Lecture 2 DATA ENVELOPMENT ANALYSIS - II

Transcription

1 Lecture DATA ENVELOPMENT ANALYSIS - II Learning objective To eplain Data Envelopent Anali for ultiple input and ultiple output cae in the for of linear prograing

2 .5 DEA: Multiple input, ultiple output Now, conider the cae when there i ore than one input affecting ore than one output. The efficienc in uch cae cannot be deterined like in the previou cae of ingle input and ingle output. Let u ee below an illutration. The owner of a chain of coffee hop Coffee and ore want to evaluate the efficienc of the hop in one cit. The ite old a output in hi tore are in two categorie: Beverage Coffee, Tea and other drink and Snack Sandwiche, Bicuit and o on. The input leading to the ale of thee ite are: Available floor pace in the tore and the nuber of eploee. The data regarding input and output are given below in Table.6. Table.6: Input and output data for Coffee and ore Store Location A B C D E F G Nuber of eploee Floor pace in Unit of Snack Unit of Beverage To find the efficienc of each of the unit and to deterine the relative efficienc, we need to give weight to each input and to each output. Thee weight are the coefficient for each input and output variable. The coefficient pertaining to output variable eaure the relative decreae in efficienc with each unit reduction of output variable. The coefficient pertaining to input variable eaure the relative increae in efficienc with each unit reduction of input variable. Thee weight can be either provided b the uer or can be deterined optiall b uing oe optiization technique like DEA.

3 .5. Uer provided weight Suppoe we attach weight to input and output a given below: Input - V (weight for eploee trength): V (weight for floor pace) = 5: Output - U (weight for nack ite): U (weight for beverage) = : 3 Thu, the efficienc of unit A = (u 00)+(u 80) (v 40)+(v 30) =.47 (no unit) The efficienc of other unit are deterined in iilar fahion which are preented in Table.7. Relative efficienc of each unit can be deterined uing the ratio of efficienc of that unit and efficienc of ot efficient unit a hown in Table.7. Table.7: Relative efficienc of DMU Coffee tore A B C D E F G Efficienc Relative efficienc Deterination of weight uing CCR odel The CCR odel ue a linear prograing odel to aign weight or to deterine coefficient that are choen in a anner that aign a bet et of weight/coefficient to each of the unit. CCR tand for Charne, Cooper and Rhode, who introduced DEA in 978. In CCR odel we arrange the inforation available fro the data into a atri forat with X to be input atri and Y to be output atri. The data i for n deciion aking unit, input and output iilar to the one preented below. The input and output are aued to be known and all poitive.

4 Input X Y = Output = Data Data Matri Matri n n n n n n Now, the efficienc of each of the DMU j (Deciion Making Unit the coffee tore in thi cae), with j =,,.n i to be eaured o that we can deterine relative efficienc and identif inefficient unit. If there are n DMU, we need n optiization, one for each DMU uing following notation. DMU o : DMU which will be evaluated in a particular trial, with o=,,,n. Thi notation will be ued for the DMU in the objective function. The ae DMU in the contraint will follow the notation DMU j a defined net. DMU j : DMU with j=,,..n v i : Coefficient for input i, with i =,,.. u r : Coefficient for output r, with r =,,... The DEA odel can be preented to aiize the efficienc of DMU o, θ 0,b writing the objective function of o th DMU a given below. Ma θ = o u v o o + u + v u v o o o o

5 The odel will be ubjected to the contraint; () efficienc of all other DMU will be le than or equal to and () non negativit contraint. The odel will take fractional prograing for a given below. : Ma θ = u (for o th DMU) o o v o Subject to contraint: u v j j v, v u, u,..., v + u + v j j,..., u u v + u + v j o j o u v The fractional prograing odel can be converted into a linear prograing odel. Thi i done b caling each of the input to and rewriting the contraint a entioned below. o o ( j =,,..., n) Ma θ = u v o o Subject to + v o o + u o v u o = o (for o th DMU) u v, v u, u j,..., v,..., u + u j u The preented odel find the bet et of weight or coefficient pertaining to each input and output variable while aiu rating of efficienc i aigned to o th DMU. After olving linear prograing odel for DMU o, the o th DMU will be efficient onl if the odel reult in: j v ( j =, j + v,..., j v n) j

6 ) The optial efficienc of o th DMU i equal to ) All lack are zero The efficient DMU will for efficient frontier. If the optial efficienc for an DMU becoe le than, then that DMU i inefficient. The lack variable for inefficient unit will not be zero, which ean that the DMU i utilizing one of the input in ece than the efficient unit to produce the ae aount of output. Each inefficient DMU will have a et of efficient unit, which would act a a reference et to iprove the perforance of inefficient unit. When we will olve the linear prograing odel for DMU o, the hadow price of the contraint pertaining to efficient unit in that odel will help in deterining the point on the efficient frontier where the inefficient unit can be projected to ake it efficient. The projected point i a linear cobination of efficient unit where the coefficient of thoe linear variable are the hadow price of efficient contraint. Eaple: Multiple Input and Multiple Output Super lab are in Reearch and Developent of electronic good. The founder, in the ear 00 decided to et-up their own copan to produce conuer electronic good that ue the tate-of-the art technologie that have been developed b the. The copan ha et up buine in North Aerica and quickl oved to Aia, Europe, Africa and South Aerica. The ain product of the copan are it ipreive range of Sart phone, Tablet PC and Laptop. At the beginning of the ear 0, the CEO- global operation of the copan quickl pulled report of Input and Output that are being ued and produced at variou region in which copan operate. He arrived at the following figure:

7 INPUTS ($ in illion) OUTPUTS (nuber in '000) DMU Product Developent Marketing Sart Phone Tablet Laptop N.Aerica Europe Aia Africa S.Aerica The data tell hi how uch each of the region conue for the two ain activitie, nael Product developent and Marketing, and how an unit of it product are being old. However, he i unable to figure out, which are the bet perforing unit and how uch of increae or decreae the Regional director are to be advied to ake on invetent. One of hi analt recoend the ue of Data Envelopent Anali to analze the perforance of the different unit and then ake a concluion. The analt alo entioned that the tool can be bet ued a the input and output to the variou Deciion Making Unit are the ae unit ($ pent of product developent, arketing and nuber of unit of each product old). The analt quickl arrived at the following forulation: For each of the input, aign weight u, u and aign weight v, v and v 3 for the output. Therefore the forulation for the five DMU would be a follow: For Unit (N.Aerica): Maiize 9v+ 4v + 6v3 Subject to the contraint: 5u + 4u =

8 (9v+ 4v + 6 v3) (5u+ 4 u) 0 (8v+ v + 9 v3) (0u+ 8 u) 0 (9v+ 4v + 0 v3) (9u+ 6 u) 0 (6v+ v + 8 v3) (7u+ u) 0 (9v+ 4v + 6 v3) (5u+ 4 u) 0 (0v + 4v + 4 v ) (9u + 6 u ) 0 u, u For Unit (Europe): v, v and v3 Maiize 8v+ v + 9v3 Subject to the contraint: 0u + 8u = For Unit 3 (Aia): (9v + 4v + 6 v ) (5u + 4 u ) 0 (8v + v + 9 v ) (0u + 8 u ) 0 (9v + 4v + 0 v ) (9u + 6 u ) 0 (6v + v + 8 v ) (7u + u ) 0 (9v + 4v + 6 v ) (5u + 4 u ) 0 (0v + 4v + 4 v ) (9u + 6 u ) 0 u, u v, v and v3 Maiize 9v+ 4v + 0v3 Subject to the contraint: 9u + 6u =

9 (9v+ 4v + 6 v3) (5u+ 4 u) 0 (8v+ v + 9 v3) (0u+ 8 u) 0 (9v+ 4v + 0 v3) (9u+ 6 u) 0 (6v+ v + 8 v3) (7u+ u) 0 (9v+ 4v + 6 v3) (5u+ 4 u) 0 (0v + 4v + 4 v ) (9u + 6 u ) 0 u, u v, v and v3 For Unit 4 (Africa): Maiize 6v+ v + 8v3 Subject to the contraint: 7u + u = (9v + 4v + 6 v ) (5u + 4 u ) 0 (8v + v + 9 v ) (0u + 8 u ) 0 (9v + 4v + 0 v ) (9u + 6 u ) 0 (6v + v + 8 v ) (7u + u ) 0 (9v + 4v + 6 v ) (5u + 4 u ) 0 (0v + 4v + 4 v ) (9u + 6 u ) 0 u, u v, v and v3 For Unit 5 (S.Aerica): Maiize 8v+ v + 9v3 Subject to the contraint: 0u + 8u =

10 (9v+ 4v + 6 v3) (5u+ 4 u) 0 (8v+ v + 9 v3) (0u+ 8 u) 0 (9v+ 4v + 0 v3) (9u+ 6 u) 0 (6v+ v + 8 v3) (7u+ u) 0 (9v+ 4v + 6 v3) (5u+ 4 u) 0 (0v + 4v + 4 v ) (9u + 6 u ) 0 u, u v, v and v3 The above preented odel can be olved uing an optiization oftware or uing ecel. On olving the above linear prograing odel in Ecel for all the unit, the following efficiencie for the DMU were found. DMU Efficienc N.Aerica Europe 0.67 Aia Africa 0.75 S.Aerica Now, CEO like to copare the efficienc of the unit in Europe (inefficient unit) with other efficient unit. It i found that the hadow price of the efficient unit that are N. Aerica and S. Aerica to be 0.57 and repectivel. So, N. Aerica and S. Aerica for the reference et for Europe. The linear cobination of a particular input for two efficient unit uing hadow price a ultiplier, will give the point on the efficient frontier where the inefficient unit can be projected. That i Europe hould brought down the product developent epene fro 0 to ( ).

11 Input and Output Product Developent North Aerica South Aerica Value Weight Total Value Weight Total Su of Total and Europe Ece ued b the unit Europe Marketing Sart Phone Tablet Laptop Thu on coparing the input and output of Europe with that of N and S Aerica, Europe to reduce Product developent and Marketing cot b 3.30 and ( 000 of dollar value) and increae ale of Tablet and Laptop b.56 and.7 unit repectivel to reach the value of the efficient unit. The ale of Sart phone i however to the level of the efficient unit N and S Aerica. Thu the change to be ade to the other inefficient DMU can be analzed.