A DEA approach for merging dairy farms

209 Agricultural Economics – Czech, 67, 2021 (6): 209–219 Original Paper https://doi.org/10.17221/418/2020-AGRICECON Supported bythe Spanish Ministry ofScience and the European Regional Development Fund (ERDF) (Grant No.DPI2017- -85343-P). ADEA approach for merging dairy farms Sebastián Lozano1, Belarmino Adenso-Díaz2* 1Department ofIndustrial Management, Engineering School, University ofSeville, Spain 2Department ofIndustrial Engineering, Engineering School, University ofOviedo, Spain *Corresponding author: [email protected] Citation: Lozano S., Adenso-Díaz B. (2021): ADEA approach for merging dairy farms. Agric. Econ. –Czech, 67: 209–219. Abstract: This paper proposes amodel for determining the most advantageous merger within aset ofdairy farms. Ituses data envelopment analysis (DEA) toestimate the total technical efficiency improvement that the merger would produce and for decomposing itinto alearning effect and apure merger effect. Adesign ofexperiments has also been carried totest the effects ofvarious factors (the total number offarms, the standard deviation ofherd size, the percentage offarms exhibiting increasing returns toscale, the standard deviation ofthe current technical efficiency ofthe farms) ondifferent response variables (the percentage offarms involved inthe merger, the reduction ofherd size and the efficiency improvement obtained bythe merger). Theresults show that the disparity inthe herd size ofthe farms inaregion and the percentage offarms that exhibit increasing returns toscale increase the number offarms that enter into the most advantageous merger. Thedisparity ofherd size also increases the number ofcows that are not needed after the merger. Finally, the expected efficiency improvement increases with the total number offarms. Keywords: dairy farming; land consolidation; learning effect; pure merger gains; technical efficiency According toFAO (2014), the vast majority offarms around the world (more than80%) are family-run and operate atasmall scale (less than2 ha). Agriculture atasmall scale has many advantages, including providing employment and food security tomillions ofpeople and improving environmental sustainability. However, the small scale implies many sources of inefficiency, which makes this type ofagriculture less competitive inthe market. Innovation, technology and resource access are challenges difficult toovercome when operating atsmall scale, making itmore difficult tocompete inprice long-term (Raup 1972). Given the importance offixing the rural population and guaranteeing a sustainable agriculture system, some governments have passed various land consolidation policies torevert that situation since mid-last century. Promoting group farming and farmers' co-ops has led individual farmers topool their resources, sharing costs and profits, and gaining alarger scale and greater competitiveness. Agarwal and Dorin (2019) present many cases ofgroup farming incountries such asNorway and Ireland for the case ofmilk, Japan with rice crops, different former socialist countries orIndia. Piet etal. (2012) report that the number ofFrench farms decreased one third inthe period 1970–2007, increasing the farms average size from 18.84ha to53.96ha during that period. Monke etal. (1992) remarked onthe viability and economic benefit obtained byfarm size enlargement inthe case ofnorthern Portugal. Inthe case ofcow-calf farms, inSweden Holmström etal. (2018) analysed how creating larger pastures from scattered plots yields positive effects, thanks to lower operating costs resulting from economies of scale. For the case ofmilk production, similar results were observed byCorral etal. (2011) inSpain, with higher profitability for dairy farms (DFs) with lower land fragmenta- 210 Original Paper Agricultural Economics – Czech, 67, 2021 (6): 209–219 https://doi.org/10.17221/418/2020-AGRICECON tion. Different studies conclude that inindustrialised countries, there has been aclear tendency over the last decades todecrease the number offarms and increase their size, although the growth isnot uniform across all countries (Piet atal. 2012). This paper deals with a specific consolidation case, i.e.merging DFs inalimited geographic area. Thegoal is efficiency improvement. Gloy et al. (2002) reported that larger farm size and milking parlours inthe dairy sector significantly increase profitability. Infact, the number ofsmall DFs has decreased atahigh rate intheUSA inrecent decades (Tauer and Mishra 2006). One ofthe first studies onthe consolidation ofDFs was Corral etal. (2011). According totheir analysis, reducing the average number ofplots per farm by80% increases profits by12%, with greater effect when the milk price islow. The goal of the paper is to develop an approach toidentify the most promising mergers among aset ofDFs, assessing and decomposing the efficiency improvements that these mergers would bring about. This research topic has not been dealt with inthe literature. Inorder toassess the relative efficiency ofthe existing DFs aswell asofthe unit that would result from merging them, data envelopment analysis (DEA) isproposed. DEA isanon-parametric methodology that, from the observed data about input consumption and output production and making some simple assumptions, infers the production possibility set (PPS; also known as DEA technology). PPS contains all the operating points that are deemed feasible. The non-dominated region ofthe PPS corresponds tothe efficient frontier, and the DFs whose operating points lie onthat efficient frontier are said toexhibit the best practices (Leetal. 2019). Conversely, those DFs that can reduce their inputs without reducing their outputs orincrease their outputs without increasing their inputs are deemed inefficient, and an inefficiency score that measures the margin for improvement can becomputed. Apart from the technical efficiency improvement that each DF can individually achieve byadopting the best practices oftheir peers (i.e.the so-called learning effect), often the DF can achieve additional gains bymerging. This isthe case when their relative sizes are small, and they operate inaregion ofthe PPS inwhich increasing returns toscale (IRS) prevail. This isthe problem addressed inthis paper, inwhich anew DEAmodel toidentify the most advantageous DFmerger, quantifying the potential efficiency gains, isproposed. Alarge number ofstudies have applied DEA toDFefficiency assessment (Iribarren etal. 2011; Mugera 2013; Mu et al. 2018; Siafakas et al. 2019). Some of these studies have used second-stage regression totest the influence offarm size ontechnical efficiency. Ingeneral, itseems that technical efficiency inDFs ispositively affected byfarm size measured byherd size orby dairy production (Barnes 2006; Demircan etal. 2010). Hansson (2008), however, found anon-linear relationship sothat this positive effect applies only for larger farms and linked this with the capacity for adopting technological innovations. The DEA methodology can also be used to study the desirability of mergers from a technical efficiency point ofview. Adistinction can bemade between those approaches that aim topre-evaluate the potential gains ofagiven merger from those approaches that aim toidentify the units tobemerged. Inthe first category ofstudies (Bogetoft and Wang 2005), the units tobemerged are known beforehand. Itiscommon that all possible combinations (often pairwise) are considered and pre-evaluated (Lietal. 2018). Sometimes only certain mergers fulfilling certain geographical restrictions are meaningful (Zschille 2015), and sometimes such constraints are considered toreduce the number ofpossible combinations (Mydland 2020). Depending onthe application, specific types ofmergers may beinteresting tostudy (e.g.merging hospitals inthe same city ormerging rural hospitals inthe same region). Insome cases, specific mergers that had taken place (Mattson and Tidanå 2019) orthat had been proposed inofficial reports (Flokou etal. 2017) are studied. There are other approaches toestimate efficiency gains ofmergers that are not based onBogetoft and Wang (2005), such asLozano and Villa (2010) orBlancard etal. (2016). All the above DEA merger approaches have one thing incommon: the units tobemerged are given. However, there are afew approaches that aim toidentify the best partner for amerger and set the corresponding input and output targets (Lozano 2013; Zhu et al. 2017). The mixed-integer linear programming (MILP) model used inBlancard etal. (2016) toidentify the firms toenter into amerger with agiven unit isalso ofthis type. Those authors also impose anupper bound onthe number ofunits tomerge. This isanindirect way oftaking into account the unobserved transaction costs derived from areorganisation involving asignificant number ofunits. An important aspect that has tobetaken into account when studying mergers isthe returns toscale (RTS). Scenarios, consistent with the additivity assumption [constant returns toscale (CRS) and non- -decreasing returns toscale (NDRS)], guarantee that the operating point that results from just adding the 211 Agricultural Economics – Czech, 67, 2021 (6): 209–219 Original Paper https://doi.org/10.17221/418/2020-AGRICECON inputs and outputs ofany subset ofunits isalways feasible. Often, however, variable returns toscale (VRS) are considered more realistic (Mydland 2020), but inthat case, not all mergers generate feasible operating points. Inany case, the estimated efficiency gains from mergers are smaller, and hence, fewer mergers appear advantageous inthe VRScase than inthe CRScase (Wuetal. 2011). MATERIAL AND METHODS In this paper, anew DEAmodel for identifying the most advantageous dairy farm (DF) merging is presented. Theproposed approach belongs tothe second category of DEA merger approaches, i.e.the units tomerge are not given but are determined bythe model based onthe observed data. Adifference with other approaches is that no constraint is imposed on the number of units that should enter into the merger, i.e.the optimal number isleft tothe model todecide. TheVRStechnology considered precludes the merger ofalarge number ofunits unless the sample contains units ofwidely diverging sizes. Also, different tothe radial or cost efficiency approaches generally considered inthe literature, the proposed approach uses aslacks-based inefficiency (SBI) measure (Fukuyama and Weber 2009). DEA merger model. Theproposed SBI DEA merger model is[Model(1)]: (1) where: j,r=1,2,…,n–indexes ondecision making units (DMUs); i=1,2,…,m –index oninputs; k=1,2,…,s –index onoutputs; xij–amount ofinput i consumed byDF j; ykj–amount ofoutputk produced byDFj; SBIR –estimated merger efficiency gains; ( ) , x y ms g gg + + = ∈  –SBI directional vector (i.e.slacks-normalisation coefficients); (λ1,λ2,…,λn) –intensity variables used tocompute the merger operating point asalinear combination ofthe observed DMUs; r δ –binary variable indicating ifDFrwill enter into the merger ornot; { } :1 r Rr= δ= –set ofDMUs that will enter into the merger; x iR s –slack (i.e.potential improvement) ofinput i after the merger; y kR s –slack (i.e.potential improvement) ofoutput k after the merger; ˆ iR x –target value ofinput i after the merger; ˆ kR y–target value ofoutput k after the merger. The above model computes input and output targets ofafeasible operating point within the VRStechnology. That target operating point maximises the sum of the normalised input and output improvements over the aggregated virtual point that results from merging the DMUs inthe setR. Theslacks normalisation coefficients x i g and y k g are given and can bechosen, for example, as: max max x i ij j y kj kj g xi g yk = ∀ = ∀ (2) The above MILPmodel always has afeasible solution. Ifweadd the constraint that atleast two units should beselected for merging: 1 2 n r r= δ≥ ∑ (3) Then feasibility isno longer guaranteed asitmay occur, insome cases, that none ofthe possible mergers lies within the VRSPPS. This does not occur if CRS orNDRS are assumed. If sodesired, wecan impose anupper bound onthe maximum number of DF entering into the merger oron the maximum size (in terms ofone ormore inputs oroutputs) ofthe operating point after the merger. Itisalso possible toinclude incompatibility constraints preventing, for example, two specific DFs r'andr" from entering both into the merger, i.e. ' '' 2 rr δ +δ ≤ . It isalso possible toexplore the merger options available toaspecific DFr' byforcing the model toinclude itin the merger compulsorily, i.e. ' 1 r δ= . Another possible scenario isthe case that only DFs that belong toacertain subsetAare allowed tomerge. For example, Amay contain the indexes ofthe DFs that belong toacertain region. Making the model consider the potential mergers ofonly those units can beimposed with 0 r rAδ= ∀∉ . 11 1 Max ms y x iR kR Rxy ik ik s s SBI ms gg = =  = +   + ∑∑ s.t. 11 ˆ nn x iR j ij r ir iR jr x x xs i = = =λ=δ− ∀ ∑∑ 11 ˆ nn y kR j kj r kr kR jr y y ys k = = =λ=δ− ∀ ∑∑ 1 1 n j j= λ= ∑ { } 0 0 0 0,1 y x iR j r kR s is k j r≥∀ ≥∀ λ≥∀δ∈ ∀ 212 Original Paper Agricultural Economics – Czech, 67, 2021 (6): 209–219 https://doi.org/10.17221/418/2020-AGRICECON These are just examples of the multiple modelling possibilities that the proposed approach allows, which can becustomised toeach specific application. Finally, note that all DF DEA applications consider animportant input: the herd size, which isaninteger variable. For that variable (and any other integer variable considered), integrality constraints should be imposed (Lozano and Villa 2006). Merger efficiency gains decomposition. The optimal value of the objective function of Model (1) SBIRrepresents the average input and output improvement that can beobtained from the merger. Thus, if: 11 ,, nn r ir r kr ir kr r r rR rR x y xy = = ∈∈    δ δ=          ∑ ∑ ∑∑ isthe operating point that results from aggregating the inputs and outputs ofthe DMUs prior tothe merger, then the input and output slacks computed bythe model correspond tothe input and output improvements that can bemade from that aggregated operating point. That iswhat the first two constraints ofModel(1) mean. Inother words, the estimated merger efficiency gains SBIR correspond tothe technical inefficiency ofthat aggregated operating point measured with respect tothe target ( ) ˆˆ , iR kR xy computed bythe Model(1). Moreover, since Model(1) projects onto the efficient frontier, the target ( ) ˆˆ , iR kR xy istechnically efficient. However, itmay occur (it isquite likely) that the DFs that will enter into the merger were not technically efficient. That means that part ofthe estimated efficiency gains can bedue toremoving the technical inefficiency present inthe observed DFs prior tothe merger. Thetechnical inefficiency ofany observed DFrcan becomputed using the following conventional SBIDEAmodel [Model(4)]: (4) where: SBIr –slacks-based inefficiency ofDFr. Following the above reasoning, the estimated merger efficiency gains SBIR can be decomposed into a technical efficiency (or learning effect) component LE R SBI that measures the technical efficiency improvementsthat can beachieved without carrying out the merger, only bymaking the inefficient DFs adopt the best practices available, plus the pure merger efficiency gains * R SBI , i.e.: LE Rr rR SBI SBI ∈ =∑ where: LE R SBI –technical efficiency (or learning effect) component; * R SBI –pure merger efficiency gains. A merger R is said to be advantageous if *0 R SBI > , i.e.ifthere are pure merger efficiency gains, and not all the merger efficiency gains are due tolearning effects that are independent ofthe merger. APPLICATION OF THE PROPOSED APPROACH Let us consider a set of 20 dairy farms (DFs) that consume twoinputs [herd size(HS), and operating costs(OC)] toproduce asingle output [milk production (MP)]. Thecorresponding data are shown inTable1 and have been generated randomly using asimilar procedure tothe one described inthe next section for the Monte Carlo experiments. Table1 also shows the results ofprojecting each DMU onto the efficient frontier using Model(4) and using Equation(2) asslacks- -normalisation coefficients. It can be seen that eight ofthe DFs (namely, DF1, DF4, DF7, DF8, DF10, DF11, DF16, and DF20) are efficient. The rest have technical inefficiencies measured by their SBIr inefficiency score. Table1 shows the corresponding input and output slacks. Thesum ofthe reductions ofthe input herd size is147, which represents 6.77% ofthe total herd size ofthe sample. Similarly, the sum ofthe reduction inoperating costs isEUR246288.5 (9.1% ofthe totalOC ofthe sample). Thesum ofthe output increases ismuch more modest, just 12366L ofmilk (0.12% ofthe total milk production ofthe sample). Solving Model(1) with the slacks-normalisation coefficients [Equation(2)] and imposing [Equation (3)] toidentify mergers ofatleast twoDFs, the merger shown asIteration1 inTable2 isobtained. This represents the *LE RRR SBI SBI SBI= − 11 1 Max y ms x ir kr rxy i ik k s s SBI ms gg = =  = +   + ∑∑ s.t. 1 ˆ n x ir j ij ir ir j x xxs i = =λ=−∀ ∑ 1 ˆ n y kr j kj kr kr j y yys k = =λ=−∀ ∑ 1 1 n j j= λ= ∑ 00 0 y x ir j kr s is k j≥∀ ≥∀ λ≥∀ 213 Agricultural Economics – Czech, 67, 2021 (6): 209–219 Original Paper https://doi.org/10.17221/418/2020-AGRICECON most advantageous merger and involves four relatively small DFs, namely DF3, DF11, DF15, and DF20. Note that ofthese four DFs, two are technically inefficient. Adding the corresponding SBIrinefficiency scores ofthe learning effect component can beobtained and subtracting itfrom the overall merger efficiency gains SBIR=0.563 the pure merger efficiency gains can beobtained, i.e.: 3 11 15 20 0.086 + 0.000 + 0.063 + 0.000 = 0.149 LE R DF DF DF DF SBI SBI SBI SBI SBI=+++= = Assuming that this most advantageous DF merger takes place, wecan identify the second most advantageous merger inthe sample. This can bedone using the proposed Model(1) imposing the constraints: * 0.563 0.149 0.414 LE RRR SBI SBI SBI=−=−= The results obtained are shown inTable2 onthe Iteration2row. Inthis case, the merger involves just two DFs (DF2 and DF13). Theestimated overall efficiency gains of this merger and its corresponding learning effect and pure merger efficiency gains are, respectively, 0.292 R SBI = , 0.199 LE R SBI = and * 0.093 R SBI = . Inthis case, most ofthe merger efficiency gains are due tolearning effect. This isdue tothe fact that both DFs entering into the merger are rather inefficient. Note that although the objective function ofModel(1) looks for the most advantageous merger from the point ofview ofthe overall merger efficiency gains, the criterion chosen may be, alternatively, the merger that leads to the maximum pure merger efficiency gains. Table 1. Illustrative dataset: Observed DF data and conventional SBI projections DFrHSr (n)OCr (EUR) MPr (L) SBIr ,, , xx HS r OC r ss (n) ,, , xx HS r OC r ss (EUR) , y MP r s (L) DF1 169 216323.4 1 036 024 0.00000.0 0.0 DF2 114 161 603.0 464 592 0.123 17 57784.9 1891.0 DF3 114 114932.5 377 504 0.086 28 18936.9 5294.6 DF4 97 103818.1 466 483 0.000 00.0 0.0 DF5 140 189119.0 742 424 0.064 12 26386.6 0.0 DF6 102 122630.5 450 775 0.045 720234.7 492.7 DF7 106 136479.2 592 224 0.000 00.0 0.0 DF8 105 152980.8 605 904 0.000 00.0 0.0 DF9 90 102748.5 335 773 0.037 10 11019.7 1379.5 DF10 141 155444.0 729 543 0.000 00.0 0.0 DF11 51 71105.7 116 530 0.000 00.0 0.0 DF12 95 127311.4 449 217 0.039 024915.5 2050.7 DF13 131 131973.3 529 924 0.076 29 12197.3 0.0 DF14 104 112771.4 504 157 0.007 3646.1 0.0 DF15 92 122900.0 358 718 0.063 929037.9 1257.6 DF16 110 162605.3 652 393 0.000 00.0 0.0 DF17 121 160817.5 586 152 0.070 15 26408.4 0.0 DF18 112 116091.4 507 186 0.026 12 1799.0 0.0 DF19 124 167765.8 677 611 0.036 516921.5 0.0 DF20 53 75843.0 137 952 0.000 00.0 0.0 Maximum 169216323.4 1 036 024 – – – – Sum 2 171 2 705263.9 10 321 086 –147 246288.5 12366.0 DF –dairy farm; HSr–herd size; MPr–milk production; n–number; OCr–operating cost; SBI –slacks-based inefficiency; SBIr –slacks-based inefficiency ofDFr; ,, , xx HS r OC r ss , ,, , xx HS r OC r ss –input slacks; , y MP r s –output slacks Dataset generated randomly based onactual unit costs and production rates and using the procedure described inthe text; SBI efficiency scores and input and output slacks computed byModel(4); the input and output data and decision variables are explained inthe text (HSr, OCr, MPr, SBIr, ,, , xx HS r OC r ss , , y MP r s ) Source: Authors' own calculations { } 0 DF3, DF11, DF15, DF20 r rδ= ∀∈ 214 Original Paper Agricultural Economics – Czech, 67, 2021 (6): 209–219 https://doi.org/10.17221/418/2020-AGRICECON That would just require substituting objective function ofModel(1) by: * 11 1 1 Max – y ms x iR kR Rxy i ik k n rr r s s SBI ms gg SBI = = =  = +−   + δ ∑∑ ∑ (5) Of course, this requires having previously solved Model(4) for each DFr. Proceeding as before, we can compute the third, fourth, ..., most advantageous mergers, evaluating each ofthem interms oftheir overall and pure merger efficiency gains. For the dataset considered inthis illustration, the3rd,4th,and 5throunds ofmergers involve each ofthe twoDFs, technically inefficient inall cases. After those five mergers take place, ofthe original 20DFs, only eight remain unmerged, and ofthose eight, all are efficient except two (DF5 and DF19). These unmerged, inefficient DFs can still benefit from the efficiency improvements computed byModel(4). EXPERIMENTAL RESEARCH Experimental design. Tofurther explore the influence ofdifferent factors onsome characteristics ofdairy farm (DF) mergers, aMonte Carlo experiment has been designed. This section presents the factors and response variables considered and the resultsofthe experiments carried out. For the sake ofresults consistency, inall the random instances generated, the total number ofcows inthe whole set ofDFs was constant (N=10000cows). TheDFs were assumed tobelong toacertain geographical region, and this was the total number ofcows inthat region. For each instance, the characteristics of the various farms inthe sample were determined byfour experimental factors which were considered to have apossible influence. As shown inTable3, two levels were considered for each experimental factor. Thefirst Table 2. Illustrative dataset: Five most advantageous DF mergers Merger iteration R , ir kr rR rR xy ∈∈     ∑∑ ( ) ˆˆ , iR kR xy ( ) ,y x iR kR ss SBIR LE R SBI * R SBI 1{DF3, DF11, DF15, DF20} HS (n)310 163 147 0.563 0.149 0.414 OC (EUR) 384781.3 207678.2 177103.1 MP (L) 990704.0 990704.0 0.0 2{DF2, DF13} HS (n)245 163 82 0.292 0.199 0.093OC (EUR) 293576.3 209218.5 84357.8 MP (L) 994516.0 994516.0 0.0 3{DF9, DF17} HS (n)211 153 58 0.219 0.107 0.112OC (EUR) 263566.0 195572.3 67993.7 MP (L) 921925.0 921925.0 0.0 4{DF6, DF12} HS (n)197 150 47 0.183 0.084 0.099 OC (EUR) 249941.9 191497.6 58444.4 MP (L) 899992.0 899992.0 0.0 5{DF14, DF18} HS (n)216 166 50 0.125 0.033 0.092 OC (EUR) 228862.7 211635.7 17227.1 MP (L) 1 011343.0 1 012293.1 950.1 DF–dairy farm; HS–herd size; MP–milk production; n–number; OC–operating cost; R–merging DFs; SBIRslacks- -based inefficiency; LE R SBI –technical efficiency (or learning effect) component; * R SBI –pure merger efficiency gains; x iR s –slack (i.e.potential improvement) ofinput i after the merger; y kR s –slack (i.e.potential improvement) ofoutputk after the merger; xir–amount of inputi consumed byDFr; ˆiR x –target value ofinput i after the merger; ykr–amount ofoutputk produced byDFr; ˆ kR y –target value ofoutputkafter the merger Successive optimal mergers identified using Model(1); ineach case, the setRcorresponds tothe merging DFs; the results show the total inputs and outputs ofthose DF aswell asthe corresponding targets for the merged unit ( ) ˆˆ , iR kR xy ; the resulting improvements (i.e.input and output slacks), the total merger efficiency gain and the proposed decomposition ofthe merger efficiency gains (i.e.learning effect and pure merger efficiency gain) are also shown Source: Authors' own calculations 215 Agricultural Economics – Czech, 67, 2021 (6): 209–219 Original Paper https://doi.org/10.17221/418/2020-AGRICECON two factors refer tothe number ofDFs and the herd size distribution, which correspond tothe first input ofthe DEAmodel. Thus, experimental factorF1 isthe number offarms tobeconsidered (with 100or 500farms aslow and high levels, respectively), while experimental factorF2 controls the similarity ofthe farms regarding the herd size. Theherd size ofeach farm isgenerated using anormal distribution with mean μ=N/F1, and astandard deviation ofσ=μ/5 orσ=μ/3 (low and high levels ofF2, respectively). This was done sothat the sum ofthe herd sizes ofall the farms equalled N. The second input of the DEAmodel [operating costs(OC)] results from multiplying the DF herd size byanOC percapita ratio randomly generated using auniform distribution cμ±0.2cμ, where cμisthe mean operating cost peranimal inthe area. According tothe data gathered inDFs inthe north ofSpain, avaluecμ ofEUR1250peryear was used. The third and fourth experimental factors, F3 and F4, are related tothe output ofthe DEAmodel (milk production). Togenerate this output, asingle-output Cobb-Douglas production function was considered (Chen and Delmas 2012): 12 12 /Y KX X F αα = ε (6) where: K–constant (total factor productivity); X1,X2 –the two inputs; ε–noise random variable (mean value=1); F(≥1) –random inefficiency variable that reduces the actual output obtained with respect tothe maximum technical production achievable. Taking logarithms, 112 2 ln ln ln lnY K X X ef= +α +α + − (7) It has been assumed that e~N(0;0.03) and f~|N(0;σf )| isanon-negative half-normal. Note that the RTS depend onthe value ofα1+α2 sothat CRS apply ifα1+α2=1 while ifα1+α2>1 the production function exhibits IRS (Coelli etal. 2005). Therefore, the last two factors ofthe experiments characterise the efficiency ofthe farms involved. Experimental factorF3 takes into account the percentage offarms exhibiting IRS (with their output using α1+α2>1); for the remaining DFs, half are generated using CRS and the other half using DRS). Finally, factorF4 controls the values ofthe standard deviation ofthe inefficiency variableσf (Table3). Regarding the complexity ofthe experiment, for each ofthe 24factor level treatments inthis full factorial design, 10random instances were generated, which makes atotal of160problem instances. For each instance, the single most advantageous merger was calculated according tothe proposed DEAmodel, and three measures ofthe solution found were recorded asresponse variables for the statistical analysis. Theresponse variable considered were: i)num_cows: the reduction ofthe herd size due tothe merger, ii)perc_farms: the percentage offarms that are chosen toenter into the merger, and iii)var_eff: the expected efficiency improvement due the merger, using the aggregated inputs and outputs , ir kr rR rR xy ∈∈     ∑∑ asslacks-normalisation coefficients. RESULTS AND DISCUSSION The proposed DEA merger model was solved for each ofthe 160instances using Lingo10.0, each run requiring anegligible amount ofcomputation time (less than 1second). Theeffects ofthe four experimental factors onthe three response variables mentioned above were statistically analysed. Figure1 shows some boxplots corresponding tothe two more significant measures. Abigger influence ofthe factors can beseen inthe percentage of farms that merge (perc_farms), mainly by the influenceoffactorsF2 (disparity insize among the farms) and F1 (total number offarms inthe region). Regarding the efficiency improvement response (var_eff), there isaclear influence ofF1 inthe results, with significant higher values for instances with ahigher number offarms involved. Table 3. Factors levels considered inthe experiments Factor Factor level 1 (low) Factor level 2 (high) F1. Total number offarms 100 500 F2. Standard deviation ofherd size σ = (1/5)N/F1 σ = (1/3)N/F1 F3. Percentage offarms exhibiting IRS 1/3 1/2 F4. Standard deviation ofefficiency variable σf = 0.07 σf = 0.15 IRS–increasing returns toscale; N–total number ofcows Source: Authors' own calculations 216 Original Paper Agricultural Economics – Czech, 67, 2021 (6): 209–219 https://doi.org/10.17221/418/2020-AGRICECON Focusing on these two measures, Figure2 shows their average value for each level ofthe four factors. The percentage of farms involved in the merger increases upto1.71points when there isahigher disparity inthe herd size ofthe farms (F2). Also, asthere are 5times more farms inthe higher level ofF1 with respect tothe lower level, the percentage offarms involved inthe merger decreases by0.73points onaverage. Regarding the num_cows measure, its mean value increases slightly (by an amount of 0.21) when the number offarms (F1) ishigher, not showing any relevant effect for the other three factors. Ahalf-normal plot confirms these results for F2 inthe case ofmeasure perc_farms and F1 for var_eff (Figure3). AWilcoxon rank-sum test was performed with the results shown inTable4. Asitcan beseen, the factors considered have aninfluence, mainly affecting the percentage offarms involved inthe merger (perc_farms). Not only F2, asseen before, but also F1 and F3 give results that differ depending onthe level ofthose factors. For the number ofcows merged (num_cows), only the disparity inherd size seems tobesignificant. Regarding the var_eff measure, only the number offarms (F1) has aninfluence. F4 does not seem tosignificantly influence the results for any of the three response variables considered. However, ananalysis ofthe interactions among the four factors does not show relevant interactions, except for F1 and F4. Figure4 shows the presence ofinteraction between those two factors for num_cows and perc_farms. Inthe latter case, which seemed tobestatistically significant, the reduction in perc_farms for higher values ofF1 issmaller when factorF4 (the technical inefficiency ofthe farms) ishigher. Summarising, the results indicate that: i)the greaterthe disparity inherd sizes, the more farms enter into the merger and the larger the reduction inthe number ofcows that the merger can obtain; ii)the number offarms that enter into the merger, expressed inrelative terms asapercentage ofthe total number offarms, de- 2 4 6 8 Perc_farms (%) 10 Var_eff 0.1 0.3 0.5 2 4 6 8 10 2 4 6 8 10 2 4 6 8 10 L H L H L H L H F1 F2 F3 F4 L H L H L H L H F1 F2 F3 F4 0.1 0.3 0.5 0.1 0.3 0.5 0.1 0.3 0.5 Figure 1. Boxplots for the two more significant response variables (perc_farms and var_eff) F1, F2, F3, F4–factors; H–high; L–low; perc_farms –percentage offarms that are chosen toenter into the merger; var_eff –expected efficiency improvement due the merger Source: Authors' own calculations Figure 2. Average value of(A)perc_farms and (B)var_eff for each level ofthe four factors considered F1, F2, F3, F4–factors; H–high; L–low; perc_farms –percentage offarms that are chosen toenter into the merger; var_eff –expected efficiency improvement due the merger Source: Authors' own calculations 4.5 Perc_farms 4.0 3.5 3.0 2.5 F1 F2 F3 F4 (A) 4.29 3.80 3.07 2.58 3.05 3.82 3.48 3.38 H H H HL LLL (B) F1 F2 F3 F4 Var_eff 0.50 0.29 0.40 0.39 0.39 0.40 0.41 0.39 H H H HL LLL 0.55 0.50 0.45 0.40 0.35 0.30 0.25 217 Agricultural Economics – Czech, 67, 2021 (6): 209–219 Original Paper https://doi.org/10.17221/418/2020-AGRICECON creases asthe total number offarms increases; iii)the number offarms that enter into the merger increases when more farms exhibit IRS; and iv)the expected efficiency improvement due tothe merger increases asthe number offarms inthe region increases. CONCLUSION This paper has proposed anew DEA-based approach for identifying the most advantageous dairy farm (DF) merger. Themodel isflexible inthe sense that itallows including additional constraints toreflect the preferences and circumstances ofeach specific application. It can also be used iteratively to identify multiple merger possibilities. For each merger identified, the proposed approach provides the estimated merger efficiency gains aswell asits decomposition into alearning effect and apure merger efficiency gain. The proposed approach has been illustrated using a small dataset, showing the simplicity of the methodology and its usefulness. Thedataset was generated randomly using actual unit costs and production rates. Table 4. Corresponding P-value ofthe Wilcoxon rank-sum test for each factor and response Factor Num_cows Perc_farms Var_eff F1. Total number offarms 0.490 0.002* 0.000* F2. Standard deviation ofherd size (s) 0.040* 0.000* 0.077 F3. Percentage offarms exhibiting IRS 0.097 0.045* 0.479 F4. Standard deviation ofefficiency variable (σf )0.338 0.898 0.464 *Significant effects atthe 0.05 significance level; IRS–increasing returns toscale; num_cows –reduction ofthe herd size due tothe merger; perc_farms –percentage offarms that are chosen toenter into the merger; var_eff –expected efficiency improvement due the merger Source: Authors' own calculations Half-normal scoresHalf-normal scores 0 1 2 0 1 2 0.0 0.5 1.0 1.5 2.0 B 0.00 0.05 0.10 0.15 0.20 0.25 Absolute effects A = F1 B = F2 C = F3 D = F4 A (A) (B) Absolute effects Num_cows 160 170 180 130 140 150 120 Perc_farms (%) 3.8 3.2 3.4 3.6 3.0 H L F4 F4 F4 H L HL F1 HL F1 HL F1 Var_eff 0.50 0.35 0.40 0.45 0.30 H L Figure 3. Half normal plots for measures (A)perc_farms (alpha=0.05) and (B)var_eff (alpha=0.05) F1, F2, F3, F4–factors; perc_farms –percentage offarms that are chosen toenter into the merger; var_eff –expected efficiency improvement due the merger Source: Authors' own calculations Figure 4. Interactions between F1 and F4 for the three measures considered F1, F4–factors; H–high; L–low; num_cows –reduction ofthe herd size due tothe merger; perc_farms –percentage offarms that are chosen toenter into the merger; var_eff –expected efficiency improvement due the merger Source: Authors' own calculations