SOLVING MULTI-OBJECTIVE INTERVAL TRANSPORTATION PROBLEM USING GREY SITUATION DECISION-MAKING THEORY BASED ON GREY NUMBERS

Transcription

1 Internatonal Journal of Pure and Appled Mathematcs Volume 113 No , ISSN: (prnted verson; ISSN: (on-lne verson url: do: /pam.v PApam.eu SOLVING MULTI-OBJECTIVE INTERVAL TRANSPORTATION PROBLEM USING GREY SITUATION DECISION-MAKING THEORY BASED ON GREY NUMBERS Jgnasha G. Patel 1, Jayesh M. Dhodya 2 1,2 Appled Mathematcs and Humantes Department S.V.N.I.T, Ichchhanath Surat, , Guarat, INDIA AMS Subect Classfcaton: 90B06 Key Words: mult-obectve nterval transportaton, effect measure, effcent soluton, decson weght, grey stuaton 1. Introducton In mathematcal nterval programmng models deal wth uncertanty and nterval coeffcents [8]. An nterval transportaton problem construct the data of supply, demand and obectve functons such as cost, tme etc n some ntervals. Ths problem can be converted nto a classcal MOTP by usng the concept of rght lmt, half-wdth, left lmt, and center of an nterval [1]. Receved: September 21, 2016 Revsed: January 18, 2017 Publshed: March 19, 2017 c 2017 Academc Publcatons, Ltd. url: Correspondence author

2 220 J. Patel, J. Dhodya Das at al. (1999 used fuzzy programmng technque to solve MOITP n whch cost coeffcent, destnaton and source parameters are n nterval form. Sengupta, Pal, and Chakraborty (2001 also provde soluton of lnear nterval number programmng problems n whch ther coeffcents are n nterval form. For better soluton of MOITP Sengupta and Pal (2003 developed fuzzy technques based soluton n whch md pont and wdth of the nterval they consder and provde effcent soluton. In 2006 they developed algorthm for fnd the soluton of shortest path problem of transportaton network n whch coeffcents are n nterval numbers. For nterval valued obectve functon problem Wu (2009 derved KarushKuhnTucker condtons and by usng ths condton he fnd the soluton of problem. P. Pandan and G. Nataraan (2010 utlzed separaton method and obtaned compromse soluton of nteger transportaton problems n whch transportaton cost, supply and demand are n uncertan form. Sudhakar, V.J. and Navaneethakumar, V. (2010 developed nnovatve approach to fnd an effcent soluton for IITP (nteger nterval transportaton problems. Dutta and A. Satyanarayana Murthy (2010 solve a fuzzy TP wth addtonal restrctons by ntroducng a lnear fractonal programmng method and they consder transportaton costs n nterval form. P. Pandan and D. Anuradha (2011 acheve an optmal soluton for fully nteger nterval transportaton problems by usng splt and bound method. In ths approach they have not consder mdpont and wdth of the ntervals but based on floatng pont method they fnd the soluton. S.K. Roy and D. R. Mahapatra (2011 developed a method based on weghted sum and solved mult-obectve stochastc transportaton problem n whch parameters are lognormal random varables and the coeffcents of the obectves are nterval numbers wth nequalty type of constrants. Arpta Panda and Chandan Bkash Das (2013 [2] focused on the soluton of Cost Varyng Interval Transportaton Problem under Two Vehcles by applyng fuzzy programmng technque. Abdusalam mohmed khalfa, E. E. Ammar (2014 have used the concept of UUIT, ULIT, LUIT and lower nterval LLIT to solve Rough Interval Mult-obectve Transportaton Problems (RITP. In 2015 Vncent F. Yu, Kuo-Jen Hu & An-Yuan Chang obtaned compromse soluton for the MOITP by applyng an nteractve approach. Dalbnder Kaur, Sath Mukheree and Kala Basu (2015 [5] present the soluton of a mult-obectve and mult-ndex real-lfe transportaton problem by applyng an exponental membershp functon n fuzzy programmng technque.

3 SOLVING MULTI-OBJECTIVE INTERVAL TRANSPORTATION General Form of Interval Transportaton Problem Wth Multple Obectves The formulaton of MITP s the problem of mnmzng k nterval valued obectve functons wth nterval supply and nterval destnaton parameters s gven and an effcent algorthm s presented to fnd the optmal soluton of MITP. The mathematcal model of MITP when all the cost coeffcent, supply and demand are nterval-valued s gven by: Mnmze Z k = m n ] [C k,c k x L R where k = 1,2,...,K, =1=1 Subect to n x = [a L,a R ], = 1,2,...,m, =1 m x = [ ] b L,b R, = 1,2,...,n, =1 x 0, = 1,2,...,m, = 1,2,...,n, Wth m a L = n b L and m a R = n =1 =1 =1 b R =1 Where the source parameter les between left lmt a L and rght lmt a R. Smlarly, destnaton ] parameter les between left lmt b L and rght lmt b R and [C k,c k, (k = 1,2,...,Ksannterval ndcatngtheuncertancost for L R the transportaton problem; t can exemplfy delvery tme, quantty of goods delvered, under used capacty, etc. [4]. 3. A Method to Solve Mult-obectve Interval Transportaton Problem Consder some notatons to defne the varables and the sets n mult-obectve nterval transportaton problem. Let A = A 1,A 2,...,A m be the set of m-orgns havng a ( = 1,2,...m unts of supply respectvely.let B = B 1,B 2,...,B n be the set of n-destnatons wth b ( = 1,2,...,n unts of requrement respectvely. There s a penalty c such as cost or delvery tme or safety of delvery etc. assocated wth transportng a commodty from th source to th destnaton. A varable x represent the unknown quantty to be shpped from th source to th destnaton.

4 222 J. Patel, J. Dhodya The problem s to determne the transportaton schedule when multple obectves wth nterval parameters exst. Grey stuaton decson makng theory s used to mnmze or maxmze the total transportaton penalty accordng to the problem whch satsfyng supply and demand condtons. Assume that the set of m-orgns A = a 1,a 2,...,a m as the set of events, the set of n-destnaton B = b 1,b 2,...,b n as the set of countermeasures, the penalty c as the stuaton set denoted by C = c = (a,b a A, b B. Frst of all confrm the decson makng goals (obectves, try to fnd the correspondng effect measure matrx U (k as Ũ(k = [ ũ (k ] = ũ (k 11 ũ (k ũ (k 1m ũ (k 21 ũ (k ũ (k ũ (k n1 ũ (k n2 2m ũ (k nm Here the data of decson makng goals for transportng a product s the effect sample value ũ (k ( = 1,2,...,n; = 1,2,...,m of stuaton c C wth obectve k = 1,2,...,s. due to that the decson nformaton s a grey number but not an exact number, the effect sample value ũ (k of the stuaton c wth [ ] obectve k s a grey number. Let ũ (k = u (kl,u (ku where u (kl and u (ku are respectvely the upper and the lower lmt of effect value of stuaton c wth obectve k. Now, fnd the upper effect measure and lower effect measure usng formula [7]: 1 For obectves whch nvolve the effect sample value fulfllng the bgger, the better and the more, the better, we establsh the effect value to measure the upper effect measure for upper and lower lmt respectvely and the upper effect measure try to fnd the maxmum varaton data such as speed, safety etc. = max ũ (k max ũ (k 2 For obectves that necesstate the effect sample value fulfllng the less, the better, we establsh the effect value to measure the lower effect measure for upper and lower lmt respectvely and the lower effect measure seek out to fnd the mnmum varaton data such as travel tme, cost etc. mn mn = ũ (k ũ (k

5 SOLVING MULTI-OBJECTIVE INTERVAL TRANSPORTATION [ ] Due to that the above two effectve measures = r (kl,r (ku can satsfy (1 Dmensonless r (kl,r (ku (2 r (kl,r (ku [0,1]; (3 The more deal the effect s, the bgger the s. and acheve the consstent effect measure matrx R (k of stuaton C wth obectve k usng upper effect measure and lower effect measure as ( R (k = = m n1 n2 2m r nm (k Subtract each data from 1 of the consstent matrx of effect measure. There are two dfferent way to obtan obectve weghts η 1,η 2,...,η s. 1 Decson maker gve a weght to the obectves accordng to ther frst preference. 2 Decson maker apply some methods to obtan the obectve weghts. Here, fortheobectve weghts η 1,η 2,...,η s theoptmzaton modelofobectve weght s appled by constructng lagrange functon as L(η,λ = µ µ n m s =1 =1 k=1 n m s =1 =1 k=1 +(1 2µ d +(, ṽ(k+ η k d (, ṽ(k η k ( s s η k lnη k λ η k 1 =1 After smplfcaton we obtang the formula to fnd the obectve weght whch s as follow [ µ n m ( d, ṽ(k µ n m ( ] d +, ṽ(k+ =1=1 =1=1 exp (1 2µ 1 η k = [ s µ n m d (, ṽ(k µ n m d +( ], ṽ(k+ =1=1 =1=1 exp (1 2µ 1 k=1 k=1 (1

6 224 J. Patel, J. Dhodya From the obectve weghts and consstent matrces of effect measure acqure the comprehensve effect measure of stuaton c s r = s η k r (k and acheve the comprehensve matrx of effect measure [3]. R = ( r = r 11 r 12 r 1m r 21 r 22 r 2m r n1 r n2 r nm Fnd solutons for mult-obectve transportaton problem from comprehensve matrx r of effect measure usng modfed dstrbuton method n LINGO package and here supply and demand shpped from orgns to destnatons s already gven. k=1 (2 4. Developed Algorthm Input ( Effect measure matrx Ũ(k = Ũ(1, Ũ(2,..., Ũ(s ;n m Output Soluton of MITP Compute the effcent soluton of MITP usng the optmzaton model of obectve weght. Solve MITP begn Step 1 Read: Example whle example = MITP do for k=1 to s do enter effect measure matrx Ũ(k Step 2 Fnd the lower effect measure and upper effect measure for both upper and lower [ lmt ] and accomplsh the consstent matrx of effect measure R (k =. For k=1 to s do

7 SOLVING MULTI-OBJECTIVE INTERVAL TRANSPORTATION = max ( R (k = ũ (k max ũ (k = and = mn mn ũ (k ũ (k m m r nm (k n1 n2 Step 3 Subtract each value from 1 of consstent matrx of effect measure. for k=1 to s do R (k = 1 R (k Step 4 Fndthepostveandnegatve dealvector ofeffectmeasure ṽ (k+ ṽ (k for both upper and lower lmt respectvely. for k=1 to s do ṽ (k+ = max and ṽ (k = mn and ( ( Step 5 Fnd the devaton d + ṽ(k+ and d ṽ(k of upper and lower lmt for obectve k = 1,2,,s after that fnd the total devaton under all targets. for( k=1 to s do d + =, ṽ(k+ and( d, ṽ(k for k=1 to s do D + = n m =1=1 = ( d +, ṽ(k+ ṽ(k+ ṽ(k and D = n m =1=1 ( d, ṽ(k Step 6 Select the balance coeffcent µ (1 µ 1/2 between the obectves and compute the obectves weght η 1,η 2,...,η s. µ = 1/3 for k=1 to s do η k

8 226 J. Patel, J. Dhodya Step 7 Get the comprehensve effect measure matrx for stuaton s s R = [ r ]. for k=1 to s do R = [ r ] = s η k k=1 Step 8 Fnd solutons for mult-obectve nterval transportaton problem from comprehensve matrx R = [ r ] of effect measure usng modfed dstrbuton method n LINGO package. 5. Illustraton Examples To llustrate the above method, consder the followng examples of mult-obectve transportaton problem. Numercal llustraton 1: A company has three producton facltes (orgns A 1, A 2 anda 3 wth producton capacty of 8, 19 and 17 unts of a product respectvely. These unts are to be shpped to four warehouses B 1, B 2, B 3 andb 4 wth requrement of 11, 3, 14 and 16 unts respectvely. The transportaton cost and transportaton tme between companes to warehouses are gven below [6]. Ũ(1 = Ũ(2 = [1,2] [1,3] [5,9] [4,8] [1,2] [7,10] [2,6] [3,5] [7,9] [7,11] [3,5] [5,7] [3,5] [2,6] [2,4] [1,5] [4,6] [7,9] [7,10] [9,11] [4,8] [1,3] [3,6] [1,2] Soluton: Consder case set, counter set and stuaton set. Producton facltes of company are the case. Let A = A 1, A 2, A 3 s the case set and A 1, A 2 anda 3 on the behalf of three producton facltes of company (orgns. Destnaton s the counter. B = B 1, B 2, B 3, B 4 s the counter set and B 1, B 2, B 3 andb 4 on the behalf of four destnatons. Stuaton set C = c = (A, B A A, B B s structured by A and B. 1 Authentcate the decson-makng goals tme and cost and after that consder the upper lmt of ntervals of the matrces as a one matrx and smlarly

9 SOLVING MULTI-OBJECTIVE INTERVAL TRANSPORTATION consder the matrces for the lower lmts. So the effect measure matrces under two goals are gven below. U (1L = U (2L = U (1U = U (2U = For transportng a product, tme and cost are the less, the batter, so use lower effect measure. Here, the lower effect measure for frst data r (1L 11 = mn mnu L = 1 u L 1 = 1. Smlarly obtan lower effect measure for each data of the 11 matrces. Therefore the consstent matrces of effect measure are gven below. U (1L = U (1U = U (2L = U (2U = Subtract each value from 1 of consstent effect measure matrx under target k = 1,2. U (1L = U (1U = U (2L = U (2U =

10 228 J. Patel, J. Dhodya 4 The postve and the negatve deal vector of effect measure for both obectves are gven below: The postve deal vector of effect measure for two obectves For frst obectve k=1 v (1L+ 1 = max r (1L 1.8 v (1L 1 = mn v (1L+ 2 = max r (1L v (1L 2 = mn v (1L+ 3 = max r (1L v (1L 3 = mn v (1U+ 1 = max r (1U v (1U 1 = mn v (1U+ 2 = max r (1U 2.8 v (1U 2 = mn v (1U+ 3 = max v (1U 3 = mn 1 = max 2 = max 3 = max 1 = max 2 = max 3 = max v (2L+ v (2L+ v (2L+ v (2U+ v (2U+ v (2U d ( r (1L =1 =1 3 =1 =1 3 4 =1 = =1 =1 d ( r (1U d ( r (2L d ( r (2U r (1U 3 r (2L 1 r (2L 2 r (2L 3 r (2U 1 r (2U 2 r (2U 3 The negatve deal vector of effect measure for two obectves r (1L 1 r (1L 2 r (1L 3 r (1U 1 r (1U 2 r (1U 3 For frst obectve k= v (2L 1 = mn v (2L 2 = mn.75 v (2L 3 = mn.6 v (2U 1 = mn v (2U 2 = mn.75 v (2U 3 = mn v (1L = , v (1U = , v (2L = , v (2U = , 3 4 =1 = =1 =1 =1 =1 3 4 d +( r (1L d +( r (1U d +( r (2L 4 =1 =1 d +( r (2U r (2L 1 r (2L 2 r (2L 3 r (2U 1 r (2U 2 r (2U v (1L+ = v (1U+ = v (2L+ = v (2U+ = Gve the equlbrum coeffcent µ = 1/3 and calculate the weght of the obectves η 1, η 2,...,η s usng equaton (2. The weghts of the obectve are η L , η L , ηu ,ηU

11 SOLVING MULTI-OBJECTIVE INTERVAL TRANSPORTATION The comprehensve matrx of effect measure s got accordng to r = s η k r (k ; k=1 r = [ , ] [0.3355, ] [0.5987, ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [0.282, ] [0.557, ] [0.1316, ] 8 Soluton of comprehensve matrx of effect measure usng modfed dstrbuton method n LINGO package. 1. Soluton of comprehensve matrx of effect measure for lower lmt x 12 = 2,x 13 = 6,x 21 = 11,x 23 = 8,x 32 = 1,x 34 = Soluton of comprehensve matrx of effect measure for upper lmt x 12 = 3,x 13 = 5,x 21 = 11,x 23 = 8,x 33 = 1,x 34 = 16 Comparson U (1L = = 146 U (2L = = 133 U (1U = = 241 U (2U = = 222 Deepka Ran [4] Deepka Ran [4] Deepka Ran [4] S. K. Das, A. Developed (Lnear (Exponental (Hyperbolc Goswam and Method membershp membershp membershp S. S. Alam [6] functon functon functon [172.2,222.55] [171.50,221.63] [172.2,222.55] [119.14,214.42] [146,241] [206.1,252.75] [207.54,254.36] [206.1,252.75] [180.64,241.1] [133,222] Numercal llustraton 2: A company has three producton facltes (orgns A 1, A 2 anda 3 wth producton capacty of [7, 9], [17, 21] and [16, 18] unts of a product respectvely. These unts are to be shpped to four warehouses B 1, B 2, B 3 andb 4 wth requrement of [10, 12], [2, 4], [13, 15] and [15, 17] unts respectvely. The transportaton cost and transportaton tme between companes to warehouses are gven below [4]. Ũ(1 = Ũ(2 = [1,2] [1,3] [5,9] [4,8] [1,2] [7,10] [2,6] [3,5] [7,9] [7,11] [3,5] [5,7] [3,5] [2,6] [2,4] [1,5] [4,6] [7,9] [7,10] [9,11] [4,8] [1,3] [3,6] [1,2]

12 230 J. Patel, J. Dhodya Soluton: Solutons of comprehensve matrx of effect measure usng LINGO Software. 1. Soluton of comprehensve matrx of effect measure for lower lmt x 12 = 1, x 13 = 6, x 21 = 10, x 23 = 1, x 32 = 1, x 34 = 1 2. Soluton of comprehensve matrx of effect measure for upper lmt x 12 = 2, x 13 = 5, x 21 = 12, x 23 = 5, x 33 = 3, x 34 = 15 Comparson U (1L = = 137 U (2L = = 119 U (1U = = 225 U (2U = = 202 Deepka Ran [4] Deepka Ran [4] Deepka Ran [4] Developed (Lnear (Exponental (Hyperbolc Method membershp membershp membershp functon functon functon [159.02, ] [158.95,204.90] [159.02,205.03] [137,225] [176.54, ] [186.48,227.95] [176.54,227.95] [119,202] Numercal llustraton 3: A company has three producton facltes (orgns A 1, A 2 anda 3 wth producton capacty of [7, 9], [17, 21] and [16, 18] unts of a product respectvely. These unts are to be shpped to four warehouses B 1, B 2, B 3 andb 4 wth requrement of [10, 12], [2, 4], [13, 15] and [15, 17] unts respectvely. The transportaton cost and transportaton tme between companes to warehouses are gven below [4]. Ũ(1 = , Ũ(2 = Soluton: Solutons of comprehensve matrx of effect measure usng LINGO Software. x 12 = 2, x 13 = 5, x 21 = 12, x 23 = 5, x 33 = 3, x 34 = 15 U (1 = = 168 U (2 = = 158 Comparson Numercal llustraton 4: A company has three producton facltes (orgns A 1, A 2 anda 3 wth producton capacty of 12, 16 and 20 unts of

13 SOLVING MULTI-OBJECTIVE INTERVAL TRANSPORTATION Deepka Ran [4] Deepka Ran [4] Deepka Ran [4] Developed (Lnear (Exponental (Hyperbolc Method membershp membershp membershp functon functon functon a product respectvely. These unts are to be shpped to four warehouses B 1, B 2, B 3 andb 4 wth requrement of 15, 12, 10 and 11 unts respectvely. The transportaton cost and transportaton tme between companes to warehouses are gven below. Ũ(1 = Ũ(2 = [0.5, 1.1] [0.8, 1.2] [1.3, 1.7] [1.8, 2.2] [1.2, 1.6] [1.3, 1.5] [1.5, 1.9] [2.0, 2.4] [1.6, 2.0] [1.8, 2.2] [2.2, 2.4] [1.2, 1.6] [0.4, 0.9] [0.7, 1.1] [1.2, 1.5] [1.6, 2.0] [1.1, 1.8] [0.9, 1.4] [1.3, 1.8] [2.1, 2.5] [1.9, 2.2] [1.5, 2.0] [2.3, 2.7] [1.4, 2.1] Soluton: Solutons of comprehensve matrx of effect measure usng LINGO Software. 1. Soluton of comprehensve matrx of effect measure for lower lmt x 11 = 12, x 22 = 6, x 23 = 10, x 31 = 3, x 32 = 6, x 34 = Soluton of comprehensve matrx of effect measure for upper lmt x 11 = 12, x 22 = 12, x 23 = 4, x 31 = 3, x 33 = 6, x 34 = 11 U (1L = = 57.6 U (2L = = 53.3 U (1U = = 76.8 U (2U = = 80.7 Numercal llustraton 5: A company has three producton facltes (orgns A 1, A 2 anda 3 wth producton capacty of 18, 10 and 22 unts of a product respectvely. These unts are to be shpped to four warehouses B 1, B 2, B 3 andb 4 wth requrement of 16, 13, 12 and 9 unts respectvely. The transportaton cost and transportaton tme between companes to warehouses are gven below. Ũ(1 = [4, 6] [5, 8] [7, 9] [8, 12] [6, 8] [7, 13] [10, 14] [15, 19] [8, 12] [10, 14] [16, 20] [8, 10]

14 232 J. Patel, J. Dhodya Ũ(2 = [2, 5] [4, 7] [6, 8] [7, 11] [6, 10] [8, 11] [9, 12] [12, 16] [10, 14] [12, 15] [14, 18] [18, 20] Soluton: Solutons of comprehensve matrx of effect measure usng LINGO Software. 1. Soluton of comprehensve matrx of effect measure for lower lmt x 11 = 16, x 12 = 2, x 23 = 10, x 32 = 11, x 33 = 2, x 34 = 9 2. Soluton of comprehensve matrx of effect measure for upper lmt x 11 = 16, x 13 = 2, x 23 = 10, x 32 = 13, x 34 = 9 U (1L = = 388 U (2L = = 452 U (1U = = 526 U (2U = = Concluson Ths paper present the compromse soluton of mult-obectve nterval transportaton problem obtaned usng grey stuaton decson makng theory based method wth obectve weghts. The comparson shows that the compromse soluton s better nd acceptable n real lfe stuaton when more than one obectve avalable n transportng a product. References [1] A. Panda and C. B. Das, Cost Varyng Interval Transportaton Problem under Two Vehcles, Journal of New Results n Scence, 3 (2013, [2] S. Murugesan and B. Ramesh Kumar, New Optmal Soluton to Fuzzy Interval Transportaton Problem, Internatonal Journal of Engneerng Scence and Technology, 3(1 (2013, [3] Dang Yao-guo, Wang zheng-xn, L Xue-me, Xu nng, The Optmzaton Model of Obectve Weght n Grey Stuaton Decson, Proceedngs of IEEE Internatonal Conference on Grey Systems and Intellgent Servces, Nanng, Chna (2009. [4] Deepka Ran, Fuzzy Programmng Technque for Solvng Dfferent Types of Multobectve Transportaton Problem, Thapar Unversty, Punab (2010. [5] Kaur, Dalbnder, Sath Mukheree, and K. Basu, Soluton of a Mult-Obectve and Mult-Index Real-Lfe transportaton problem usng dfferent fuzzy membershp functons, Journal of Optmzaton Theory and Applcatons, 164(2 (2015,

15 SOLVING MULTI-OBJECTIVE INTERVAL TRANSPORTATION [6] S. K. Das, A. Goswam and S. S. Alam, Mult-obectve transportaton problem wth nterval cost, source and destnaton parameters, Elsever, European ournal of operatonal research, 117(1 (1999, [7] Sfeng Lu and Y Ln, Grey Informaton: Theory and Practcal Applcatons, Sprnger (1990. [8] Vncent F. Yu, Kuo-Jen Hu and An-Yuan Chang, An nteractve approach for the multobectve transportaton problem wth nterval parameters, Taylor & Francs, nternatonal ournal of producton research, 53(4 (2014,

16 234