CET-vol 105 DOI: 10.3303/CET23105065 Paper Received: 19 January 2023; Revised: 18 April 2023; Accepted: 8 October 2023 Please cite this article as: Vladova R.K., Kirilova E.G., Petrova T.S., Kirilov K.T., 2023, A Sustainable Multi-objective Optimization Model for Design of Supply Chain Under Uncertain Products Demands and Products Prices: a Real Case of Dairy Industry, Chemical Engineering Transactions, 105, 385-390 DOI:10.3303/CET23105065 CHEMICAL ENGINEERING TRANSACTIONS VOL. 105, 2023 A publication of The Italian Association of Chemical Engineering Online at www.cetjournal.it Guest Editors: David Bogle, Flavio Manenti, Piero Salatino Copyright © 2023, AIDIC Servizi S.r.l. ISBN 979-12-81206-04-5; ISSN 2283-9216 A Sustainable Multi-Objective Optimization Model for Design of Supply Chain Under Uncertain Products Demands and Products Prices: a Real Case of Dairy Industry Rayka K. Vladovaa, Elisaveta G. Kirilovaa,*, Tatyana S. Petrovaa, Kiril T. Kirilovb,c aInstitute of Chemical Engineering at Bulgarian Academy of Sciences, Akad. G. Bontchev Str., Bl. 103, Sofia 1113, Bulgaria bNew Bulgarian University, Montevideo Str., 21, Sofia 1618, Bulgaria cInstitute of Molecular Biology, Bulgarian Academy of Sciences, Akad. G. Bontchev Str., Bl. 21, 1113 Sofia, Bulgaria e.kirilova@iche.bas.bg Dairy production has a significant environmental impact related with wastewater and air pollution. In addition, it is a large consumer of water and energy. The most effective way to improve its sustainability is through analysis of the food-water-energy nexus, which can be done by optimizing all activities in the supply chain (SC) from the raw materials to the end user while meeting environmental, economic and social criteria. However, the presence of uncertainties regarding the main parameters of the SC would lead to problems related to the implementation of processes and the operating the system as a whole. To solve these problems an implementation of models for optimal design of dairy SCs that handle these uncertainties is needed. The present study proposes an extended version of already developed mixed integer non-linear programming (MINLP) approach to optimal design of a sustainable SC for the production of different dairy products according different recipes under uncertain products demands and products prices. The model has taken into account economic, environmental and social aspects. The obtained results show that the increase in the uncertainty level leads to an increase of values of economic, environmental and social costs as well as the profit, with the standard deviation of the same being the highest at the lowest level of uncertainty, while at the others it is preserved relatively constant. The number of workers employed by the supply centers, dairies and markets is kept constant regardless of the variation in the level of uncertainty of products demands and products prices. 1. Introduction Given the development of the dairy sector and the new challenges it faces, the effective management of dairy SCs has become an attractive topic for researchers and practitioners. In this context, the integration of uncertain aspects is constantly gaining importance for management decision-making, as it can lead to an increase in their sustainability and ultimately competitiveness in the market (Mishra & Shekhar, 2011). One of the most effective ways to deal with these uncertainties is by applying approaches for quantitative modeling of dairy SCs operating under uncertain conditions, (Sel & Bilgen, 2015). These approaches can be classified in terms of three most relevant recognized characteristics: types of uncertainty modeling, programming approaches, and functional areas of application, (Borodin et al., 2016). In the developed approaches as uncertainties operational costs, prices of raw materials and products, products demands, energy consumption, transportation, the products lifetime, have been considered Li et al., (2008). Yang et al., (2015) have proposed a new two-stage optimization method for multi-objective SC design problem with uncertain transportation costs and uncertain products demands. Yavari and Geraeli, (2019) have developed a mixed-integer linear programming (MILP) model for optimal design of multi-period and multi-product dairy SC satisfying economic and environmental criteria. As uncertain parameters the products demands, the rate of return and the quality of returned products have been considered. Dutta and Shrivastava, (2020) have proposed a stochastic programming approach for achievement of the optimal facility location and shipment decisions in distribution network operating on retailer level under demand, supply and process uncertainties while minimizing the total costs. Jouzdani & Govindan, (2021) have investigated the interrelations and interactions of the three aspects of sustainability through developing and 385 implementation of a multi-objective stochastic mathematical programming model for optimization of the costs, energy consumption, and the transport traffic associated with dairy SC operations. In their model, as an uncertain variable, the product lifetime has been considered. The latter it is assumed to be affected by vehicle refrigerator utilization, which is considered as a decision variable. From the literature review, it can be seen that a large part of the developed approaches aim managing the food-water-energy nexus in dairy SCs operating under uncertainties regarding products demands, transport and production costs, facility capacity, life (durability) of products, transport traffic, etc. On the other hand, most approaches aim to achieve sustainability in terms of mostly environmental and economic assessments, and there are also approaches that take into account all three aspects of sustainability. The approaches are implemented for optimization of material and energy flows. Maximization of profit from production, minimization of total costs, minimization of pollution, most often in terms of carbon emissions related to the transport of raw materials and products, reduction of energy consumption from production, achievement of social satisfaction are considered as optimized criteria.There are no approaches in the literature that consider the three aspects of sustainability in designing optimal dairy SCs for the production of different products according to different technologies, operating under conditions of uncertainty regarding product demands and product prices in the market. As well as those that take into account the environmental impact not only in terms of CO2 related to transport and energy consumed, but wastewater related with used raw materials and production of the dairy products.The present study proposes an extended version of the already developed approach of Kirilova et al. (2022a) where the deterministic model for the optimal design of sustainable dairy SC has been extended with a Robust Counterpart (RC) for defining of the uncertain products demands and products prices using the approach of Ben-Tal et al. (2005). 2. Formulation of a robust optimization problem for design of a sustainable dairy SC taking into account economic, environmental and social criteria The approach of Kirilova et al. (2022a) extended with RC for formulation of the uncertain parameters (Kirilova et al. 2022b) has been implemented on a three-echelon dairy SC including milk suppliers, dairies and markets for production of different products according to different technologies using different types of milk as raw materials. The products should be produced in given quantities in the dairies for a given time horizon. The latter should satisfy predefined product demands for the markets.The approach involves four models for the production of dairy products; the SC design, the SC environmental and social impact. The mathematical model of production of dairy products according production technologies includes: 1). Models of the used production technologies for the production of the dairy products. In the considered dairy SC of two types of cottage cheese - low-fat content and high-fat content have been produced according to different technologies using as raw materials standardized whole milk and skimmed condensed milk. The first production technology involves the following production tasks: pasteurization of standardized whole milk; acidification to obtain both dairy products; draining to produce the target dairy products. The second production technology includes one more production task related with the dilution of skimmed condensed milk. Milk fat content has been considered as a key decision variable which determine as the amounts of the produced dairy products as well the environmental impact of the consider production technology. The mathematical models of the production technologies involve equations for determination of the concentrations of protein, casein and lactose in used raw materials. The production technologies model has been added with equations defining the products yield as functions of the milk fat content in the used raw materials and equations for determination of specific indicator accounting for the quality of the produced products. Model of the considered three-echelons dairy SC includes equations determining the quantities of raw materials provided by all suppliers and products delivered on the markets. Model of the environmental impact performance of the considered dairy SC includes equations for: 1). BOD5 related with wastewater generated during conducting production tasks for the production of both products according both technologies as well as those related with pre-processing of used raw materials – different types of milk. 2). Equations for determination of the quantities of CO2 emissions related with energy consumed by the dairies for the production of the products. 3). Equations for determination of the CO2 emissions generated by the trucks used for transporting the raw materials and products. Mathematical model of the economic performance of the considered dairy SC includes equations for all costs related with: the production of the products in the dairies; purchasing the required quantities of milks by the suppliers; transporting of raw materials and products between milk suppliers, dairies and markets. The model of SC social impact includes the equations about the numbers of employee who will be hired by the suppliers, dairies and markets. They depend on average quantities of raw materials/products processed by employees in suppliers, dairies and markets. The environmental impact is assessed by CO2 emissions generated due to transportation and energy consumed and BOD5 (biochemical oxygen demand for 5 days) related with generated wastewater generated during the production of dairy products. The dairy SC social impact is related to the employees hired by suppliers, dairies and markets. The optimization criterion includes all aspects of sustainability - environmental, economic and social defined in terms 386 of costs. For the purpose of implementation of the considered approach four groups of data are required. They are related with: 1). The composition of the used raw materials and the dairy products produced; 2). The production system, capacity of milk suppliers, selling prices of milk and products, production costs, distances between milk suppliers, dairies and markets, transporting costs and payload capacity of used transportation trucks; 3). The environmental impact related to pollutants generated in air and water and 4). The 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. An optimization framework has been formulated which includes the following decision variables: 1). Binary variables for definition of the links between suppliers, dairies and markets in the considered dairy SC; 2). Continuous variables for definition of the raw materials and products flows between different sites of the SC. They are introduced to account for the amounts 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 for definition of the milk fat content in the used raw materials; 4). Integer variables for definition of the number of employees (job positions) depending on the processed amounts of raw materials/products in suppliers, dairies and markets. The milk fat content of the raw materials is included as a decision variable in both the models of dairy products production, as well as in a model of the SC environmental impact. On the other hand, obtaining larger quantities of products is associated with employing more people at all sites of the SC. In this sense, this variable affects the overall sustainability of the resulting solutions. The proposed optimization framework includes the constraints related with: 1). Conducting production of the products in the dairies in the predefined time horizon (for their calculation, size factors, the quantities of the produced products and the production tasks are used) 2). Capacities of considered milk suppliers; 3). Capacities of the markets to deliver the produced products and 4). Environmental impact costs for treatment of the generated in air and water emissions of pollutants.All equations are referred to 1 kg milk and 1 kg target product. The connection between the production tasks is provided by so called size factors. They determine the quantities of processed in each production unit materials so as 1 kg product to be produced. Information about size factors, production units, production tasks, processing times, etc. is given in Kirilova and Vaklieva-Bancheva, (2017).Several optimization problems have been formulated and solved at different realizations of the nominal values for the products demands and products prices generated randomly with uniform probability distribution under four levels in specified uncertainty set, (Ben-Tal et al. 2005; Kirilova et al. 2022b).The used optimization criterion represents the optimal profit obtained after reduction of the revenue from the sale of the products on the markets with all economic, environmental and social costs. Detailed description of formulated optimization problems including needed data, all mathematical models, constraints and optimization criterions is given in Kirilova et al., (2022a); Kirilova et al., (2022b) and Kirilova and Vaklieva-Bancheva, (2017). 3. Case study The presented above robust optimization approach is applied in a real case study from Bulgaria including production of two types of cottage cheese according two production technologies using standardized whole milk and skimmed condensed milk as raw materials. The production of both types of products is conducted in two dairies. Three suppliers provide dairies with the raw materials. The produced products deliver on three markets. The products production is realized over time horizon of one month. Used production units for conducting the production tasks and theirs summarized volumes are listed in Table 1. Table 1: Equipment units with summarized volumes (m3) Milk tanks Pasteurizers Curd vats Drainers Dairy 1 1,450 800 950 300 Dairy 2 1,450 950 1,050 340 Capacities of the three suppliers (kg), milk prices (BGN/kg) are presented in Table 2. Table 2: Capacities of suppliers (kg) and milk prices (BGN) Capacity (kg) Milk price (BGN) Milk 1 Milk 2 Milk 1 Milk 2 Supplier 1 97,000 57,000 0.90 2.7 Supplier 2 100,540 54,500 0.80 2.4 Supplier 3 113,000 78,000 1 3 387 Distances (km) between suppliers, dairies and markets are listed in Table 3. Table 3: Distances between suppliers, dairies and markets (km) Supplier 1 Supplier 2 Supplier 3 Market 1 Market 2 Market 3 Dairy 1 10 15 20 98 136 46 Dairy 2 20 10 15 22 23 75 In Table 4 are presented the data about vehicles used for transportation of raw materials and products and their characteristics. They are used for calculation of the CO2 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). Table 4: Vehicles used for transportation of raw materials and products and their characteristics Vehicles used for transportation Payload capacity, (L/kg) Energy of fuel, (kWh/L) CO2 emissions generated from fuel combustion, (kg CO2/kWh) Fuel consumption, (L/100km) Fuel price, (BGN/L) Milk tanker truck with petrol engine 1,500 8.056 0.249 32 2.22 Refrigenerator truck with diesel engine 3,500 9.583 0.267 23 2.27 The environmental costs associated with transportation are obtained using data given in Table 4 and the price of CO2 emissions which is 0.174 BGN/kg CO2. 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 CO2 emissions associated with both processes is 0.46 kg CO2/kWh. The price of BOD5 paid to wastewater treatment plants from Dairy 1 is 2.9 BGN/kg, while from Dairy 2 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 5 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 (BGN), working clothes (BGN), social benefits (BGN) and medical insurance (BGN). The same table also shows the average quantities (kg) of raw materials or products that employees can process per day in the different echelons of the SC. Table 5: Data related with employees Employees Costs for salaries, (BGN) Working clothes, (BGN) Social benefits, (BGN) Medical insurance, (BGN) Average quantities, (kg) Suppliers 1,300 50 100 90 1,000 Dairies 2,300 50 200 90 300 Markets 1,700 50 100 90 80 4. Results and discussions The proposed robust optimization approach has been implemented in a real case study from Bulgaria. The dairy SC includes three suppliers, two dairies and three markets. All needed data are given in Kirilova et al., (2022a); Kirilova et al., (2022b) and Kirilova and Vaklieva-Bancheva, (2017). Several robust optimization problems were formulated and solved under nominal product requirement data and four different uncertainty levels (UL) (ρ = 0.25; 0.5; 0.75; 1). Under each uncertainty level, ten random realizations have been uniformly generated in a box uncertainty set, as follows: [𝑛𝑜𝑚𝑖𝑛𝑎𝑙 𝑣𝑎𝑙𝑢𝑒 ± 𝜌 ∗ 𝐺]. The performance of the robust optimization method strongly depends on the choice of the type of uncertainty set. Using a box uncertainty set ensures the generation of the full set of realizations of the uncertain parameters that never violate the constraints of this set. On the other hand, using this set may lead to generating realizations of uncertain parameters with worse values and therefore developing smaller uncertainty sets that still ensure that the "almost never" constraint is violated. The optimization models have been solved using GAMS® optimization software-BARON solver as all calculations have been carried out on an AMD 7 3700X 8-CORE (3.6/4.4. GHz, 32 MB, AM4) CPU with 16 GB DDR4 3600 MHz RAM. Two performance measures have been used to evaluate the models: mean and standard deviation of the obtained results. The optimization problems have been formulated and solved at given boundaries of 388 varying of the products demands: Product 1, Market 1 – 25,000 ÷ 35,000 kg; Product 1, Market 2 – 15,000 ÷ 25,000 kg; Product 1, Market 3 – 20,000 ÷ 30,000 kg; Product 2, Market 1 – 5,000 ÷ 15,000 kg Product 2, Market 2 – 35,000 ÷ 45,000 kg; Product 2, Market 3 – 20,000 ÷ 30,000 kg. On the other hand, the products prices vary in the following boundaries, as follows: Product 1, Market 1 - 11 ÷ 17 BGN; Product 1, Market 2 – 19.50 ÷ 25.50 BGN; Product 1, Market 3 – 14.90 ÷ 20.90 BGN; Product 2, Market 1 – 14.30 ÷ 20.30 BGN; Product 2, Market 2 – 11.6 ÷ 17.6 BGN; Product 2, Market 3 – 22.70 ÷ 28.70 BGN. The results from the optimization models are listed in Table 6 and Table 7. One can see from Table 6 that as the level of uncertainty of varying products demands and products prices increases, the values for the economic, environmental and social costs as well as the profit also increase, with the standard deviation of the same being the highest at the lowest level of uncertainty of ρ=0.25, while at the others it is preserved relatively constant. At an uncertainty level of ρ=0.75, a significantly larger mean and standard deviation for the SC profit can be observed. It can be seen from Table 7 that as the uncertainty levels of varying the uncertain variables increase, the number of employed people from the three echelons of the SC – suppliers, dairies and markets does not change significantly. Table 6: Obtained results from the optimization problems solutions for the SC economic, environmental, social costs and profit in mean and standard deviation values for the different uncertainty levels and scenarios Uncertainty level Economic costs, (BGN) Environmental costs, (BGN) Social costs, (BGN) Profit, (BGN) ρ=0.25 Mean 507,365 20,979 123,968 786,484 Standard deviation 54,117 1936 11,540 155,744 ρ=0.5 Mean 522,335 21,511 127,147 818,704 Standard deviation 25,148 720 5,063 142,383 ρ=0.75 Mean 521,726 21,461 127,005 1,133,985 Standard deviation 26,436 777 5,334 923,414 ρ=1 Mean 523,938 21,544 127,464 833,210 Standard deviation 26,725 778 5,385 158,372 Table 7: Obtained results from the optimization problems solutions the number of employees hired from the suppliers, dairies and markets for different uncertainties levels and scenarios Uncertainty level Employees S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 ρ=0.25 Suppliers Dairies Markets 12; 10; 39 13; 11; 43 8, 7, 29 12, 10, 40 12, 10, 39 12, 10, 39 12; 10; 40 13; 11; 43 12; 10; 38 12; 11; 41 ρ=0.5 Suppliers Dairies Markets 12; 10; 39 13; 11; 43 12, 11, 42 12, 10, 40 12, 10, 39 12, 10, 39 12, 10, 39 13; 11; 43 12; 10; 38 12; 10; 41 ρ=0.75 Suppliers Dairies Markets 12; 10; 39 13; 11; 43 13, 11, 42 12, 10, 40 12, 10, 38 12, 10, 39 12, 10, 39 13, 11, 43 12; 10; 38 12; 10; 41 ρ=1 Suppliers Dairies Markets 12; 10; 40 13; 11; 43 13; 11; 43 12; 10; 40 12; 10; 38 12; 10; 39 12; 10; 39 13, 11, 42 12; 10; 38 12; 10; 41 From the results obtained for the product portfolios of the two dairies, it can be seen that they also do not change significantly when varying products demands and products prices at the different uncertainty levels. For the lowest uncertainty level of ρ=0.25, Dairy 2 produces both products as Product 1 is produced only according to the Technology 1 and supplies to Market 2 and Product 2 is produced according to both production technologies (with a significantly larger amount being produced according to Technology 1) and delivered to Market 2. For all uncertainty levels, in Dairy 2, both products are produced as Product 1 is produced by both recipes (with a significantly larger amount being produced according to Technology 1) and delivered to Market 2, and Product 2 is produced by Technology 1 and delivered to Market 1. For all uncertainty levels, in terms of environmental costs, the biggest share is the costs related to the energy consumed by production (about 75%), followed by 389 the costs related to wastewater (about 25%), while the environmental costs related to transport are negligibly small (about 1%). Regarding the number of employees, it can be seen from Table 7 that for all uncertainty levels, 12 workers are employed in the supply centers, 10 in dairies and 40 in markets. 5. Conclusions The research represents an implementation of the robust optimization approach of Kirilova et al., (2022b) for optimal design of sustainable dairy SC for production of two products according two technologies in a production complex comprising three milk suppliers, two dairies and three markets operating under uncertain product demands and products prices. Several robust optimization problems have been formulated and solved under different random realizations of products demands taking into account economic, environmental and social considerations. The nominal data for the product demands and product prices were randomly generated at four levels using uniform distribution in a specified uncertainty set. Two performance measures were used to evaluate the optimization models: the mean and standard deviation of the objective function values under random realizations. The obtained results show that the increase in the uncertainty level leads to an increase of values of economic, environmental and social costs as well as the profit, with the standard deviation of the same being the highest at the lowest level of uncertainty of ρ=0.25, while at the others it is preserved relatively constant. At an uncertainty level of ρ=0.75, a significantly larger mean and standard deviation for the SC profit can be observed. The number of workers employed by the supply centers, dairies and markets is kept constant regardless of the variation in the level of uncertainty of products demands and products prices. The approach has been applied to another case study (Kirilova et al. 2022b), as in this case study as well as in the present one, the results obtained for profit, environmental and economic cost have shown robustness to varying uncertainty level of products demands and product prices. In this sense, the model can be applied on a more general scale. Acknowledgments The study represents results obtained with the financial support of the Bulgarian Science Fund under Contract No. КΠ-06-Н37/5/06.12.19. References Ben-Tal A., Golany B., Nemirovski A., Vial J.P., 2005, Retailer-supplier flexible commitments contracts: a robust optimization approach, Manufacturing and Service Operation Management, 7, 248-271. Borodin V., Bourtembourg J., Hnaien F., & Labadie N., 2016, Handling uncertainty in agricultural supply chain management: A state of the art, European Journal of Operational Research, 254(2), 348-359. 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