This paper describes two methods for using the single -objective solver that comes with Microsoft Excel and the TOPSIS decision -making technique for multi -objective optimisation of a two-stage vapour -compres sion refrigeration (VCR) system . The Excel -aided m odel developed for analysing the exergetic and economic performance of the sys tem wa s first used to obtain six optimised solutions by using the two solution methods provided by S olver that separately maximise d the system’s exergetic efficiency (ε) and mini mise its total cost -rate ( Ctotal) and equipment cost rate ( Cequip). A dual -objective optimised solution that simultaneously maximises ε and minimises Ctotal was also obtained by using the MIDA CO solver. The first method applies TOPSIS to rank the seven optimised solutions by using five weighting schemes. As should be expected, the results of this method show that MIDACO ’s dual -objective solution achieved the first rank , while t he two Solver solutions that minimised the equipment cost rate occupied the lowest two ranks. According to this method, t he two Solver solutions that maximise d the exergetic efficiency closely satisfy the dual objective of the 2E optimisation. The second method applies TOPSIS in an active manner by using Solver to adjust the relevan t design variables so that the parameter Ci that measures the closeness of the base design to the ideal dual -objective target is maximised . The results obtained show that this method cannot only produce a multi - objective solution, but that its solution is closer to the ideal dual -objective target than MIDACO’s 2E solutio n. Keywords: m ulti-objective optimisation, sing le-objective optimisation, multi -stage vapour compression refrigerati on, Microsoft Excel Solver, MIDACO solver Multi -objective optimisation of VCR systems by applying TOPSIS to the single -objective solutions obtained with Excel Solver Mohamed M. El -Awad Independent Researchers and Academic mmelawad09@gmail.com Abstract This paper describes two methods for using the single -objective solver that comes with Microsoft Excel and the TOPSIS decision -making technique for multi -objective optimisation of a two-stage vapour -compres sion refrigeration (VCR) system . The Excel -aided m odel developed for analysing the exergetic and economic performance of the sys tem wa s first used to obtain six optimised solutions by using the two solution methods provided by S olver that separately maximise d the system’s exergetic efficiency (ε) and mini mise its total cost -rate ( Ctotal) and equipment cost rate ( Cequip). A dual -objective optimised solution that simultaneously maximises ε and minimises Ctotal was also obtained by using the MIDA CO solver. The first method applies TOPSIS to rank the seven optimised solutions by using five weighting schemes. As should be expected, the results of this method show that MIDACO ’s dual -objective solution achieved the first rank , while t he two Solver solutions that minimised the equipment cost rate occupied the lowest two ranks. According to this method, t he two Solver solutions that maximise d the exergetic efficiency close ly satisfy the dual objective of the 2E optimisation. The second method applies TOPSIS in an active manner by using Solver to adjust the relevan t design variables so that TOPSIS parameter that measures the closeness of the base design to the ideal dual -objective target is maximised . The results of this method show that it can produce a solution that is closer to the ideal dual -objective target tha n MIDACO’s 2E solutio n. Keywords: multi-objective optimisation, sing le-objective optimisation, multi -stage vapour compression refrigerati on, Microsoft Excel Solver, MIDACO solver 1. Introduction The significant share of vapour -compression refrigeration (VCR) system in the energy consumption of residential, commercial, and industrial sectors necessitates improving the efficiency of these systems. One method for increasing the systems’ coeffici ent o f performance (C OP) is to use multi -stage compression [1,2] . Since the improved systems cost more than the simple systems, their feasibility depends on careful compromise s between their electrical energy consumption and cost . The increasing concern about t he environmental change due to global warming and the ozone -layer depletion added the need for replacing the conventional synthetic refrigerants with more environment -friendly fluids as a third factor [3 ]. This inspired many researchers to be involved in d eveloping suitable multi -objective optimisation (MOO) method s for optimising the energetic, economic, and environmental (3E) performance of VCR systems u sing natural or environment -friendly refrigerants [4]. Ahmed et al. [1 ] analysed a two -stage VCR system by using a novel hybrid multi - objective grey wolf optimizer (HMOGWO) algorithm. The system wa s model led using response surface methods (RSM) to investigate the impacts of design variables on the set responses. Three conflicting scenarios in bi-objective o ptimis ation were built focusing on the overall system following the TOPSIS and LINMAP decision - making methods. The optimal solutions indicate that for the first to third scenarios, the exergetic efficiency ( ε) and capital expenditure are optimis ed by 33.4% and 7.5%, and the ε and operational expenditure are improved by 27.4% and 19.0%. The ε and global warming potential are also optimis ed by 27.2% and 19.1%, where the proposed HMOGWO outperforms the MOGWO and NSGA -II. Based on the research outcomes, they co ncluded that the co mbined RSM and HMOGWO technique is an excellent solution to simulate and optimis e two -stage VCR systems . Singh et al . [4] analysed an ammonia -based multi -stage VCR system incorporated with a flash intercooler which also works as a sub -cooler. They carried out a thermo - economic optimisation of the system in order to maximise its exergetic efficiency and minimise its total capital cost rate. The e vaporator temperature, condenser temperature, subcooling parameter, and de -superheating parameter were considered as design variables for their MOO analysis . They also employed the multi -objective genetic algorithm tool provided with MATLAB to carry out the optimisation analysis and used EES to determine the thermodynamic properties of the refrigerants. TOPSIS [5] was used to select unique solutions for five different weighting factors of exergetic efficiency and total cost. Their results reveal ed that the exergetic efficiency and total capital cost of the system at the thermo -economic optimal operating conditio ns were 41.76% and $223,717.6 , respectively . Being widely -available general -purpose software with powerful analytical tools, Microsoft Excel allows more researchers and engineering students to join the search for alternative environment -friendly refrigerants and contribute to the development of innovative VCR systems. Excel is supported by a versatile solver for single - objective optimisation (SOO) analyses and the VBA programming language that comes with MS applications can be used for developing property functions for various conventional and alternative refrigerants [ 6]. However , currently Excel doesn’t have its own solver fo r MOO analyses . Although a free version of the MIDACO solver [7] is available for Excel users, it allows only four design variables to be considered in the analysis ; which is not adequate for analy sing multi -stage compression systems with various design parameters such as compressor efficiency and sub -cooling and superheating degrees . This paper shows how Excel’s Solver can be used with TOPSIS for MOO analyses of two-stage compression VCR systems . The idea of using a SOO solver for a MOO analysis is not new. Balabanov [8] demonstrated the method by solving a mathematical multi -objective problem by using the NLP Solver submodule in LibreOffice Calc. However , there is no published work on the use of similar methods for MOO analyses of VCR systems on any platform . The paper presents two methods for using TOPSIS with Solver to conduct MOO analyses the first of which applies TOPSIS simply to identify a Solver solution(s) that is close to satisfy the multi - objective requirement. The second method uses Solver and TOPSIS to improv e a base design so as to achieve the multi -objective requirement . The results of the two methods are compared with a real MOO solution obtained by using the limited - version of the MIDACO solver . 2. The a nalytical model for the two -stage compression VCR system Figure s 1 and 2 show schematic and T-s diagram s of the two-stage compression VCR system which is to be analysed with the assumed input paramet ers shown on Table 1. The liquid refrigerant expands in the first expansion valve to the flash chamber pressure (pFC). Part of the liquid vaporises during this process and t he resulting saturated vapour (state 3) is mixed with the superheated vapour from the low - pressure compressor (st ate 2) before entering the high -pressure compressor ( state 9 ). Although the refrige rant’s flow to the evaporator is reduced, the total compression work is also reduced and the net effect is an increase in the system’s COP * 9]. Since the COP depends on the flash -chamber pressure, this pressure has to be optimised. Figure 1 . Schematic of the two -stage VCR system Figure 2 . T-s diagram of the two -stage VCR system Condenser Compressor 2 Direct contact heat exchanger Compressor 1 Evaporator 8 1 2 3 9 4 5 6 7 Conde Throttle valve 1 Throttle valve 2 Flash chamber Table 1. Assumed input parameters for thermodynamic analysis of the syst em [10 ] Parameter Value Cooling capacity of the system, CC 10 kW Overall heat transfer coefficient for evaporator , UEvap 0.03 kW/m2.K Overall heat transfer coefficient for condenser , UCond 0.04 kW/m2.K Ambient temperature, T0 25oC Temperature change for air in evaporator and condenser ± 5oC Temperature of the inlet air to evaporator 0oC 2.1. The thermodynamic model The system will be analysed for a constant condenser temperature of 40oC, but the evaporator temperature is allowed to vary in the range −25oC ≤TEvap ≤−15oC. The inter - stage pressure, pFC, is initially determined as: Cond Evap FC p p p  (1) The i sentropic efficiencies of the two compressor s are assumed to vary with the corresponding suction and discharge pressures as follows [4]: In Out Comp P/P . . 046670850 (2) Table 2 shows the mass, energy and exergy balance equations for the different system components. Table 2. Equations for mass and energy balances and rates of exergy destruction in the individual system components Mass balance Energy balance Exergy destruction Evaporator rm m m 8 1 CChmhm 88 11 8 11 0 ssmT EvapT/CC Compressor 1 1 2m m 1 2 1 hhm Wr Comp  1 210 ssmT Compressor 2 6 9 4 1x /m m mr 3 4 3 2 hhm WComp  9 4 40 ssmT Condenser 6 4 5 1x /m m mr CondQhmhm  44 55 4 5 4 0 ssmT Cond cond T/ Q Throttle valve 1 6 5 6 1x /m m mr 5 6hh 5 6 05 ssTm Throttle valve 2 rm m m 7 8 7 8hh 7 807 ssTm Flash chamber rmx mx m 6 56 3 rmx m  6 71 77 83 66 hmhmhm   6 610 1sxmT 7 6 36 1 sx sx Direct contact heat exchanger rm m m m  3 2 9 99 3 22 hmhmhm   6 901 1x /sTm   6 36 2 1x /sxs  The system’s total compression work , COP , and ex ergetic efficiency ( ε), are given by: 2 1 Comp Comp Total W W W  = 6 9 4 1 2 1x /hhm hhmr r   (3) TotalW/CC COP  (4) TotalD Total Total W/ E W  (5) Where CC is the system’s cooling capacity and D TotalE is its total ex ergy destruction rate which is the summation of the exergy destruction ra tes in the eight components: D DCHXD FCD TVD TVD CondD CompD CompD EvapD Total E E E E E E E E E  2 1 2 1 (6) Table 2 also shows the equations used to determine the exergy destruction rates . 2.1. The economic model The total annualised cost rate of the system is given by [10]: env op k total C C C C   4 1 (7) where, kC is the capital and maintenance cost rate of individual components, opC is the operational cost rate of the system, and envC is the CO 2 penalty cost rate of the system. The total capital and maintenance cost rate is calculated by adding the capital and maintenance cost rate s of the indivi dual component given by [11]: CRF..C Ck k (8) where, ϕ is the maintenance factor and CRF is the capital recovery factor defined as : 1 1 1 n ni /ii CRF (9) Where i is the interest rate and n is the system’s expected lifetime . The capital costs of the various system components are estimated using th e relations shown on Table 3. Table 3. Capital cost functions of the various system components [10] Component Capital cost function Evaporator 8901397. eva eva A C  Low -temperature compressor 460 1 1 5 10167. Comp Comp W. C  High -temperature compressor 460 2 2 2 9624. Comp comp W. C  Throttle valve 1 5 1 5114 m. CTV Condenser 8901397. con con A C  Throttle valve 2 7 2 5114 m. CTV Flash chamber and direct contact heat exchanger 6703280. r FC m. C  [12] The heat -transfer areas of the evaporator and condenser given in Table 3 are obtained by using the log -mean temperature difference (LMTD) method. The o perational cost rate of the system is mainly the cost of electricity as given by: elec op c.N.W C (10) Where N is the annual operational hours and celec is the cost of electricity in $/kWh. Following Wang et al . [13], the CO 2 penalty cost rate of the systems is calculated from: 2 2 COeCO env c. m C (11) Where, 2COc is the penalty cost of the avoided CO 2 emission and eCOm 2 is the amount of annual CO 2 emission from the system that can be estimated by: annual eCO eCO E. m 2 2 (12) Where eCO2 is the region al (country) electricity conversion factor and annualE is the annual amount of energy consum ed by the system. The values of N, eCO2 , celec, and 2COc used in the present ana lysis are shown on Table 4 . Table 4. Assumed input parameters for economic analysis of the system [10] Parameter Value Maintenance factor, ϕ 1.06 Interest rate, i 14% Plant life time, n 15 Years Annual operation hours, N 4266 hour s Electrical power cost, celec 0.09 $/kWh Emission factor, eCO2 0.968 kg/kWh Cost of CO 2 avoided, 2COc 0.09 $/kg of CO 2 emission 2.2. Development of the Excel -aided model Figure 3 show s the front sheet (Sheet 1) of the Excel -aided model developed for the VCR system. The data part on the left side of the sheet shows the specified values of the evaporator and condenser temperature s (T_E and T_C), the values of the flash - chamb er pressure and temperature (P_fc and T_f c), the isentropic efficiencies of the two compressor s (η_c1 and η_c2), and the system’s cooling capacity (C C). The calculations part in the central part of the sheet determines the enthalpy and entropy of the refrigerant at all the 8 states by using VBA functions [6 ]. Note that the refrigerant nam e is stored as a variable (Fluid) so that the same model can be used for other refrigerants without modification . The results part on the right -hand side of the sheet determines the refrigerant’s mass flow rate, the compressors’ work, the rates of exergy d estruction in the eight system components, the COP , and the overall exergetic efficiency of the system. Finall y, the sheet displays the total cost rate and the total equipment cost rate as calculated by the back sheet (Sheet 2 ) that applies the economic model as shown on Figure 4 . Sheet 2 determines the areas of the evaporator and condenser by using the LMTD method to calculate their costs from the relations shown on Table 3 . Values of the temperatures involved are imported from Sheet 1 . Figure 3 . The f ront sheet of the model for the thermodynamic calculations Figure 4 . The back sheet of the model for the economic calculations 3. Single -objective optimisation by using Solver Two design variables are considered in the present analysis for optimising the VCR system ; which are the inter -stage pressure , pFC, and evaporator temperature, TEvap. The thermodynamic and economic performance of the VCR system is measured by two performance indicators ; which are the exergetic efficiency , ε, and total cost rate Ctotal. Figure 5 show s the variation of the two indicators with pFC. Figure 5.a shows that the exergetic efficiency reach es its maximum value at an inter -stage pressure of about 400 kPa, while Figure 5 .b shows that the total cost rate increases gradually with the pressure and does not reach a maximum within the specified range of pFC. Determining the value s of pFC and TEvap that optimise each of the two performance indicators requires the use of a single -objective solver such as Excel’s Solver. Figure 6 shows Solver’s set -up for determining the values of pFC and TEvap that maximise the system’s exergetic efficiency. Four constraints are imposed on the solution to keep the values of pFC and TEvap within the range s 200 ≤ pFC ≤800 kPa and −25oC ≤ TEvap ≤ −15oC, respectively . (a) (b) Figu re 5. Variation of the exergetic efficiency and total cost rate with the inter -stage pressure Figure 6. Solver set -up for maximising the exergetic efficiency Figure 7 shows the solution obtained by the GRG Nonlinear method of Solver . According to this solution, the maximum exergetic efficiency is 64.77 % which is achieved at TEvap of -15oC and pFC of 338.914 kPa. Another solution was obtained for maximising ε with the Evolutionary method and four other solutions were obtained for minimising the total cost rate or the equipment cost rate by using Solver’s two solution method s. Table 5 shows the six values of the exergetic efficiency and total 40455055606570 0 200 400 600 800Exergetic efficiency (%) Inter -stage pressure (kPa) 12000125001300013500140001450015000 0 200 400 600 800Total cost rate ($/y) Inter -stage pressure (kPa) cost rate as d etermined by the six solutions that will be used for obtaining a dual - objective solution by using the proposed Solver -TOPSIS technique to be discussed in Section 5 . A real MOO solution is needed as a reference for verifying and also clarifying the proposed method. This is dealt with in the following section. Figure 7 . Solver solution for maximising the exergetic efficiency with the GRG Nonlinear method Table 5 . Results of the single -objective optimisation analyses using Solver Solution method Objective ε Ctotal GRG Nonlinear Maximise ε 64.770 12961.882 Minimise Ctotal 63.292 12546.965 Minimise Cequip 56.688 12493.246 Evolutionary Maximise ε 65.172 13114.228 Minimise Ctotal 59.722 12342.795 Minimise Cequip 56.688 12493.246 4. Dual -objective optimisation by using MIDACO MOO analyses , in which two or more conflicting performance indicators are to be simultaneously optimised, require a multi -objective sol ver to be used. This is illustrate d by using the MIDACO solver [ 7] for a dual -objective ( 2E) optimisation analysis of the two -stage compression system. The conflict between the two objectives , which are maximising the system’s total exergy efficiency and minimising its total cost rate, is clearly shown on Figure 5. The MIDACO solver can be used with Excel as an add-in just like Solver. Figure 8shows the set -up for the MIDACO solver for this analysis. As Figure 8 shows, there are two changing variable s in the analysis which are the evaporator temperature TEvap stored in cell B3 and the inter -stage pressure pFC stored in cell B9. Since a MOO analysis does not lead to a single solution but a Pareto front, a de cision -making method has to be applied to select the most desirable solution. Figure 9 shows the Pareto front obt ained by MIDACO and Figure 10 shows the selected solution. According to this solution , the optimum evaporator temperature and inter -stage pressure are -17.7oC and 302.95 kPa, respectively, at which the values of the exergetic efficiency and total cost rate are 63.447% and $12544.7/y , respectively. Figure 8. MIDACO set -up for the dual -objective optimisation analysis Figure 9. Pareto front of the 2E solution obtained by MIDACO Figure 10. The dual -objective optimised solution obtained by MIDACO Figure 11 shows the values of the two performance indicators as obtained by the dual -objective solution of MI DACO together with the six single -objective solutions of Solver. The figure shows that Solver’s solution that m inimise d Ctotal with the GRG Non linear method (the blue colour ) is practically the same as MIDACO’s entry (the red colour) . The following section shows how Solver and the TOPSIS decision - making method [5] can be used to obtain a solution that satisfies the dual -objective requirement by applying a systematic procedure . Figure 11. The seven optimised solutions obtained by Solver and MIDACO 5. Dual -obje ctive optimisation by using Solver solutions and TOPSIS Being developed a s a tool for SOO analyses, Solver does not produce a Pareto front but its two solution methods, the GRG Nonlinear method and the Evolutiona ry method, provide multiple optimised solutions for the various performance indicators. The TOPSIS technique [5, 14 ] allows th e solution that is closest to satisfying the dual -objective requirement to be identified. Although , the number of 12,961.88 12,546.97 12,493.25 13,114.23 12,342.80 12,493.25 12,546.98 12300124001250012600127001280012900130001310013200 56 58 60 62 64 66Total cost rate ($/y) Exergetic efficiency (%) optimised solutions that can be available for the selection process will be far less than a MOO solver can generate , the decision -making proce dure that follows allows various weighting factors to be tested ; something that may not be allowable by a proper MOO solver. TOPSIS follows six well -defined steps explained in the a ppendix to rank the different solutions depend ing on the weighting factors provided to it (refer to Equation A.3) . Five schemes of weighting factors are applied as shown on Table 6 to compare the seven optimised solutions shown on Figure 11 . Schemes 1 and 5 are strongly unbalanced schemes that give a large weight factor of 0.75 t o one performance indicator and only 0.25 to the other indicator. Schemes 2 and 4 are weakly unbalanced schemes (0.6/0.4 or 0.4/0.6 ), while Scheme 3 (0.5/0.5) is a balanced scheme that gives the same weight to both performance i ndicators. Table 6. The fiv e weighting schemes for applying TOPSIS method Weighting scheme Maximise exergetic efficiency (benefit) Minimise total cost rate (non -benefit) 1 0.75 0.25 2 0.6 0.4 3 0.5 0.5 4 0.4 0.6 5 0.25 0.75 Figure 12 shows the Excel sheet developed for applying the TOPSIS method by modifying an example sheet available at [15]. Note that the ‘benefit’ objective for this analysis is maximising the exergetic efficiency (ε), while the ‘non -benefit’ objective is minimisin g the total cost rate (C_total) . The sheet shown on F igure 12 applies Scheme 3 that gives the same weight to the benefit and non -benefit objectives . As the formula bar shows, ranking of the seven solutions is done by using Excel’s function ‚Rank‛ so that the different schemes can be applied by simply adjusting the values of the weight factors W1 and W2 stored in cells B4 and C4 . Figure 12. TOPSIS sheet using Scheme 3 to rank the seven solutions According to Scheme 3, the solution closest to the ideal configuration (rank 1) is that obtained with the MIDACO solver . The two solution s obtained by the GRG Non linear method and the Evolutionary method of Solver for maximising the exergetic efficiency occupied the second and third rank s, respectively , while t he two solutions tha t minimised the total cots rate occupied the fourth and fifth ranks. The two solutions that minimised the equipment cost occupied the two lowest ranks. Table 7 summarises TOPSIS results with the fiv e schemes applied to the seven solutions with the two Solver solutions obtained by the GRG Nonlinear method listed first followed by the two solutions obtained by the Evolutionary method and then the 2E solution obtained by MIDACO. The figure s on Table 7 show that the five weighting schemes give different ranks to each solution , but MIDACO’s 2E solution achieved the highest average rank . Also, note the high ranks reached by the two solution s obtained with GRG Nonlinear method that minimised the total cost rate and maxi mised the exergetic efficiency with the corresponding solutions obtained with the Evolutionary method occupying the fourth and fifth ranks. In agreement with Figure 11, the method gi ves the lowest two ranks to Solver’s solutions that minimised the equipment cost. Table 7 . Results of applying the TOPSIS method to Solver and MIDACO optimised solutions with the five weighting schemes Schemes 1 2 3 4 5 Average Solver1_ε 6 3 3 1 1 2.8 Solver1_Ctotal 2 2 2 3 4 2.6 Solver1_Cequip 4 6 6 6 6 5.6 Solver2_ε 7 4 4 4 2 4.2 Solver2_Ctotal 3 5 5 5 5 4.6 Solver2_Cequip 4 6 6 6 6 5.6 MIDACO 1 1 1 2 3 1.6 A more effective application of the Solver -TOPSIS technique is to use TOPSIS , not only in a passive manner to rank the different Solve r solutions , but in an active manner to improve the base design variables so as to achieve a multi -objective target . For that purpose, the sheet that applies TOPSIS is integrated with the analytical model as the third sheet and Solv er is used to maximise the value of Ci in Equation (A.8) by adjusting the base -design variables . Figure 13 shows the first sheet of the integrated model that copies th e value of Ci from sheet 3 as shown on the formula bar. The third sheet (not shown) copies the values of the exergetic efficiency and total cost rate from sheet 1 as the seventh option (replacing the MIDACO ’s2E solution). Note that the value of Ci for the base design (0.771 ) is lower than that of the 2E solution obtained by MIDACO (0.786 ). Solv er can now be used to maximise the value of Ci by adjusting the value s of Tevap and pFCin Sheet 1 using the same set -up shown on Figure 6. Figure 14 shows the third sheet of the solution obtained in which the value of Ci increased to 0.807 which is higher than that of MIDACO ’s 2E solution . According to this solution, Tevap increased to -18.07oC and pFC increased to 336.5kPa. Figure 15 compares the rates of exergy destruction in the system as determined by the present method to those of the base desi gn and the 2E solution obtained by MIDACO. Figure 13 . Sheet 1 of the TOPSIS -integrated analytical model ( before maximising Ci) Figure 14. Sheet 3 of the TOPSIS -integrated analytical model ( after maximising Ci) Figure 15. Comparison of the rates of exergy destruction in the system -0.20.00.20.40.60.81.01.21.41.6Rates of exergy destruction (kW) Base MIDACO Solver_TOPSIS Figure 15 shows that the main rates of exergy destruction occur in the two compressors , valve 1, the condenser, and then valve 2. MIDACO’s solution reduced the rate of exergy destruction of compressor 1, but increased that of compressor 2. By comparison, the solutio n obtained by the present technique reduced the rates of exergy d estruction i n both compressors and also in the condenser and valve 1. Although it increased the rate of exergy destruction in valve 2, its total rate of exergy destruction in the system is lower than that of the base design and that determined by MIDACO. 6. Conclusions Single -objective optimisation of VCR systems can easily be performed by using Excel’s Solver with any practical number of changing variables, but a solver for multi -objective optimisati on is currently un available for the general Excel users . Although the free version o f the MIDACO solver can be used , its capability is limited to four changing variables. This paper presents two method s for utilising the TOPSIS decision -making method and Solver so as to achieve multi -objective optimisation of VCR systems . In the first method, Solver’s two solution methods are used to generate multiple optimised solutions for the key performance indicators and TOPSIS is then used to rank the various solutions according to the designer’s preference. In the second method, Solv er is used to maximise the value of Ci that measures the closeness of the design to the ideal dual -objective target by adjusting the relevant design variables. The two methods are a pplied for a dual -objective optimisation of a two -stage compression VCR sys tem that simultaneously maximises its exergetic efficiency and minimises its total cost rate by changing the values of the evaporator temperature and inter -stage pressure. The reliability of the first method is tested by including a true 2E solution for th e optimisation problem obtained by using the MIDACO solver. The results of this method show that MIDACO’s 2E solution obtained the first rank closely followed by Solver’s solutions with the two solution methods for maximising the exergetic efficiency. As s hould be expected, the method shows that Solver’s solutions for minimising the equipment cost rate are the farthest from satisfying the dual -objective requirement of the optimisation . The result of the second method shows that it is more effective than the first one since it could produce a solution that is closer to the dual -objective target than MIDACO’s 2E solution. 7. Nomenclature Latin Letters A Heat -transfer area (m2) C Annual cost rate ($/year) DE Rate of exergy destruction (kW) m Mass flow (kg/s) Q Heat -transfer rate (kW) W Compressor power (kW) CC Cooling capacity (kW) c Unit cost (e.g. electricity price) ($/kWh) COP Coefficient of performance ( -) CRF Capital recovery factor ( -) E Electrical energy consumption (kWh) h Specific enthalpy (kJ/kg) i Interest rate (%) N Number of operation hours per year (hours) n Plant life time(years) p Pressure (kPa) s Specific entropy (kJ/kg) T Temperature (K or oC) U Heat transfer coefficient ( kW/m2.K) x Vapour quality ( -) Greek Letters  Compressor ’s isentropic efficiency (%) eCO2 Regional (country) electricity conversion factor ϕ Maintenance factor Indices k System component number Abbreviations MOO Multi -objective optimisation SOO Single -objective optimisation TEWI Total equivalent warming impact VBA Visual Basic for Applications language VCR Vapour -compression refrigeration Subscripts 0 Ambient Casc Cascade CO 2 Avoided carbon dioxide emission Comp Compressor Cond Condenser DCHX Direct contact heat exchanger elec Electrical env Environmental evap Evaporator FC Flash chamber op Operation r Refrigerant Total Total work or exergy destruction TV Throttle valve