Mat Seminar on optimization

Transkriptio

1 Mat-.44 Seminar n ptimizatin Data enelpment analsis and its applicatins The Basic CCR Mdel S steemianalsin Labratri Teknillinen krkeakl The Basic CCR Mdel Tapani Hämäki Optimintipin seminaari - Sks 007

2 Cntents. The basic cncepts f DEA needed fr CCR mdel. Reqirements fr data 3. The CCR mdel frmlatin 4. Eample case f sing CCR S steemianalsin Labratri Teknillinen krkeakl The Basic CCR Mdel Tapani Hämäki Optimintipin seminaari - Sks 007

3 . The basic cncepts needed fr CCR mdel CCR is an etensin t DEA which is sed mainl when dealing with 'Mltiple Inpt Mltiple Otpt' -mdels. In CRR inpts and tpts f a DMU Decisin Making Unit) are called Virtal Inpt and Virtal Otpt which can be nderstd as the ttal inpt and tpt. S steemianalsin Labratri Teknillinen krkeakl The Basic CCR Mdel Tapani Hämäki Optimintipin seminaari - Sks 007

4 Virtal Inpt and Virtal Otpt can be frmlated as a sm f inpts i and tpts r weighted with weights i and r respectiel) Virtal Virtal inpt tpt m s m s ) The weights are assigned in sch wa that the efficienc rati scre) Virtal Virtal tpt inpt ) is maimized. The weight ales sall ar frm DMU t anther. S steemianalsin Labratri Teknillinen krkeakl The Basic CCR Mdel Tapani Hämäki Optimintipin seminaari - Sks 007

5 The weights can be determined in tw was:. Weights can be fied in adance accrding t preailing assmptins.. Optimal ales t weights can be determined with linear prgramming. In general the alternatie tw is als called the CCR mdel. S steemianalsin Labratri Teknillinen krkeakl The Basic CCR Mdel Tapani Hämäki Optimintipin seminaari - Sks 007

6 . Reqirements fr data In DEA the rganizatins) nder std is called DMU Decisin making nit). DMU can be defined as the entit that is respnsible fr cnerting inpts int tpts. In general DEA is ealating and cmparing the perfrmances f the DMUs. Let s sa there is n DMUs inled in a DEA prcedre and dente them as DMU DMU... DMU n. S steemianalsin Labratri Teknillinen krkeakl The Basic CCR Mdel Tapani Hämäki Optimintipin seminaari - Sks 007

7 Fr each DMU =...n) there is inpt items i i=...m) and tpt items r r=...s) which flfills the fllwing cnditins. Fr each inpt i and tpt r there mst be nmerical data aailable which is assmed t be psitie. S steemianalsin Labratri Teknillinen krkeakl This data describes the amnt f resrce inpt) cnsmed b DMU and the amnt f tcme tpt) prdced b DMU.. The items inpts tpts and chice f DMUs) shld reflect an analst's r a manager's interest in the cmpnents that will enter int the relatie efficienc ealatin f the DMUs. 3. In principle smaller inpt amnts are preferable and larger tpt amnts are preferable which shld als be reflected t efficienc scres. 4. The measrement nits f different inpts and tpts need nt t be cngrent. Sme ma inle nmber f persns r areas f flr space mne epended etc. The Basic CCR Mdel Tapani Hämäki Optimintipin seminaari - Sks 007

8 Cnsider the inpt data and tpt data fr DMU as ectrs =... m ) and =... s ) respectiel. Nw we can write the inpt and tpt data mre frmall as matrices X m m n n mn and Y s s n n sn where th inpt/tpt ectr is the th clmn ectr f the crrespnding matri. S steemianalsin Labratri Teknillinen krkeakl The Basic CCR Mdel Tapani Hämäki Optimintipin seminaari - Sks 007

9 S steemianalsin Labratri Teknillinen krkeakl The Basic CCR Mdel Tapani Hämäki Optimintipin seminaari - Sks The CCR mdel frmlatin T determine the ptimal weight ectrs and we need t find the maimm fr the efficienc rati defined in eqatin ). This prdces a fractinal prgramming prblem FP ) 3) 4) 5) 0 0 ) s.t ma m m m m s s m m s s n

10 The cnstraint 4) means that the efficienc rati Virtal tpt s. Virtal Inpt mst nt eceed fr each DMU A fractinal prgram prblem can be qite cmplicate t sle that is wh we transfrm this prblem int a linear prgramming prblem. This can be dne b mltipling eqatins 3) and 4) with the denminatr plnmial. ma LP ) s.t 7) m 0 m s s m s s m 0 m m n ) 6) 8) S steemianalsin Labratri Teknillinen krkeakl The Basic CCR Mdel Tapani Hämäki Optimintipin seminaari - Sks 007

11 The basic assmptin in DEA is that the mst efficient DMU reaches the ale ths in the maimm θ * = The Unit Inariance Therem see. [] Therem.) states that we can d the calclatins with data determined in an nits. S if we calclate the distance in miles r in kilmeters r the lme in gallns r liter we alwas end p t same reslts in efficienc ales becase f the nrmalizatin dne b the weight ectrs and. S steemianalsin Labratri Teknillinen krkeakl The Basic CCR Mdel Tapani Hämäki Optimintipin seminaari - Sks 007

12 CCR-efficienc: DMU is CCR-efficient if θ * = and there eists at least ne ptimal * * ) > 0 0) CCR-inefficienc: DMU is CCR-inefficient if either i) θ * < r ii) θ * = and at least ne element in * * ) is zer. The set f efficient DMUs spans the efficient frntier f DMUs. S steemianalsin Labratri Teknillinen krkeakl The Basic CCR Mdel Tapani Hämäki Optimintipin seminaari - Sks 007

13 4. Eample case f sing CCR Table n the right shws an eample case f 6 DMUs with inpts and tpt each In this case the ales f tpts are nrmalized Inpts cld be cnsidered fr eample as emplees nmber ) and flr space ) Otpt cld be sale in 0000$ S steemianalsin Labratri Teknillinen krkeakl The Basic CCR Mdel Tapani Hämäki Optimintipin seminaari - Sks 007

14 S steemianalsin Labratri Teknillinen krkeakl The Basic CCR Mdel Tapani Hämäki Optimintipin seminaari - Sks 007 The linear prgram LP ) fr DMU A is nw <A> This prblem can be sled with a linear prgramming algrithm. B ding s we btain the ptimal sltin: 0 F) 0 E) 4 D) 4 C) 8 B) 3 7 A) s.t ma 0.857) ) * * * *

15 S steemianalsin Labratri Teknillinen krkeakl The Basic CCR Mdel Tapani Hämäki Optimintipin seminaari - Sks 007 Similarl linear prgram LP ) fr DMU B is <B> This prblem can be sled with a linear prgramming algrithm. B ding s we btain the ptimal sltin: 0 F) 0 E) 4 D) 4 C) 8 B) 3 7 A) s.t ma 0.636) ) * * * *

16 Sling the linear prgram fr all the rest in similar wa we get the reslts shwn in the table belw. In the table we can see that DMUs C D and E are CCRefficient becase fr eachθ * =. The als frm the efficient frntier f DMUs. DMU F is nt CCR-efficient becase its = 0 S steemianalsin Labratri Teknillinen krkeakl The Basic CCR Mdel Tapani Hämäki Optimintipin seminaari - Sks 007

17 Graphical illstratin f the DMUs in a plane f inpts can be seen belw. S steemianalsin Labratri Teknillinen krkeakl The Basic CCR Mdel Tapani Hämäki Optimintipin seminaari - Sks 007

18 Hme Assignment Sle the CCR-efficient DMUs sing the fllwing data A B C D E F emplees in) raw material csts in) infrastrctre csts in) sale t) Inclde t r answer The frmlatin f the linear prgramming prblem Calclate the scre ales θ * and the crrespnding weight ectrs and and determine the CCR-efficient DMUs Y can se ecel t sle the linear prblem S steemianalsin Labratri Teknillinen krkeakl The Basic CCR Mdel Tapani Hämäki Optimintipin seminaari - Sks 007

19 References [] Intrdctin t Data Enelpment Analsis and its ses Cper William W 005 S steemianalsin Labratri Teknillinen krkeakl The Basic CCR Mdel Tapani Hämäki Optimintipin seminaari - Sks 007