Kirilova and Vaklieva-Bancheva 2017) according to different production recipes (Kirilova et al. 2020) in a production complex formed by suppliers, dairies and markets. It includes the social criteria such as jobs opportunities and related with them social costs into consideration along with the environmental and economic ones. The environmental criterion includes assessments of pollutant emissions in relation to two areas of impact-air and water.
The effectiveness of the proposed approach is proved on a real example from Bulgaria. The obtained results show how the inclusion of social criteria affects the products portfolio, environmental and economic costs and can be used in the decision-making process to achieve social sustainability of the considered supply chain.

General formulation of the optimization problem
The proposed approach has been developed to plan the activities in a three-echelon SC including milk suppliers, dairies and markets, to satisfy given consumer demands (shortterm) for a group of products at different recipes, where different raw materials are used. It is represented in Fig. 1.
As a result of its implementation, the optimal sustainable production portfolios of the dairies, satisfying the trade-off between environmental, economic and social objectives, are found. It includes four models for: (i) the production of the products using different recipes; (ii) the SC design; (iii) description of the environmental impact; (iv) description of the SC social impact.
The SC environmental impact is assessed in terms of two areas: wastewater generated at each processing task of the production recipes, including those related to the used raw materials; CO 2 emissions related to the energy consumed by the dairies; and CO 2 emissions produced during transportation of raw materials and products between suppliers, dairies  Fig. 1 Optimization approach for a sustainable decision-making in product portfolio design of dairy supply chain and markets. Biochemical Oxygen Demand during 5 days (BOD 5 ) is used as a main indicator for the wastewater assessment. Environmental pollution taxes are imposed on dairies to keep the wastewater and CO 2 emissions below given acceptable levels. The SC social impact is related to the employees hired by suppliers, dairies and markets. The four models together with constraints on the realization of the production portfolio over time horizon, the planned amounts of products and environmental impact costs which should be paid for the treatment of the pollutants are included in an optimization framework. As an optimization criterion, the total site profit is used. It is represented by the income from the market sale of products after the deduction of all expenses incurred such as production costs, raw materials costs, transportation costs, environmental costs and social costs so as the best trade-off between them to be obtained.

Data
In order to develop the mathematical models, three groups of data should be known: (1). Raw materials and products data-the composition of used raw materials and target products; (2). SC data-data for the production system; markets' demands; capacities of the milk suppliers; selling prices of milk and products; production costs, distances between milk suppliers, dairies and markets; transportation costs; vehicles' types capacities; (3). environmental impact data-related to the environmental impact of pollutants obtained from the implementation of the SC activities in relation to two areas of impact-air and water. For the assessment as indicators, BOD 5 for the wastewater and CO 2 for the air emissions of pollutants are used; (4). Social impact data-social costs related with the employees (job positions) hired by suppliers, dairies and markets. They are costs for salaries, social benefits as food, working clothes, medical care and insurance and the average quantities of raw materials/products processed by employees in suppliers, dairies and markets. between suppliers and dairies and dairies and markets. They are introduced to account for the quantities of both types of milk bought by dairies from the suppliers and the quantities of products produced in dairies and sold at markets; (3). Continuous variables to follow for milk fat content in the used raw materials; (4). Integer variables for number of employees (job positions) depending on the processed quantities of raw materials/products in suppliers, dairies and markets.

Models of production recipes
The productions of two types of curd-low-fat content and high-fat content in two different recipes, each of which uses as a raw material -standardized whole milk (raw material 1-RM1) and skimmed condensed milk (raw material 2-RM2), are considered. The production recipes comprise different production tasks performed in units of different types. The first recipe includes three production tasks: milk pasteurization (Task 1); acidification to produce a raw dairy product (Task 2); draining to produce target dairy product (Task 3). The second one includes four production tasks: milk dilution (Task 1); milk pasteurization (Task 2); acidification (Task 3) and draining (Task 4). The mathematical description of the production recipes includes: (1) Dependencies for determination of the protein, casein and lactose concentrations in raw materials: Production recipe 1 Skimming of whole standardized milk.
Production recipe 2 Dilution of skimmed condensed milk.
where MF (%), MP (%), MC(% ) and ML(%) are the concentrations of milk fat content, proteins, casein and lactose in the used raw materials. CF(%) is cream fat content. MP(x(r p )) (%), MC(x(r p )) (%) and ML(x(r p )) (%) are the concentrations of proteins, casein and lactose in the skimmed milk. r p is the recipe used for the production of the dairy product p.
(2). Equations for target products yield YP(x(r p )) (kg) describing the compositions as functions of the fat content in the used raw materials (Johnson, 2000):

Decision variables
To describe the problem for optimal sustainable portfolio design, the following groups of decision variables should be introduced: (1). Binary variables to structure the SC between suppliers, dairies and markets; (2). Continuous variables to follow the transfer of raw materials and products flows where PS p (%) is the solids' content in products and RC p (%) and RS p (%) are the recovery factors for casein, and all solids. RF(x(r p )) (%) is the milk fat recovery factor.
(3). Equations for the quality of target products-Determination of Fat in Dry Matter-FDM p (%) (Johnson 2000) used as an indicator for curd quality: where PF p is fat content of the product (%).
Data about processing times and equipment used for realization of production tasks as well as the fractions of the processed raw materials and raw products and target products, referred to 1 kg milk and 1 kg curd-target product in production recipe 2, are listed in Table 1.
In it, y(x(r p )) represents the used milk and YP1(x(r p )) is the yield of curd-raw product (containing residual whey) which is produced according production recipe 2, as a function of milk fat content x(r p ) in used milk.
The represented above dependencies Eqs.
(1-4) are referred to 1 kg milk and 1 kg target product. The models of the production recipes provide connection between the production tasks by calculating the size factors representing the "volumes" of materials that have to be processed in production tasks so as to produce 1 kg from target products. The size factors together with the quantities of produced products related with the products portfolio and the production tasks are used for determination of constraints for realization of the production portfolio in the time horizon. They are described in details in Kirilova and Vaklieva-Bancheva (2017). (3)

Model of supply chain
The mathematical description of the three-echelon supply chain includes: (1). Mass balance equations for the subsystem's suppliers-dairies and dairies-markets to prevent from the accumulation of raw materials QM(r p ) i (kg) in the suppliers and products QP(r p ) i (kg) in the dairies. YY(r p ) i,s (kg) are the quantities of raw materials bought by dairies i from the suppliers s , XX(r p ) i,m (kg) are the quantities of products p produced in dairies i and sold at markets m , i,s and i,m are binary variables to structure the SC between suppliers and dairies and dairies and markets.

Model of supply chain environmental impact.
The environmental impact model includes equations for: (1). BOD 5 associated with wastes generated during conducting all production tasks in both recipes and introduced from outside related to the pre-processing of used raw materials. (5) +0.69ML(x(r p )) 10 −2 , (kg O 2 /kg milk) ∀r p , r p ∈ R p , ∀p, p ∈ P BOD cu (x(r p )) = BOD M (x(r p )) YP(x(r p )) (kg O 2 /kg curd), ∀r p , r p ∈ R p , ∀p, p ∈ P where BOD M (x(r p )) is the BOD load related spills of skim milk during implementation of Task 1 in Recipe 1 and Task 2 in Recipe 2 as a function of milk fat content x(r p ) . BOD Cu (x(r p )) is the BOD load related to losses of curd during implementation of Task 3 in Recipe 1 and Task 4 in Recipe 2 as a function of milk fat content x(r p ).
The total environmental impact assessment PBOD p for production of 1 kg of each type of curd is: where m(x(r p )) w,l (∀w, w ∈ W ; ∀l, l ∈ L(r p );∀r p , r p ∈ R p ;∀p, p ∈ P ) are environmental impact indices determining the mass of each type of waste w generated in any production task l related to 1 kg target product. For their determination, In/Out fractions listed in Table 1, target products yield (Eq. 3) and the eligible levels of losses listed in Table 2 are used (Kirilova and Vaklieva-Bancheva (2017)).
BOD 5 load related to the wastes, production tasks and eligible levels of losses is listed in Table 2. Table 2 shows that BOD Pa is the BOD load related to deposits on pasteurizers walls during pasteurization of the skim milk (Task 1 in Recipe 1 and Task 2 in Recipe 2), BOD Wh is the BOD load related to spills of whey produced as by-product during discharging of curd vats (Tasks 2, 3 in Recipe 1 and Tasks 3, 4 in Recipe 2). LS SM represents losses of skim milk in milk pasteurization (Task 1 in Recipe 1 and Task 2 in Recipe 2). LS Wh represents losses of whey in acidification and draining of the produced curd (Tasks 2, 3 in Recipe 1 and Tasks 3, 4 in Recipe 2), while LS2 Cu and LS3 Cu represent losses of curd (Tasks 2, 3 in Recipe 1 and Tasks 3, 4 in Recipe 2).
(2). Equations for the impact of CO 2 emissions associated with the heating and cooling of milk.
referred to 1 kg milk as follows: where EH and EC are the energy required for realization of the processes of heating and cooling in the pasteurization in (kWh/kg milk); ECO 2 is the mass of CO 2 emissions associated with the energy (kg CO 2 /kWh).
3). Equations for the impact of CO 2 emissions associated with the transport of raw materials and products, referred to 1 kg from both:(kg CO 2 /km·kg curd): where TCO 2 is the quantity of CO 2 emissions produced by fuel combustion (kg CO 2 / km) and VCm (kg) and VCp (kg) are the payload capacities of used vehicles for transportation of raw materials and products.

Model of supply chain social impact
The model of supply chain social impact includes the equations about the numbers of employee who will be hired by the suppliers Eq. (11), dairies Eq. (12) and markets Eq. (13). They depend on average quantities of raw materials/products processed by employees in suppliers, dairies and markets.
(9) EIMCO 2 (x(r p )) = (EH + EC)ECO 2 CF−MF CF−x(r p ) (kg CO 2 /kg curd) ∀r p , r p ∈ R p , ∀p, p ∈ P (10) TMCO 2 = 2 TCO 2 VCm (kg CO 2 /km kg milk)  XX i,p,m,r (kg) are the quantity of product p produced according recipe r in dairy i and sold at market m . i,p,m,r is binary variable used to structure the supply chain between the dairies and markets. It takes a value of "1" when the dairy i , where the product p is produced according recipe r , is connected with market m . Otherwise, it takes the value "0."

Constraints
To estimate the feasibility of the obtained sustainable product portfolios, the following constraints are introduced for: (1). Realization of the production portfolio in the time horizon; (2). Capacity of the suppliers of raw materials; (3). Capacity of the markets for realization of the planned amounts of products; (4). Environmental impact costs which should be paid for the treatment of the pollutants. These are the costs for BOD removal in the WWTPs and the CO 2 costs related with the production of the products in a dairies and transportation of raw materials and products.

Optimization criterion
As an optimization criterion a single objective optimization function is used F Profit (BGN). It represents the difference between the production profit and the economic, environmental and social costs, as follows: where F R (BGN) is the revenue from the sale of the products at the markets; F P_Costs (BGN) are the total production costs for the dairies; F M_Costs (BGN) are the total costs incurred by the dairies for purchasing the required quantities of both types of milk from suppliers for the production of the products; F T_Costs (BGN) are the total costs for the transportation of the milk and products between suppliers, dairies and markets; F BOD_Costs (BGN) are the total BOD 5 costs paid for treatment of the wastewater generated during the production of the products; F CO 2 _E_Costs (BGN) are the total CO 2 emissions costs associated with the energy consumed by pasteurization process; F CO 2 _T_Costs (BGN) are the total CO 2 costs associated with emissions of pollutants generated during milk and products transportation; F Social_Costs_Suppliers , F Social_Costs_Dairies and F Social_Costs_Markets are costs related with the number of employees (job positions) who should be hired by the suppliers, dairies and markets.
The latter terms of the optimization criterion (14) is the following: where CS , CWCl , CSB , CMI are costs for salaries, working clothes, social benefits and medical insurance related with suppliers, dairies and markets.
The objective function (14) is subject of maximization: The formulated optimization problem belongs to the mix integer nonlinear programming (MINLP). It contains both binary and continuous variables, sets of modeling equations and inequality constraints. A detailed description of the presented above optimization approach with the models for (i) the production of the products; (ii) planning of activities in SCs; (iii) describing the environmental impact of SCs are given in Kirilova and Vaklieva-Bancheva (2017).

Case study
The approach is implemented on a real case study from Bulgaria comprising the production of two types of curd with low and high fat content (P1 and P2) using two production recipes (PR1 and PR2) with two types of raw materials (RM1 and RM2). The products are produced in two dairies (D1 and D2) supplied with RM1 and RM2 by two suppliers (S1 and S2). The products are sold on two markets (M1 and M2). The planned quantities of the products that should be produced are 30,000 kg per product. The products production is realized over time horizon of one month (720 h).
The equipment units for performing the production tasks and theirs summarized volumes are listed in Table 3. To formulate the portfolio feasibility constraints, the processing times should also be known.
Capacities of suppliers (kg), prices of RM1 and RM2 (BGN/kg), market demands (kg) and prices of products (BGN/kg) are presented in Table 4.
In Table 5, distances (km) between suppliers, dairies and markets are presented. It also includes data about the type of the used vehicles (V)-Milk tanker truck with petrol engine-V1 and Refrigerator truck with diesel engine-V2 such as: payload capacity-PC (L); energy of fuel-FE (kWh/L); FCO 2 (kg CO 2 /kWh) generated from fuel combustion; fuel consumption-FC (L/100 km) and fuel price-FP(BGN/L).
They are used for calculation of the CO 2 emissions associated with transportation and transportation costs. The latter in BGN/kg.km are calculated by multiplication of the vehicle's fuel consumption (L/100 km), the vehicle's fuel price (BGN/L) and the number of vehicles' courses. The latter is divided by the total quantities of raw materials or products produced (kg).
The environmental costs associated with transportation are obtained using data given in Table 5 and the price of CO 2 emissions which is 0.1278 BGN/kg CO 2 . The energy consumed in both recipes for heating of 1 kg milk is 8.333 × 10 −3 kWh/kg milk and for cooling is 6.333 × 10 −2 kWh/kg milk. The CO 2 emissions associated with both processes is 0.46 kg CO 2 /kWh. The price of BOD 5 paid to wastewater treatment plants from D1 is 2.9 BGN/kg, while from D2 it is 3.5 BGN/ kg. The production costs are obtained based on the energy used for production of the products using the price of energy, which is 0.14072 BGN/kWh.
In Table 6, the average costs (BGN) related to the number of employees (job positions) who should be hired by the suppliers, dairies and markets are given. They include costs for salaries-CS, working clothes-CWCl, social benefits-CSB and medical insurance-CMI. The same table also shows the average quantities (kg)-AQ of raw materials or products that employees can process per day in different echelons of the SC.

Results and discussions
Four MINLP optimization models were defined-one without social impact consideration and three with social impact consideration. They were solved by BARON solver. MINLP optimization models are coded and run in GAMS optimization software, GAMS Release: 32.2.0 rc62c018 WEX-WEI × 86 64bit/MS Windows on a AMD 7 3700X 8-CORE (3.6/4.4.GHz, 32 MB, AM4) CPU with 16 GB DDR4 3600 MHz RAM. The optimization problems were calculated in a few seconds. Table 7 presents of obtained optimal values of revenues, economic, environmental and social costs as well as the production profits for the solutions.
The four solutions meet the market demands for the supply of 30,000 kg of each of the two products or a total of 60,000 kg. The first three solutions were obtained at different values of the environmental constraints imposed in terms of fees that should be paid for BOD removal in the WWTPs and CO 2 emissions due to transport of raw materials and products and consumed energy from the dairy production.
The fourth solution was obtained with constraint imposed on both environmental and social costs of 95,220 BGN. All solutions were obtained at the upper boundaries of the set constraints, the values of which are shown in Table 7. As it can be seen, from the solutions with included social criterion, Solution 4 has the highest value for the environmental costs and profit and lowest economic costs. On the other hand, Solution 3 has the lowest value for the profit and the environmental costs and highest economic costs. The differences in the values of economic costs are due to different quantities of raw materials used for the production of the products according to two recipes. Table 7 shows that in all four solutions the greatest share in the environmental impact has BOD 5 related with the wastewater generated from dairy production, as their percentage varies in different solutions from 52% for Solution 1 to 72% for Solution 3. They are followed by CO 2 emissions related with energy consumed from dairy production from 37% for Solution 1 to 45% for Solution 3. The environmental costs associated with the CO 2 emissions due to transport of raw materials and products have the smallest values from 11% for Solution 2 to 14% for Solution 3. The 1 3 optimal number of employees for suppliers, dairies and markets in Solutions 2 and 3 is: 7, 9 and 34, respectively. The social costs related with them are 87,140 BGN from which: 70,600 BGN for salaries; 7500 BGN for working clothes; 4540 BGN for social benefits and 4500 BGN for medical insurance. In the Solution 4, the optimal number of jobs obtained is 8, 9 and 34 with social costs of 88,830 BGN. The social costs account for 38% of all costs. From them, 81% are for salaries.
The obtained optimal products portfolios for the Solution 1, Solution 2, Solution 3 and Solution 4 are represented in Figs. 2,3,4,5. Red arrows in Figs. 2,3,4,5 show the material flows between suppliers, dairies and markets related to the production of the two products-P1 and P2 from the first raw material-RM1. Suppliers and dairies that do not participate in the products portfolio are marked in grey. Figure 2 shows Solution 1, which does not include social assessment. One can see from the figure that D1 produces P1 and P2 according to PR1 and PR2, which are delivered to both markets. D2 produces P1 and P2 according to PR2 which is delivered to both markets. S1 supplies both dairies with RM2 and S2 supplies D1 with RM1 and RM2. Figures 3 and 4 show the two solutions (Solution 2 and Solution 3) with included social assessments, with different constraints imposed on the environmental impact assessments. One can see from the figures, that for Solution 2, D1 produces P1 and P2 according to PR1 and PR2, which are delivered to both markets. D2 produces P2 according to PR2 which is delivered to both markets. S1 supplies D2 with RM2 and S2 supplies D1 with RM1 and RM2. For the Solution 3, one can see that D1 produces P1 and P2 according to PR1 and PR2, which are delivered to both markets. D2 produces P1 and P2 according to PR2 which is delivered to both markets. S1 supplies both dairies with RM2 and S2 supplies D1 with RM1 and RM2. Figure 5 shows the solution, in which a constraint regarding to both environmental and social impact is imposed. Figure 5 shows that although this solution satisfies market demands of 30,000 kg of each of the products, it considers the production of the two products only in a Dairy 1. Presented in Figs. 2,3,4, optimal product portfolios exclude from consideration the supply of RM1 from S1 and the production of P1 and P2 in Dairy 2 according to PR1. Moreover, all three portfolios (Solutions 1, 2, 3) include the supply of raw material RM1 from supplier S2 for the production of P2 according to production recipe PR1 in dairy D1, which is supplied on the market M2.
The solutions presented show that the inclusion of social impact assessment into consideration affects the obtained optimal products portfolios, and hence the profit from the dairy production. On the other hand, when a social criterion is included, varying the values of environmental constraints also affects the obtained optimal products portfolios and profits, respectively, without affecting the values of social costs. In these cases, the imposition of stricter environmental impact constraints is associated with higher economic costs and leads to lower profit. Conversely, high values of environmental constraints result in higher profits and lower economic costs. In the last solution, imposing a constraint on both environmental and social costs results in the highest profit, the lowest economic costs and the highest environmental costs. However, in this solution, the obtained optimal portfolio excludes the production of the products in the second dairy.
The latter three solutions can be used in the decision-making process in terms of seeking a compromise between profit, environmental and social impact, when all three aspects of sustainability have been taken into consideration.

Conclusions
The present study proposes an extended version of the approach for optimal product portfolio design of SC for production of two types of dairy products at two recipes, taking into consideration the social impact along with the environmental and economic ones. The social impact is related to the employees who should be hired by suppliers, dairies and markets for implementation of the SC activities. Four MINLP optimization problems were defined and solved using GAMS software. The first optimization problem was formulated in case without the social impact consideration, as the rest models include additional decision variables related with the employee who will be hired by the suppliers, dairies and markets and respective constraints. The first three solutions were obtained at different values of the environmental constraints related with generated wastewater and CO 2 emissions across the supply chain. The fourth solution was obtained with a constraint imposed on both environmental and social costs. The obtained results have shown that the inclusion of social impact assessment affects the obtained optimal products portfolio and the profit from the dairy production. On the other hand, when a social criterion is included, varying the values of environmental constraints also affects the products portfolios and profits, without affecting the values of social costs. The obtained results can be used in the decision-making process to achieve social sustainability of the considered dairy supply chain. In the future, the presented optimal model can be extended to account the preferences of all actors in the considered dairy supply chain, as well as uncertainties regarding the demand, transportation, production and durability of products.