DOI: 10.3303/CET23103143 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Paper Received: 17 April 2023; Revised: 24 May 2023; Accepted: 30 June 2023 
Please cite this article as: Petrova T.S., Vladova R.K., Kirilova E.G., 2023, A Robust Optimisation Model for the Sustainable Design of a Dairy 
Supply Chain under Uncertain Environmental Conditions, Chemical Engineering Transactions, 103, 853-858  DOI:10.3303/CET23103143 
  

 CHEMICAL ENGINEERING TRANSACTIONS  
 

VOL. 103, 2023 

A publication of 

 

The Italian Association 
of Chemical Engineering 
Online at www.cetjournal.it 

Guest Editors: Petar S. Varbanov, Panos Seferlis, Yee Van Fan, Athanasios Papadopoulos 

Copyright © 2023, AIDIC Servizi S.r.l. 
ISBN 979-12-81206-02-1; ISSN 2283-9216 

A Robust Optimisation Model for the Sustainable Design of a 

Dairy Supply Chain under Uncertain Environmental 

Conditions 

Tatyana S. Petrova*, Rayka K. Vladova, Elisaveta G. Kirilova 

Institute of Chemical Engineering at Bulgarian Academy of Sciences, Akad. G. Bontchev Str., Bl. 103, Sofia 1113, Bulgaria 

t.petrova@iche.bas.bg 

The production of dairy products in the relevant supply chains (SCs) is associated with the generation of 

significant amounts of pollutants in the wastewater and CO2 emissions into the air, which released into the 

environment can cause a great risk of its pollution. The presence of uncertainties regarding the released 

amounts of pollutants further exacerbates the environmental problem. One of the ways to capture these 

uncertainties and control the environmental risk of pollution is by applying approaches to optimal design of 

sustainable supply chains operating under uncertainty conditions. The present study proposes a mixed integer 

non-linear programming approach to the optimal design of a sustainable SC for the production of two types of 

dairy products with a choice of technology, which includes models of economic, environmental and social impact 

as well as dairy production. The latter takes into account uncertainties regarding the environmental impact costs 

for treatment of the generated pollutants in terms of wastewater and CO2 emissions related to the SC activities. 

The efficiency of the proposed approach has been proved in a real case from Bulgaria, which includes suppliers 

of two types of milk and dairy plants for the production of two types of dairy products and markets.  

1. Introduction 

Dairy production on a global scale is related to the generation of significant amounts of pollutants, which, 

released into the environment, lead to a great environmental risk. On the other hand, it is associated with a large 

consumption of water and energy at all stages of the dairy SC, which, in addition to the main production 

processes, also includes some auxiliary processes such as cleaning the equipment. It was found that cleaning 

operations are responsible for 70 % of the water requirements, while refrigerators, pasteurisation and packaging 

account for 70 % of the energy consumption in the dairy facility (Vasilaki et al., 2016). 

One of the most effective ways to achieve the most complete and accurate assessment of the sustainability of 

respective productions in terms of their environmental and economic impact as well as the related water and 

energy consumption is the development and application of mathematical approaches for simulating and 

optimising activities in relevant dairy SCs. For example, the developed SC energy models can provide a 

complete overview of the energy demand and the energy mix of a dairy SC (Malliaroudaki et al., 2022). 

Additionally, they can highlight the most energy-consuming processes and allow the evaluation of alternative 

energy-saving operations that can lead towards the net-zero carbon target. On the other hand, the application 

of special models to simulate the greenhouse gas emissions generated under dairy production systems enables 

the identification of the effect of small management changes that would reduce pollutants and increase farm 

profitability (Lovett et al., 2008). Cecchini et al. (2016) have investigated the environmental and economic 

sustainability of dairy farms through the implementation of life cycle environmental assessment and economic 

assessment, including the direct and indirect costs related to the production process factors. The authors have 

performed a correlation analysis and a linear regression in order to study the relationship between the carbon 

footprint and operating income. 

When applying mathematical approaches for the optimal design of dairy SCs, it is of utmost importance to take 

into account the presence of uncertainties that may arise regarding their main parameters, such as operational 

costs, prices of raw materials and products, product demands, energy consumption, transportation, the products 

853



lifetime and etc. Baghalian et al. (2013) have considered demand-side and supply-side uncertainties 

simultaneously in a stochastic mathematical formulation for the strategic design of multi-product SC involving 

capacitated production facilities, distribution centres and retailers. Jouzdani and Govindan (2021) have 

investigated the interrelations and interactions of the three aspects of sustainability through the development 

and implementation of a multi-objective stochastic mathematical programming model for optimisation of the 

costs, energy consumption, and the transport traffic associated with dairy SC operations. Ebrahimi et al. (2022) 

have proposed a multi-objective mixed-integer programming model for the design of sustainable economic dairy 

SC based on the triple bottom line strategy. Kalantari Khalil Abad et al. (2022) have developed a two-stage 

stochastic programming model for a strategic design of a green closed-loop SC. To control the uncertainty of 

strategic decisions, demand and the upper bound of emission capacity with three possible scenarios have been 

considered.  

The presented above approaches aim to achieve sustainability or considered dairy SC concerning 

environmental impact, most often in terms of the greenhouse gas emissions related to the transport, 

minimisation of consumed energy by the production of the products, maximisation of the profit, minimisation of 

the total costs, products lifetime, social satisfaction and etc. The uncertainties which have been taken into 

consideration are regarding most often product demands, transport and production costs, facility capacity, 

transport traffic, etc. 

Kirilova et al. (2022a) have proposed a deterministic approach for the optimal design of sustainable dairy SC for 

the production of different types of dairy products according to different technologies while satisfying economic, 

environmental and social criteria defined in terms of costs. The environmental criterion includes assessments 

of pollutant emissions in relation to two areas of impact - air and water. The model of Kirilova et al. (2022a) has 

been extended with a Robust Counterpart (RC) for uncertain product demands (Kirilova et al., 2022b) using the 

approach of Ben-Tal et al. (2005). It was solved considering only economic and environmental criteria for a case 

study consisting of two suppliers, two dairies and two markets. 

The present study is an implementation of the approach developed by Kirilova et al. (2022b) to a different case 

study for the SC consisting of three suppliers, two dairies and three markets, taking into account the three criteria 

of sustainability - economic, environmental and social. As an uncertain parameter, the environmental impact 

costs for treatment of the generated CO2 emissions related to the transportation and consumed energy by 

production facilities and wastewater generated during the production of the dairy products and these ones 

associated with used raw materials are considered. 

2. General formulation of the optimisation problem for the design of a sustainable dairy SC 
under uncertain environmental costs 

The considered approach has been developed to plan the activities in a SC, including milk suppliers, dairies and 

markets, to satisfy given consumer demands (short-term) for the production of two types of cottage cheese - 

low-fat content and high-fat content according to different technologies using as raw materials standardised 

whole milk and skimmed condensed milk. 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. The approach includes three interconnected models for (i) the production of the products according to 

different recipes; (ii) SC design; (iii) the environmental impact assessment of the SC, and (iv) the social impact 

assessment related to the employees to be hired by suppliers, dairies and markets. The latter depends on the 

average quantities of raw materials/products processed by these employees. 

The environmental impact is assessed in terms of two areas: 

1). Wastewater generated at each processing task of the production technologies, including those related to 

pre-processing of used raw materials. As an environmental impact indicator of generated wastewater Biological 

Oxygen Demand for 5 days (BOD5) is used. 

2). CO2 emissions related to the energy consumption by the dairies and CO2 emissions produced during the 

transportation of raw materials and products between suppliers, dairies and markets when vehicles with different 

payload capacities and fuel engines are used.  

Wastewater produced in the dairies is treated in Wastewater Treatment Plants (WWTPs) at a given cost. The 

expenses to be paid to the WWTPs, associated with the considered products are limited for each dairy.  

Air Pollution Tax (APT) is also imposed concerning the CO2 emissions associated with the production of the 

products and transportation of raw materials and products. It aims to maintain the quantity of the emitted CO2 

below an acceptable level.  

The used optimisation criterion is represented by the income from the market sale of products after the deduction 

of all costs - production costs, raw materials costs, transportation costs, and environmental and social costs. 

The robust optimisation framework includes four groups of data: 1). Data about the contents of used raw 

materials and dairy products; 2). Data about the SC structure. The latter involves data about the used production 

854



system, the capacities of milk supply centres, 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). Data about the 

costs related to the employees (job positions) to be hired by suppliers, dairies and markets. They include costs 

for salaries, social benefits, working clothes, medical care and insurance and the average quantities of raw 

materials/products processed by employees in suppliers, dairies and markets. 

In addition, the following decision variables are defined: 1). Binary variables for the SC structuring; 2). 

Continuous variables for the raw materials and product flows between suppliers, dairies and markets. The latter 

accounts for the quantities of both types of raw materials and the quantities of dairy products; 3). Continuous 

variables for the milk fat content in the used milk; 4). Integer variables for the employees (job positions) to be 

hired by suppliers, dairies and markets. The latter depends on the processed amounts of raw materials/products. 

The proposed optimisation framework includes the constraints related to 1). Conducting production of the 

products in the dairies in the predefined time horizon; 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 air and water emissions of pollutants. The latter are considered uncertain parameters.  

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 

to produce 1 kg of product. Information about size factors, production units, production tasks, processing times, 

etc., is given in Kirilova and Vaklieva-Bancheva (2017). 

Several optimisation problems have been formulated and solved at different realisations of the nominal values 

for the environmental costs regarding wastewater and CO2 emission of pollutants generated randomly with 

uniform distribution under two levels in specified uncertainty sets. A detailed description of formulated 

optimisation problems, including needed data, all mathematical models, constraints and optimisation criteria is 

provided in Kirilova et al. (2022a). A mathematical formulation of the robust model is given in Kirilova et al. 

(2022b). 

3. Case study 

The presented above robust optimisation approach is applied in a real case study from Bulgaria, including the 

production of two types of cottage cheese according to two production technologies using standardised 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 to three 

markets. The product’s production is realised over the time horizon of one month. Used production units (UNEP, 

2000) for conducting the production tasks and their summarised volumes are listed in Table 1. 

Table 1: Equipment units with summarised volumes (m3) 

 Milk tanks Pasteurisers Curd vats Drainers 

Dairy 1 1,450 800 950 300 

Dairy 2 1,450 950 1,050 340 

Capacities of the three suppliers (kg) and milk prices (BGN/kg) are presented in Table 2 (EU PRICES of COW's 

RAW MILK, 2023). Table 3 lists product demands and product prices in the markets. Distances (km) between 

suppliers, dairies and markets are listed in Table 4.  

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 

In Table 5 are presented the data about vehicles used for transportation of raw materials and products. 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). 

855



Table 3: Products demands (kg) and products prices (BGN) 

 Product demand (kg) Products prices (BGN) 

 Product 1 Product 2 Product 1 Product 2 

Market 1 32,730 30,830 10.8 12.1 

Market 2 17,830 20,300 12.3 11.4 

Market 3 20,000 19,500 12.7 12.5 

Table 4: 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 

Table 5: Data about used vehicles for transportation of raw materials and products 

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/100 km) 

Fuel price, 

(BGN/L) 

Milk tanker truck with petrol engine 2,500 8.056 0.249 32 2.22 

Refrigerator truck with diesel engine 4,000 9.583 0.267 23 2.27 

The environmental costs associated with transportation are obtained using data listed in Table 5 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 (Maslarski and Tomova, 1980). The CO2 

emissions associated with both processes is 0.46 kg CO2/kWh (Covenant of Mayors and Joint Research Centre 

of the European Commission, 2014). The costs of CO2 are 0.00998 BGN/kg CO2. 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 6 the average costs (BGN), related to the number of employees (job positions) to 

be hired by the suppliers, dairies and markets are presented (Living and working conditions: Bulgaria, 2023). 

They include costs for salaries (BGN) (Salaries in the category: Agriculture, food industry, 2023), 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 6: Data related with employees 

Employees Costs for 

salaries, (BGN) 

Working clothes, 

(BGN) 

Social benefits, 

(BGN) 

Medical insurance, 

(BGN) 

Average quantities of 

raw materials/products 

that employees can 

process per day, (kg) 

Suppliers 1,300 50 100 90 500 

Dairies 2,300 50 200 90 150 

Markets 1,700 50 100 90 40 

4. Results and discussion 

The proposed robust optimisation approach has been implemented in a real case study from Bulgaria. The dairy 

SC includes three suppliers, two dairies and three markets. As uncertain parameters the environmental costs 

for treatment of the generated CO2 emissions related with transportation and energy consumed by product 

facilities and wastewater generated at each processing task of the production recipes, including those related 

with pre-processing of used raw materials, are considered. Several robust optimisation problems were 

formulated and solved under nominal product requirement data and two different uncertainty levels (UL) (ρ = 

0.5 and 1). Under each uncertainty level, ten random realisations (scenarios) have been uniformly generated in 

the following uncertainty set: [𝑛𝑜𝑚𝑖𝑛𝑎𝑙 𝑣𝑎𝑙𝑢𝑒 ± 𝜌∗SD], where SD is the standard deviation (SD) of the obtained 

results. The optimisation models have been solved using GAMS® optimisation 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: the mean and 

standard deviation of the obtained results. The optimisation problems have been formulated and solved at given 

boundaries of varying of the environmental costs, as follows: cost for treatment of wastewater generated in both 

856



dairies - 1,500 ÷ 5,000 BGN and costs for treatment of CO2 emissions of pollutants associated with 

transportation of raw materials and products between suppliers, dairies and markets and energy consumed by 

production facilities - 5,000 ÷ 15,000 BGN. The obtained results for model solutions are shown in Figure 1 (a) 

and Figure 1 (b). From Figure 1 (a), it can be seen that the higher level of uncertainty (ρ=1) is associated with 

greater environmental impact cost values for treating air and water generated pollutant emissions, economic, 

environmental, social cost values, and profit. The increase in the standard deviation values of the economic, 

environmental, social costs, and the profit obtained at the higher uncertainty level are 6.8 %, 6.8 %, 7.2 %, and 

7.4 %. One can see that large variations in the level of uncertainty do not lead to a large difference in the 

standard deviations of the abovementioned values. 

 

 
(a) (b) 

Figure 1 (a): Optimal solutions for the SC for economic, environmental, social costs and profit, at two UL levels 

(b) Optimal solutions for the number of employees to be hired from the suppliers, dairies and markets, at two 

UL levels for 10 scenarios S1÷S10 

The same conclusion can be made for the results presented in Figure 1 (b). In this case, the number of 

employees to be hired by suppliers, dairies and markets is kept constant at the different uncertainty levels, 

except for the case of the markets. In this case at the higher uncertainty level, the standard deviation values of 

the obtained results are about 7 %. From the resulting products portfolios of the dairies under the different 

scenarios, it can be concluded that imposing strict restrictions on the environmental costs for treatment of the 

generated emissions of pollutants in air and water result in solutions that do not satisfy predefined product 

demands. 

5. Conclusions 

The research represents an implementation of the robust optimisation approach for optimal design of 

sustainable dairy SC for production of two types of cottage cheese using two technologies at uncertain 

environmental costs. The considered SC comprises three milk suppliers, two dairies and three markets, 

operating under uncertain environmental costs for treatment of the generated CO2 emissions related to 

transportation and energy consumed by product facilities and wastewater generated at each processing task of 

the production recipes, including those related with pre-processing of used raw materials. Several robust 

optimisation problems have been formulated and solved under different random realisations of the uncertain 

parameters taking into account economic, environmental and social considerations. The nominal data for the 

environmental costs were randomly generated at two levels using uniform distribution in a specified uncertainty 

set. Two performance measures were used to evaluate the optimisation models: the mean and SD of the 

objective function values under random realisations.  

The obtained results show that a higher level of uncertainty is associated with greater environmental impact 

costs for treating pollutants released in air and wastewater, economic, environmental, and social costs, and 

profit. At the higher level of uncertainty, there is an increase in the values of the standard deviation of the 

obtained results by 6.8 % for economic and environmental costs and 7.2 % and 7.4 % for social costs and profit. 

Similar is the case with the results obtained for employees to be hired by the suppliers, dairies and markets. In 

279,520

73,880

284,680

79,290

11,990
2,840

12,170
3,050

150,280

35,700

152,580

38,460

533,840

124,880

542,620

134,820

0

50,000

100,000

150,000

200,000

250,000

300,000

350,000

400,000

450,000

500,000

550,000

Mean, ρ=0.5 SD, ρ=0.5 Mean, ρ=1 SD, ρ=1

Economic costs, (BGN) Environmental costs, (BGN)

Social costs, (BGN) Profit, (BGN)

0

10

20

30

40

50

60

70

S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 Mean SD

Suppliers, ρ=0.5 Dairies, ρ=0.5 Markets, ρ=0.5

Suppliers, ρ=1 Dairies, ρ=1 Markets, ρ=1

857



them, at the higher level of uncertainty, the largest increase in the values of the standard deviation of the 

obtained results of 7 % is related to employees hired by the markets. On the other hand, imposing strict 

restrictions on the environmental costs for treatment of the generated emissions of pollutants in air and water 

result in solutions that do not satisfy predefined product demands. The obtained optimal values of the economic, 

environmental and social costs show that the optimisation approach implementation results in sustainable 

solutions that do not change significantly with an increase in the uncertainty level of consideration of the 

environmental costs. The developed optimisation model takes into consideration environmental assessments of 

pollutants that are released only into air and water. For the future can be extended with including an assessment 

of the environmental impact of pollutants released in soil. 

Acknowledgments 

The study represents results obtained under project “Sustainable supply chains in terms of environmental, 

economic and social criteria” funded by Bulgarian Science Fund under Contract No. КΠ-06-Н37/5/06.12.19. 

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