This paper describes a method for multi -objective optimisation analyses of multi - stage vapour -compression r efrigeration (VCR) systems by utilising the single -objective solver of Microsoft Excel and the TOPSIS decision -making method . The paper illustrates the Solver -TOPSIS method by ana lysing an innovative two-stage VCR system that adds a heat -recovery intercooler to the conventional system after the first -stage compressor. The Excel -aided model developed for the system calculates its coefficient of performance (COP), exergetic efficiency, total cost rate, and total equivalent warming impact (TEWI ). Four desig n parameters are treated as variables which are the evaporator and condenser temperatures, the isentropic efficiency of the compressor, and the flow rate of the cooling water . Comparing the values of the four performance indicators as optimised by the proposed method with those optim ised by the MIDACO solver shows that the method yields higher values of the system’s COP and exergetic efficiency and lower TEWI at the expense of slightly increas ing the total cost rate. The total rate of exergy destruction in the system optimised by meth od is also lower than in the one optimised by MIDACO. KEYWORDS : Multi -stage VCR systems, multi -objective optimisation, TOPSIS, Excel INTRODUCTION The significant share of vapour -compression refrigeration (VCR) systems in the energy consumption of residential, commer cial, and industrial sectors emphasises the importance of improving the performance of these systems [1,2]. In this respect, multi - مجلة البحوث الهندسية كلية الهندسة -جامعة طرابلس Journal of Engineering Research. Faculty of Engineering -University of Tripoli 2 stage compression allows various innovative methods to be used for reducing the energy consumption of thes e systems . Since the improved systems cost more than the simple ones, their economic feasibility requires careful balance s between their electrical energy consumption and capital costs. A third factor has now become equally important due to the increasing concern about the harmful effects of global warming and ozone -layer depletion, which is the need to replace the usual synthetic refrigerants with mor e environment -friendly fluids [3 ]. The quest to design innovative VCR systems using environment -friendly refrigerants and to develop suitable methods for their energetic, economic, and environmental optimisation has inspired many researchers to be involved. Roy and Mandal [4 ] conducted a thermo -economic analysis of a simple VCR system using three refrigerants w ith low global -warming potential ( GWP ) namely, R152a, R600a and R1234ze. Developing their model with Engineering Equation Solver (EES), they evaluated the effect of evaporator and condenser temperatures on the system’s coefficient of performance (COP), exergetic efficiency , and annual plant co st rate. They carried out a multi -objective optimisat ion (MOO) analysi s by using MATLAB toolb ox and used the TOPSIS method [5 ] to select the best optimised solution. Their results show ed that R152a gave the best performance among the three investigated ref rigerants. Aminyavari et al . [6] analysed a 50-kW CO 2/NH 3 cascade refrigeration system by evaluating its exergetic, economic , and environmental performance. They also developed their model in MATLAB but employed a specific MATLAB function for calling the REFPROP data base [7] to obtain the refrigerants’ thermodynamic data. Their MOO analyses used a genetic algorithm method to achieve the optimal design parameters of the system and used TOPSIS to select the final optimum point from the set of optimal soluti ons achieved. Their resul ts showed that the optimum design results in exergetic efficiency of 45.89% and a total cost rate of 0.01099 US$/s . Singh et al . [8] analysed an ammonia -based multi -stage compression 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 evaporator temperature, condenser temperature, subcooling parameter , and de -superheating parameter were considered as design variables for their MOO analysis. They employed the multi -objective genetic algorithm tool provided with MATLAB to carry out the optimisation analysis and used EES to determine the thermodynamic pro perties of the refrigerants. TOPSIS was used to select unique solutions for five different weighting factors of exergetic efficiency and total cost. Their results revealed that the exergetic efficiency and total capital cost of the system at the thermo - economic optimal operating conditions were 41.76% and 223,717.6 USD, respectively. The above example studies show that most researchers used commercial software for their MOO analyses like MATLAB, EES, and REFPROP. However, the use of general -purpose applica tions can encourage independent researchers and engineering students to contribute to the development of innovative VCR systems using environment - friendly refrigerants . Microsoft Excel, which is an easy -to-learn general -purpose spreadsheet application, has a versatile solver for single -objective optimisation (SOO) analyses. Regarding MOO analyses, a free version of the MIDACO solver [9 ] is available for Excel users, but it allows only four design variables to be considered in the analysis; which is not adequate for analysing multi -stage compression and cascade VCR systems مجلة البحوث الهندسية كلية الهندسة -جامعة طرابلس Journal of Engineering Research. Faculty of Engineering -University of Tripoli 3 with multiple design parameters. Since the development of MOO solver s is too complex and take s a long time even for top software professionals [10], El-Awad [11] described a method for using the same SOO Solver provided by Excel for conducting MOO analyses with practically any number of design variables by utilising the TOPSIS method . The present paper applies the method for energetic, exergetic, economic, and env ironmental (4E) optimisation of an innovative two -stage compression VCR system. DESCRIPTION OF THE TWO -STAGE VCR SYSTEM The two-stage compression VCR system shown on Figure (1) adds a water intercooler to the conventional system for cooling the superheated refrigerant leaving the first-stage compressor. Apart from reducing the compression work in the second -stage compressor, the hot water exiting the intercooler can be utilised f or various needs. This system is a modified version of that describe d by Anjum et al. [12] in which the cooled refrigerant directly goes to the flash chamber which it exits as dry saturated vapour. The system shown on Figure (1) adds a direct -contact heat -exchanger (DCHX) for mixing the refrigerant leaving the intercooler with the dry saturated vapour leaving the flash chamber so that the refrigerant enters the second -stage compressor as s lightly -superheated vapour . Figure (2) shows the T-s diagram of the m odified system . Figure 1: Schematic of the two-stage VCR system with heat -recovery . Figure 2: T-s diagram of the two-stage VCR system with heat -recovery. Evaporator DCHX Condenser Flash chamb er Water intercooler with heat recovery Compressor 2 Compressor 1 Valve 1 Valve 2 7 1 2 2b 3 4 9 5 6 8 Tw2 Tw1 s 1 T 2 3 4 5 6 7 8 9 2b مجلة البحوث الهندسية كلية الهندسة -جامعة طرابلس Journal of Engineering Research. Faculty of Engineering -University of Tripoli 4 Anjum et al. [12] conducted a thermodynamic analysis of the original two-stage system by using ammonia as the refrigerant and the input parameters shown on Table 1 . The present analysis assumes the same input parameters , but deals with a multi -objective optimisation analysis of the improved system with R152a as refrigerant . Table 1 : Assumed values of the input parameters for analysing the VCR system [ 12] Parameter Value Cooling capacity of the system, CC 10 kW Evaporator temperature, TE -15oC Condenser temperature, TC 40oC Ambient temperature, T0 25oC Temperature change for air in evaporator and condenser , ΔT ± 5oC Temperature of the inlet air to evaporator , Tair 0oC Inlet temperature of cooling water , Tw1 17oC Isentropic efficiency of compressor, ηc 80% Overall heat transfer coefficient for evaporator , Ueva 0.03 kW/m2.K Overall heat transfer coefficient for condenser , Ucon 0.04 kW/m2.K Overall heat transfer coefficient for intercooler , Uintr 0.1 kW/m2.K Maintenance factor, ϕ 1.06 Interest rate, i 14% Plant life time, n 15 Years Annual operation hours, N 4266 hour Electrical power cost, celec 0.09 $/kWh Emission factor, eCO2 0.968 kg/kWh [1 3] Cost of CO 2 avoided, 2COc 0.09 $/kg of emitted CO 2 THE THERMODYNAMIC MODEL Table 2 shows the mass and energy balance equations for the different system components. The mass flow rate of the refrigerant in the evaporator is given by: 8 1hh/CC mr  (1) where CC is the cooling capacity of the system. The temperatures T2 and T4 are determined from the temperatures T2s and T4s following isentropic compression processes: c s /T T T T 1 2 1 2  (2) c s /T T T T 9 4 9 4  (3) The exit temperature of the cooling water Tw2 is determined from the specified value of the heat -exchanger effectiveness, ε: 1 2 1 2 w w w TT T T  (4) The enthalpy of the cooled refrigerant is then determined from energy balance across the intercooler as shown on Table 2 . The total compression work is given by: 2 1 comp comp Total W W W   = 6 9 4 1 2 1x /h hm h hmr r   (5) The COP and overall exergetic efficiency, ε, of the system are given by: مجلة البحوث الهندسية كلية الهندسة -جامعة طرابلس Journal of Engineering Research. Faculty of Engineering -University of Tripoli 5 Table 2 : Mass and energy balance equations and exergy destruction rates in the system components Mass balance Energy balance Exergy destruction rate Evaporator rm m m 8 1 CC hmhm 88 11   ETCCssmT8 11 0 Compressor 1 1 2m m 1 2 1 h hm Wr comp  1 210 s smT Compressor 2 6 9 4 1x /m m mr 9 4 3 2 h hm Wcomp  9 4 30 s smT Condenser 6 4 5 1x /m m mr condQ hm hm  44 55     Ccond TQssmT 4 5 4 0 Throttle valve 1 6 5 6 1x /m m mr 5 6h h 5 6 05 ssTm Throttle valve 2 rm m m 7 8 7 8h h 7 807 ssTm Flash chamber 56 3 mx m rm mx m   5 6 71 fcP@gh h3 fcP@fh h7 77 83 66 hm hm hm   6 610 1sxmT 7 6 36 1 sx sx Intercooler rm m2 2 1 w wm m 2 2h hb 2 1 2 m/T Tcmw w pww   br ww sm smT2 2 0 2 1 sm smr ww Direct heat exchanger 3 2 9 m m m  99 33 22 hm hm hmb   33 2 99 0 sm smsmTbr  TotalWCCCOP (6)      TotalE C W. T/. T CC 1 15273 15273   (7) The total exergy destruction in the system D TotalE is given by: D CondD TVD CompD DCHXD FCD ICD TVD CompD EvapD Total E E E E E E E E E E  2 2 1 1 (8) The exergy destruction rates in the different system components are shown on Table 2 . THE ECONOMIC MODEL The total annualised cost rate of the system is given by [12]: env op kk total C C C C    9 1 (9) where, kC is the capital and maintenance cost rate of individual components, opC is the operation cost rate of the system, and envC is the CO 2 penalty cost rate of the system. The total capital and maintenance cost rate of the sy stem is calculated by adding up the capital and maintenance cost rate of the individual component s which is given by: CRF..C Ck k (10) مجلة البحوث الهندسية كلية الهندسة -جامعة طرابلس Journal of Engineering Research. Faculty of Engineering -University of Tripoli 6 where, Ck is the capital cost of the component, ϕ is the maintenance factor , and CRF is the capital recovery factor obtained from:  1 11 nn iiiCRF (11) where i is the interest rate and n is the system’s expected lifetime. The values of i and n used in the present analysis are given in Table 1. The costs of the two throttle valves, flash chamber, and direct contact heat exchanger have been ignored in some analyses because they are minor compared to those of the two compressors, the eva porator, and the condenser . In the present analysis , the cost of the DCHX is taken as equal to that of the flash chamber. The capital costs of individual components are estimated by using t he relations shown on Table 3 . Table 3 : Capital cost function s of the different components [8, 12 ] Component Capital cost function Evaporator 8901397. eva eva A C  Compressor 4605 10167. comp W. C  Condenser 8901397. con con A C  Throttle valve m. CTV 5114 Flash chamber 6703280. ref FC m. C  Intercooler 6809 2382. rint rint A . C  The heat -transfer areas of the evaporator, condenser, and intercooler in the relations shown on Table 3 are determined by using the log -mean temperature difference method. The operational cost rate of the system is the cost of electricity given by: elec op c.N.W C (12) where N is the annual operational hours and celec is the cost of electricity in $ /kWh. Following Wang et al. [14 ], the penalty for the system ’s CO 2 emission is calculated from: 2 2 COeCO env c. m C (13) 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 from: annual eCO eCO E. m 2 2 (14) where eCO2 is the emission factor and annualE is the annual amount of energy consumed by the system. The values of N, eCO2 celec, and 2COc are given in Table 1. THE TOTAL EQUIVALENT WARMING IMPACT (TEWI) TEWI is a non -monetary measure of the direct and indirect global warming effects of the refrigeration systems. The direct effect results from the refrigerants being directly released or leaked into the atmosphere and the indirect effect is caused by the CO 2 مجلة البحوث الهندسية كلية الهندسة -جامعة طرابلس Journal of Engineering Research. Faculty of Engineering -University of Tripoli 7 emissions in thermal power plan ts that use fossil fuels to produce the electrical energy. The refrigerant TEWI is calculated by usi ng the following correlation [15 ].    n E mn L m GWP TEWIannual ref annual ref ref    1 (15) where GWP Ref is the GWP of the refrigerant, mRef is the total refrigerant charge, Lannual is the refrigerant leakage rate,  is the recycling factor, Eannual is the energy consumed per year, and β is the electricity regional conversion factor. Table 4 show s how mref and Lannual are calculated and gives the values of , β, and GWP ref for R 152a [15]. The underlined term on the right side of Equation ( 15) is the indirect part of the TEWI. Table 4 : TEWI analysis assumptions [15 ] Parameter mref [kg] Lannual [%]   [kg.CO 2/kWh] GWP ref Assumed value s mref240 12.5 0.7 0.65 140 THE EXCEL -AIDED MODEL AND SOO ANALYSES BY USING SOLVER Figure (3) shows the fi rst sheet of the Excel -aided model developed for the VCR system . The first column on the left side of the sheet stores the assumed data for the thermodynamic model such as the evaporator and condenser temperatures, the mass flow rate and inlet temperature of the cooling water, the intercooler ’s effectiveness, etc. The following two columns determine the temperature, enthalpy, and entropy v alues of the refrigerant and water at the various states by using Thermax property functions [16]. The formula bar reveals the formula in cell N13 that calculates ε accor ding to Equation ( 7). The fourth column from the left calculates the rates of exergy d estruction in all the system components. The last column on the right determines the mass flow rate of the refrigerant, the compression work and four performance indicators; COP, ε, CTotal, and TEWI. Figure 3: Sheet 1 of the model for the two -stage compression VCR system Figure (4) shows the second sheet of the model that calculates the cost rates of the nine system components using the relevant cost relations shown on Table 3. The formula bar shows th e formula in cell E12 that calculates the capital cost of the intercool er. Figure مجلة البحوث الهندسية كلية الهندسة -جامعة طرابلس Journal of Engineering Research. Faculty of Engineering -University of Tripoli 8 (4) show that the total cost of the system is dominated by the costs of the evaporator, the condenser, and the two compressors. Solver , the single -objecti ve solver that comes with Excel, can be u sed to optimise any of the four key performance indicators shown on Figure 3 by using either the GRG Nonlinear method or the Evolutionary method. Figure (5) shows the set -up for maximising the system ’s COP by using the Evolutionary met hod. Figure 4: Sheet 2 of the model for the two -stage VCR system Figure 5: Solver set -up for maximising the COP of the VCR system مجلة البحوث الهندسية كلية الهندسة -جامعة طرابلس Journal of Engineering Research. Faculty of Engineering -University of Tripoli 9 Figure 5 shows that the optimisation analysis involves four changing variables which are the evaporator temperature , the condenser temperature, the inter -stage pressure, and the f low rate of the cooling water. Figure 5 also shows the specified ranges within which t he four variables are allowed to vary. The other thr ee performance indicators were similarly optimised by select ing the respective objective cell and minimisin g or maximising it s value with Solver. Table 5 shows the values of the four key-performance indicators for the base design and those obtained by the four optimised solutions . Table 5: The s ingle -objective optimised solutions obtained by Solver for the system Optimisation objective COP ε [%] Ctotal [$/y] TEWI [kg CO 2/y] Base design 2.958 70.112 12337 .00 211988.7 Maximise COP 3.474 71.555 14456.96 181066.2 Maximise ε 3.445 72.142 14200.60 182500.8 Minimise total cost rate 2.296 64.777 11166.29 272165.0 Minimise TEWI 3.574 71.816 14214.03 176028.0 MULTI -OBJECTIVE OPTIMISATION OF THE SYSTEM BY USING MIDACO Although four performance indicators were considered in the previous single - objective optimisation analyses of the system, t he dominance of the indirect global - warming effect on the TEWI enables the multi -objective optimisation analysis to be conducted with either the COP or TEWI. Therefore, the multi -objective optimisation analysis presented below involves only three performance indicators which are maximising the system’s exergetic effi ciency, minimising its total cost rate, and minimising its TEWI . The best trade -off between the se three conflicting objectives can be found by using a MOO solver such as t he MIDACO solver [ 9]. Although the free version of MIDACO is limited to four changing variables to be considered in the analysis, it is adequat e for the present analysis in which the four changing variables are the same as those used for Solver and shown on Figure (5). Figure (6) shows MIDACO’s set-up for the analysis. As a multi -objective optimisation solver, MIDACO produce s a Pareto front containing a set of un-dominated optimum solutions from which the best optimum soluti on is selected. Figure (7) shows the Pareto f ront obtained by MIDACO and Figure (8) shows the selected 3E optimal solution. Figure (9) shows the percentage deviations of the four single -objective solution s obtained by Solver as shown on Table 5 from the 3E solution obtained by MIDACO . Compared to Solver’s three solution s that minimised the system’s TEWI, maximised its exergetic efficiency, or maximised its COP , the 3E solution reduced both the COP and exergetic efficiency of the system and increased the TEWI . The trade -off for degrading these three performance indicators is that the 3E solution reduced the s ystem’s total cost rate. Compared to Solver’s solution that minimised the total cost rate, the 3E solution increased the total cost rate but considera bly increased both the COP and exergetic efficiency and reduced the TEWI . It can be judged from the scale of deviations shown on Figure (9) that the closest single -objective s olution to the 3E solution is the one that maximised the system’s exergetic effic iency followed by the one that maximised its COP. مجلة البحوث الهندسية كلية الهندسة -جامعة طرابلس Journal of Engineering Research. Faculty of Engineering -University of Tripoli 10 Figure 6: MIDACO ’s set-up for the 3E optimisation of the VCR system Figure 7: The Pareto front of the 3E optimi sed solution for the VCR system مجلة البحوث الهندسية كلية الهندسة -جامعة طرابلس Journal of Engineering Research. Faculty of Engineering -University of Tripoli 11 Figure 8: The selected 3E optimised solution obtained by MIDACO Figure 9: Percentage deviations of Solver ’s four single -objective solutions from the 3E optimised solution obtained by MIDACO MOO ANALYSIS OF THE VCR SYSTEM BY USING THE SOLVER –TOPSIS TECHNIQUE The TOPSIS decision -making technique rank s the different choices under consideration by evaluating an overall index, Ci, that measures the relative distance s of these choices from the ideal choice according to t he following relationship [6, 17 ]:   j jj iS SS C (14) Where  jS and  jSare the distances from “benefit” and “non -benefit” ideal choices which are calculated based on the weighting factors provided to the method . Therefore, the method requires the benefit and non -benefit objectives as well as the weighting factors to be identified. This section shows how the single -objective solution s obtained by Solver can be used to achieve multi -objective optimisation of the VCR system by using the Solver -TOPSIS technique [11] that can be used to obtain a 3E optim ised solution with any number of design variables and various weighting factors. -40 -20 0 20 40 60 80Maximise COPMaximise εMinimise total cost rateMinimise TEWI COP ε Ctotal TEWI مجلة البحوث الهندسية كلية الهندسة -جامعة طرابلس Journal of Engineering Research. Faculty of Engineering -University of Tripoli 12 The Excel sheet shown on Figure (10) applies the TOPSIS method to the four single -objective optimised solutions obtained by Solver together with the 3E optimised solution obta ined by MIDACO. The sheet is a modified version of an example sheet available at [18]. The values of the four performance indicators obtained by the five solutions are stored as a matrix in cells B6:E10 . Note that there are two “benefit” objectives for this analysis , which are maximising the system’s COP and exergetic efficiency (ε) , and two “non -benefit” objectives , which are minimising the system’s total cost rate (C_total) and TEWI. The sheet shown on Figure ( 10) applies a balanced weight ing scheme th at gives equal weights to these four ob jectives by assigning the value 0.25 to each of the four weighting factors W1 to W4 stor ed in cells B4 to E4. Figure 10: TOPSIS sheet for ranking the different optimised solutions obtained by Solver and MIDACO As the formula bar shows, ranking of the five solutions is done by using Excel’s function “Rank”; which makes it easy to judge the different optimised solutions with different values of the four weight factors by giving more weight to any of these factors. With the balanced weighting scheme, Figure ( 10) shows that the solution that is nearest to satisfying the multi -objective requirement , i.e. the one with the largest value of Ci, is not that obtained by MIDACO but that obtained by Solver for minimising the TE WI. The figure also shows that t he solution with the smallest value of Ci is that for minimising the total cost rate. By incorporating the TOPSIS sheet with the Excel -aided model of the VCR system , the scheme can be used not only to rank the four Solver solutions, but to maximi se the value of Ci for the system by using Solver also. Figure (11) shows the third sheet that has been added to Excel-aided model shown on Figure (3) . This sheet copies the values of the four performance indicators from Sheet 1 int o its cells B9 to E9 as shown on Figure (11), while Sheet 1 copies the corresponding value of Ci from Sheet 3 into its cell N17 as shown on Figure (12). Note that the values of the fou r changing variables (the evaporator and condenser temperatures, the int er-stage pressure, and the water mass flow rate) are those of the base design that replaced the MIDACO solution in Sheet 3. Also n ote that Sheet 3 now shows the values of the four design variables , together with the value of Tw2, before Solver is used to maximise the value of Ci by adjusting the four chang ing variables in Sheet 1. Figures ( 13) and ( 14) show Sheet 1 and Sheet 3 with the solution obtained by Solver with the Evolutionary method and same constraints shown on Figure (5). مجلة البحوث الهندسية كلية الهندسة -جامعة طرابلس Journal of Engineering Research. Faculty of Engineering -University of Tripoli 13 Figure 11: Sheet 3 of the extended Excel -aided model for the system (base design) Figure 12: Sheet 1 of the extended Excel -aided model for the system (base desi gn) Figure 13: Sheet 1 of the extended Excel -aided model for the optimised system مجلة البحوث الهندسية كلية الهندسة -جامعة طرابلس Journal of Engineering Research. Faculty of Engineering -University of Tripoli 14 Figure 14: Sheet 3 of the extended Excel -aided model for the optimised system Comparison of Figure (14) with Figure ( 11) shows that Solver increased the value of Ci from 0.591 to 0.76 5 by adjusting the values of the four design variables as shown on Table 6 . Figure (15) compares the modified values of the four optimised performance indicators obtained by Solver with those of the base design and those obtained by the MOO solution of MIDACO. The figure shows that the increments in the COP and exergetic efficiency obta ined by Solver -TOPSIS solution are significantly higher than those of the solution obtained by MIDACO while the TEWI is significantly lower. T hese improvements are achieved by increasing the total cost rate which increased by 9.65% . (Actually, both the cos t of electricity and the penalty for CO 2 emissions decreased, but the purchased equipment cost increased by 12% ). With respect to the hot water, Table 6 shows that the method decreased the flow rate from 0.005 kg/s to 0.004 kg/s, but increased the exit temperature from 34.6oC to 35.99oC. Table 6: Particulars of the base design, the MIDACO solution, and the Solver - TOPSIS solution Base design MIDACO Solver -TOPSIS TE [oC] -20 -15 -15 TC [oC] 40 41.439 38.010 Pic [kPa] 331.256 351.239 377.971 waterm [kg/s] 0.005 0.001 0.004 Tw2 [oC) 34.61 33.29 35.99 Figure (16) compares the rates of exergy destruction in the different system components of the base design to those adjusted by the MIDACO solver and the Solver - TOPSIS technique. As the figure shows, the highest rate of exergy destruction for all three systems occur s in compre ssor 2 followed by compressor 1, throttle valve1, throttle valve 2 and then the condenser. By comparison, t he rates of exergy destruction in the remaining three components are negligible. The figure also shows that the total rate of exergy destruction in the system optimised by the Solver -TOPSIS technique is less than the corresponding values of the base design and that optimised by the MIDACO solver . مجلة البحوث الهندسية كلية الهندسة -جامعة طرابلس Journal of Engineering Research. Faculty of Engineering -University of Tripoli 15 Figure 15: Comparison of the four key performance indicators for the base design with those obtained by MID ACO and the Solver -TOPSIS technique Figure 16: Exergy destruction rates in the base design and at the optimal solutions obtained by MIDACO and the Solver -TOPSIS technique CONCLUDING REMARKS This paper describes a method for utilising the TOPSIS decision -making technique with the single -objective Solver of Microsoft Excel for m ulti-objective optimisation analyses of multi -stage compression VCR systems. The method is applied to analyse an innovative two -stage system that incorporates an intercooler after the first compression stage and recovers the waste energy for producing hot water. Four single -objective solutions are first obtained by using Solver to maximise the COP and exe rgetic efficiency and minimise the total cost and TEWI . Four design variables are used in these analyses which are the evaporator and condenser temperature s, the isentropic efficiency of the compressor, and the water flow rate . Solver is then used to simultaneously satisfy the four 2.963.173.47 2.62.833.23.43.6 (a) COP [-]Base MIDACO Solver-TOPSIS 70.11 69.2871.21 6869707172 (b) ε [%]Base MIDACO Solver-TOPSIS 12,33712,42513,527 115001200012500130001350014000 (c) Total cost rate, [$/y]Base MIDACO Solver-TOPSIS 211,989 198,070 181,250 160000180000200000220000 (d) TEWI [kg/y]Base MIDACO Solver-TOPSIS -0.200.20.40.60.811.2Rate of exergy destruction [kW] Base MIDACO Solver-TOPSIS مجلة البحوث الهندسية كلية الهندسة -جامعة طرابلس Journal of Engineering Research. Faculty of Engineering -University of Tripoli 16 objectives by using the TOPSIS method . Comparison of the optimised system obtained by this method with that obtained by using the MIDA CO multi -objective solver shows that the proposed method leads to higher values of the system’s COP and exergetic efficiency and a lower value of its TEWI at the expense of increas ing the total cost rate. List of Symbols and Abbreviations c - Unit cost C - Cost rate CC - Cooling capacity of the system COP - Coefficient of performance CRF - Capital -recovery factor DE - Rate of exergy destruction GWP - Global -warming potential h - Enthalpy i - Interest rate MOO - Multi -objective optimisation m - Mass of refrigerant m - Mass flow rate N - Annual operation hours n - Plant life time Q - Rate of heat transfer s - Entropy SOO - Single -objective optimisation T - Temperature TEWI - Total equivalent warming index W - Work x - Quality or dryness fraction of refrigerant Greek letters:  - Refrigerant recycling factor β - Electricity regional conversion factor ε - Exergetic efficiency η - Isentropic efficiency of compressor μ - Emission factor ϕ - Maintenance factor, Subscripts: CO 2 - Carbon -dioxide مجلة البحوث الهندسية كلية الهندسة -جامعة طرابلس Journal of Engineering Research. Faculty of Engineering -University of Tripoli 17 elec - Electricity r - Refrigerant ref - Refrigerant w - Cooling water