Sensitivity Analysis of SAW Technique: the Impact of Changing the Decision Making Matrix Elements on the Final Ranking of Alternatives

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1 Ianian Jounal of Opeations Reseach Vol. 5, No. 1, 2014, pp Sensitivity Analysis of SAW Technique: the Impact of Changing the Decision Maing Matix Elements on the Final Raning of Altenatives A. Alinezha 1, K. Saafha 2,*, A. Ai 3 Downloae fom ios.i at 4: on Fiay Apil 27th 2018 Most of ata in a multi attibute ecision maing (MADM) poblem ae unstable an changeable, an thus sensitivity analysis can effectively contibute to maing pope ecisions. Hee, we offe a new metho fo sensitivity analysis of multi-attibute ecision maing poblems so that by changing one element of ecision maing matix, we can etee changes in the esults of a ecision maing poblem. An analysis is mae fo simple aitive weighting metho (SAW) technique, a mostly use multi-attibute ecision maing techniques, an the coesponing fomulas ae obtaine. Keywos: Multi-attibute ecision maing (MADM), SAW Technique, Sensitivity analysis, Raning methos, Attibute weights. Manuscipt was eceive on 13/07/2013, evise on 04/01/2014 an accepte fo publication on 10/03/ Intouction Multi-attibute ecision maing moels ae selecto moels which ae use fo evaluating, aning an selecting the most appopiate altenative fom among altenatives. Altenatives of an MADM poblem ae evaluate by attibutes an the most appopiate altenative is selecte o, they ae ane in accoance with attibutes values fo the altenatives an the impotance of the attibutes fo the ecision mae. An MADM moel is fomulate as a ecision maing matix as follows: C 1 C 2 C A A 2 [ ] A m m1 m2 m Whee A 1, A 2, A 3,, A m ae available an peetee m altenatives an C 1, C 2, C 3,, C ae effective attibutes in ecision maing which ae use fo measuing the utility of each * Coesponing Autho. 1 Faculty of Inustial an Mechanical Engineeing, Qazvin Banch, Islamic Aza Univesity, Qazvin, Ian. alinezha_i@yahoo.com 2 Young Reseaches an Elite Club, Qazvin Banch, Islamic Aza Univesity, Qazvin, Ian. ey_saafha@yahoo.com 3 Depatment of Inustial Engineeing, Tabiat Moaes Univesity, Tehan, Ian. abbasai@yahoo.com

2 Sensitivity Analysis of SAW Technique 83 altenative an ij is special value of the attibute jth fo the altenative ith, that is, the efficiency of the ith altenative vesus the jth attibute. The most impotant issue in MADM moels is that the ata use ae unstable an changeable. Being so, sensitivity analysis afte solving the poblem can effectively contibute to maing accuate ecisions. Downloae fom ios.i at 4: on Fiay Apil 27th 2018 Ealy wo in this fiel ae the wos of Evans [3], Fishbun an Isaacs [4], Schnelle an Sphicas [11], which focuse on eteing ecision sensitivity to pobabilistic estimation eos. Soofi [13] an Baon an Schmit [2] popose sensitivity analysis fo aitive MADM moels. They assume a set of weights fo the attibutes an obtaine a new set of weights so that the efficiency of altenatives wee equal o thei oe change. Ma et al. [7] stuie the stuctue of the weight set an conitions that lea to special aning o pioities of altenatives an iscusse aitive ecision maing moels. Rios an Fench [9] by offeing a metho fo sensitivity analysis stuie the esult of changes in the weights of attibutes on the final scoe of altenatives in MADM moels an calculate the equie change the weights fo changing the optimal solution. These algoithms an methos wee evise by Rios et al. [10]. Tiantaphyllou an Sanchez [15] stuie two types of sensitivity analyse fo two MADM methos. Fist, they etee the most sensitive attibute an calculate the change in the weights that lea to change in the aning of altenatives an secon they measue the sensitivity of the ecision maing matix elements.zavasas et al. [16] popose a moel to etee sensitivity to changes of sepaate paametes to incease the eliability of the applie methos. ToloieEshlaghy et al. [14] stuie a sensitivity analysis appoach to pouce complementay infomation by eteation of citeion values in the ecision maing matix. Hsingyeh [5] pesente a new appoach to the selection of compensatoy MADM methos fo a specific cainal aning poblem via sensitivity analysis of the attibute weights. Memaiani et al. [8] offee a new metho fo sensitivity analysis of MADM poblems so that by using it an changing the weights of the attibutes one coul etee changes in the final esults of a ecision maing poblem. Simanaviciene an Ustinovichius [12] pesente sensitivity analysis of TOPSIS an SAW methos. They analyze the quantitative multiple citeia ecision maing methos an sensitivity analysis methos use in ecision suppot systems. Both methos ae stongly mathematically base. They too notice of these sensitivity methos fo the initial ata. Monte Calo metho was applie to geneate the initial ata. Alinezha an Ai [1] pesente a new metho fo sensitivity analysis in multi-attibute ecision maing poblems in which if the weights of one attibute change, then changes in the esults of the poblem was etee. These changes involve changes in the weights of othe attibutes an changes in the final ans of altenatives. In line with the context-epenent concept of infomational impotance, the appoach exae the consistency egee among the elative egee of sensitivity of iniviual attibutes using an MADM metho an the elative egee of influence of the coesponing attibutes inicate by Shannon's entopy concept. In Section 2, we eview the SAW technique an iscuss some coesponing fomulas an elations. In Section 3, we pesent a new metho fo sensitivity analysis of MADM moels. We fist stuy the esult of change in one enty of the ecision maing matix on the final scoe of altenatives an establish the esulting elations. In Section 4, by woing though a numeical example the obtaine elations an fomulas ae veifie an thei accuacies ae confime. Finally, we summaize ou conclusions an povie suggestions fo futhe eseaches.

3 84 Alinezha, Saafha an Ai 2. The SAW Technique Downloae fom ios.i at 4: on Fiay Apil 27th 2018 The SAW technique is one of the most use MADM techniques. It is simple an seves the basis of most MADM techniques such as AHP an PROMETHEE which benefits fom aitive popety to calculate final scoes of altenatives. In SAW technique, final scoe of each altenative is calculate as follows: whee ij K P i = w j ij, i = 1,, m, (1) ae nomalize values of the ecision matix elements, that is, fo pofit attibutes, o ij = ij j max, j max = max ij, j = 1,,, (2) ij = j, j = ij, j = 1,,, (3) ij Fo cost attibutes. Fo any qualitative attibutes, one can use appopiate methos to tansfom qualitative vaiables to quantitative ones. 3. Ientifying the Impact of Change in one Element of Decision Maing Matix on the Final Scoe of Altenatives Available sensitivity analysis moels fo MADM poblems mostly focus on eteing the most sensitive attibute so that with the least change, the cuent aning of altenatives is change. Hee, we consie a new metho fo sensitivity analysis of MADM poblems to calculate the change in the final scoe of altenatives when a change occus in one element of the ecision maing matix. In the SAW moel, when the th attibute in lth altenative changes, that is, when the element l in the ecision matix changes, the nomalize values at th column in the ecision matix ae change an othe values emain unchange because a linea nom is use an nomalization is applie sepaately fo the columns of the ecision matix. In SAW moel, if element l in the ecision matix changes to, then eight sepaate states aise in accoance with the followings: whethe the attibute is of pofit o cost type, whethe l is the most esiable at its column o not, afte pefog the change, whethe l o the change element emains as the most esiable element o not. The following esults istinguish these states. Theoem 3.1. Assume that the element l changes to l = l + Δ an that the th column is of the pofit type an l is the most esiable at the column, i.e., l = max i, an afte changing l to l, l is also the most esiable element at th column. Then, nomalize values of th column ae change to:

4 Sensitivity Analysis of SAW Technique 85 i = l l + Δ. i, i = 1,, m, i l, l = l = 1, an final scoes of altenatives ae: P i = P i Δ l + Δ w i, i = 1,, m, i l, P l = P l. (4) (5) Downloae fom ios.i at 4: on Fiay Apil 27th 2018 Poof. Since l is the most esiable at column, nomalize values of th column ae: Fom i i l i = i = i l l + Δ. (6), we have i i l. Replacing this in (6), we have i = l l + Δ i, i = 1,, m, i l, l = l = 1 = l, i = 1. l Theefoe, the final scoes of altenatives ae: P i = ij w j + i w = ij w j + l l i w = ij w j + (1 l l + Δ ) ij w j = ij w j Δ l + Δ i w, P l = ij w j + l w = ij w j = P l. (7) (8) Theoem 3.2. Assume that the element l changes to l = l + Δ, an that the th column is of the pofit type an element l is the most esiable one at the column, i.e., max, an afte l 1 i m changing l to l, l is not the most esiable element at th column, that is, l. Then, nomalize values of th column ae change to i < max i = i l i = i, i = 1,, m, i l, l = l + Δ, an the final scoes of altenatives ae: (9)

5 86 Alinezha, Saafha an Ai P i = P i + ( i i l ) w, i = 1,, m, i l, P l = P l + ( l + Δ 1) w. (10) Downloae fom ios.i at 4: on Fiay Apil 27th 2018 Poof. Since max i =, the nomalize values of th column ae: i = i, i = 1,, m, i l, l = l + Δ, Theefoe, the final scoes of altenatives ae: P i = ij w j + i w = ij w j + i w i w = ij w j + ( i P l = ij w j + ( i i ) w = P i + ( i i l ) w, i = 1,, m, i l, i ) w = P l + ( l + Δ 1) w. Theoem 3.3. Assume that the element l changes to l = l + Δ, an that the th column is of the pofit type an l is not the most esiable one at the column, that is, l < max i =, an afte changing l to l, l is not the most esiable element at th column, that is, l < max i =. Then, nomalize values of th column except fo the lth element will not change, i that is, we have i = i, i = 1,, m, i l, (11) (12) l = l + Δ. (13) an the final scoes of altenatives, except fo the lth altenative, will not change an we have P l = P l + Δ w. (14) Poof. Nomalize values of th column ae calculate by iviing the values of th column into an then nomalize values l ae compute as: l = l = l + Δ Theefoe, final scoe of th altenative woul be = l + Δ = l + Δ. (15) P l = lj w j + l w = lj w j + Δ w = P l + Δ w. (16)

6 Sensitivity Analysis of SAW Technique 87 Downloae fom ios.i at 4: on Fiay Apil 27th 2018 Theoem 3.4. Assume that the element l changes to l = l + Δ, an that the th column is of the pofit type an element l is not the most esiable one at the column, that is, l < max i =, an afte changing l to l, l is the most esiable element at th column that is, l < max i. Then, nomalize values of th column ae change to i l i = l + Δ. l, i = 1,, m, i l, i = 1, i = l, an the final scoes of altenatives ae: P i = P i + ( Poof. The nomalize values of th column ae: an because i i *, then i l + Δ 1) w i, i = 1,, m, i l, P l = P l + (1 l ) w. i * i i. = i l + Δ = (17) (18) = i l + Δ = i l + Δ, (19) l an by eplacing it in (19), we have l + Δ. i, i = 1,, m, i l, = l = 1, i = l. l Theefoe, the final scoes of altenatives ae: P i = ij w j + i w = ij w j = ij w j + ( l + 1) i w + l + i w ; i = 1, 2,, m, i l P l = lj w j + w = lj w j + w l w = P l + (1 l ) w ; i = l The above fou theoems emonstate the fou states coesponing to attibute of the pofit type. Next, we consie states coesponing to the cost type. (20) (21) Theoem 3.5. Assume that the element l changes to l = l + Δ, an that the th column is of the cost type an element l is the most esiable one at the column, that is, l = max i, an afte changing l to l, l is also the most esiable element at th column. Then, nomalize values of th column ae change to: i = i + Δ, i = 1,, m, i l, l (22) i = i = 1, i = l,

7 88 Alinezha, Saafha an Ai an the final scoes of altenatives ae: P i = P i + Δ i w, i = 1,, m, i l, P l = P l. (23) Downloae fom ios.i at 4: on Fiay Apil 27th 2018 Poof. We have l = max i i l i. Then, l = an nomalize values of th column ae: = l = l + Δ = l + Δ i i i Theefoe, the final scoes of altenatives ae: = l + Δ, i = 1,, m, i l, i i l = l = 1. l P i = ij w j + ij w = lj w j + i w i w (24) = lj w j + w ; i = 1,, m, i l (25) i P l = P l ; i = l Theoem 3.6. Assume that the element l changes to l = l + Δ, an that the th column is of the cost type an element l is the most esiable one at the column, that is, l = max i, but afte changing l to l, l is not the most esiable element at th column, that is, l > i l i =. Then, nomalize values of th column ae: i =, i = 1,, m, i l, i i =, i = l, l + Δ an the final scoes of altenatives ae: P i = P i + ( i l i ) w, i = 1,, m, i l, P l = P l + ( l + Δ 1) w. (26) (27) Poof. We have = max i. Then, nomalize values of th column ae: i =, i = 1,, m, i l, i i =, i = l. l + Δ Theefoe, the final scoes of altenatives ae: P i = ij w j + ij w = lj w j + i w i w (28)

8 Sensitivity Analysis of SAW Technique 89 = P i + ( i l i ) w, i = 1,, m, i l, P i = ij w j + ij w = ij w j + w i w = P i + ( l + Δ 1) w ; i = l (29) Downloae fom ios.i at 4: on Fiay Apil 27th 2018 Theoem 3.7. Assume that the element l changes to l = l + Δ, an that the th column is of the cost type an element l is not the most esiable one at the column, that is, l > i = an afte changing l to l, l is not the most esiable element at th column, that is, l i =. Then, nomalize values of th column, except fo lth element ae change to: i l = i, i = 1,, m, i l, l = (1 Δ l + Δ ) l, i = l, i an the final scoes of altenatives, except fo lth altenative, will not change an we have (30) P Δ l = P l ( l + Δ ) w l. (31) Poof. Nomalize values of th column ae calculate by iviing the values of th column into. Then, the nomalize values l ae: l = l + Δ, (32) an since l an l l l. l i, we have = i, i = 1,, m, i l, = l l l + Δ = (1 Δ l + Δ ) l, i = l. (33) Theefoe, the final scoes of th altenative ae: P l = lj w j + l w = lj w j + i Δ w ( l + Δ ) l w (34) Theoem 3.8. Assume that the element l changes to l = l + Δ, an that the th column is of the cost type an element l is not the most esiable one at the column, that is, l > i =, but afte changing l to l, l is not the most esiable element at th column, that is, l i. Then, nomalize values of th column ae change to: i l Theefoe the final scoes of altenatives ae: i = l + Δ i, i = 1,, m, i l, l = 1. (35)

9 90 Alinezha, Saafha an Ai P i = P i + ( l + Δ 1) w i, i = 1,, m, i l, P l = P l + (1 l ) w. (36) Poof. Nomalize values of th column ae: Downloae fom ios.i at 4: on Fiay Apil 27th 2018 an since i, i i i = l = l + Δ, i = 1,, m, i l, (37) i i Theefoe, the final scoes of altenatives ae: P i = ij w j + i an by eplacing it in (37), we have i i = l + Δ. i, i = 1,, m, i l, l = 1. w = ij w j + l + Δ i w i w = P i + ( l + Δ 1). w ; i = 1,, m, i l P l = lj w j + l w = lj w j + w l w = P l + (1 l ) w ; i = l Now, by consieing the above eight states that appea afte changing an element of the ecision maing matix, we can use one of the above elations an then calculate the esulting change in the final scoes of the altenatives. We now pesent the pocess in Figue 1. (38) (29) 4. Numeical Example Consie an MADM poblem with thee altenatives an fou attibutes, wheein attibutes c 1, c 4 ae of cost type an attibutes c 2, c 3 ae of pofit type:

10 Sensitivity Analysis of SAW Technique 91 Downloae fom ios.i at 4: on Fiay Apil 27th 2018 Figue 1. Flowchat to calculate the esulte change in the final scoe of altenatives R = C 1 C 2 C 3 C 4 A A 2 [ ] A

11 92 Alinezha, Saafha an Ai W t = (0.4, 0.2, 0.3, 0.1) Downloae fom ios.i at 4: on Fiay Apil 27th 2018 Using the SAW technique, the nomalize matix, accoing to the elations given in Section 2, ae: C 1 C 2 C 3 C 4 A R = A 2 [ ] A The final scoes of altenatives ae calculate by P i = w j ij, i = 1,, m, with m = 3 an = 4: P 1 = 0.758, P 2 = 0.683, P 3 = Theefoe: A 3 > A 1 > A 2. Now, we assume that the element 34 in the ecision matix is incease to 34 = 34 + Δ = = 9. Then 4th column in the new nomalize matix will change an we have: C 1 C 2 C 3 C 4 A R = A 2 [ ] A By solving the poblem again, the new scoes of altenatives ae: P i = w j ij, i = 1,, m, P 1 = 0.783, P 2 = 0.700, P 3 = Theefoe, A 3 > A 1 > A 2, an it is clea that the aning has change. Now, instea of solving the poblem again, we use the fomulas given in Section 3. Since 34 = 34 + Δ = = 9, an the egae attibute is of the cost type an this element is the most esiable one at its column, but afte the change is not the most esiable one, we use the equations coesponing to state 6 fo calculating the new scoes of altenatives: P i = P i + ( 4 34 ) w i4 4, i = 1, 2, 3, i4 Since we have P 3 = P 3 + ( Δ 1) w 4. 4 = 8, 34 = 9, w 4 = 0.1, the final change scoes of altenatives ae P 1 = 0.783, P 2 = 0.700, P 3 =

12 Sensitivity Analysis of SAW Technique 93 which ae exactly the same as the esults obtaine befoe. 5. Conclusion Downloae fom ios.i at 4: on Fiay Apil 27th 2018 Decision maing is an integal pat of human life. Regaless of the vaiety of ecision maing poblems, we can categoize them into two categoies: multi-objective ecision maing poblems in which the ecision mae must esign an appoach that has the most utility by consieing limite esouces an multi-attibute ecision maing poblems in which the ecision mae must select one altenative with most utility fom among the available altenatives. Natually, to select an altenative, one must consie seveal an often conflicting attibutes. Geneally, an MADM poblem can be epicte as a matix. Each ow of the matix coespon to one altenative an each column to one attibute an the elements of the matix ae the efficiency of altenatives against attibutes. Geneally, the attibutes that ae chosen fo ecision maing ae conflicting. This means that impovement in one attibute may esult in the eflation of othe attibutes. Consieing the elative impotance of attibutes, we can assign weights. Using a vecto of weights fo the attibutes an elements of the ecision maing matix, we can solve the MADM poblem by available techniques to an the altenatives o select the best one. In the classic techniques of MADM, it is often assume that all the use ata (such as weights of attibutes, efficiencies of altenatives against attibutes, ) ae eteistic. Then, final scoes o utilities of altenatives ae obtaine by an MADM solving techniques, wheeas in eality, the ata of the ecision maing poblem change. Afte solving the ecision maing poblem, usually a sensitivity analysis is also pefome. Most stuies on MADM poblems, often etee the most sensitive attibute in the moel. This attibute is the one that equies the least change in its weight, as compae to othe attibutes, to change a aning of the altenatives. The available stuies fequently consie attibutes sensitivity. Anothe type of sensitivity analysis, not aesse in the liteatue, is calculation of the change in the final scoes of altenatives coesponing to a change in the weight of a paticula attibute. In ou popose sensitivity analysis, fo a given change in the weight of one attibute, the changes in the scoes of altenatives ae calculate. This type of sensitivity analysis can be implemente in MADM elate softwae to solve ecision maing poblems in a way that by utilizing gaphical means, the ecision mae may one element of the ecision maing matix an obseve its effect on the final scoes an ans of the altenatives. The followings ae suggeste fo futhe eseach. Stuying the effect of the change in the weight of one attibute of the ecision maing matix on the final scoes of altenatives in the SAW technique. Stuying the effect of simultaneously changing the weight of one attibute an one element of the ecision maing matix on the final scoes of the altenatives in the SAW technique. Applying ou popose sensitivity analysis fo othe MADM techniques such as AHP an PROMETHEE.

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