Dynamic Programming and Multi Objective Linear Programming approaches

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1 0 NSP Vol. () (0), 9 Dynami Programming and Multi Objetive Linear Programming approahes P. K. De and Amita Bhinher Department o Mathematis, National Institute o Tehnology SILCHAR (Assam) India Apaji Institute o Mathematis and Applied Computer Tehnology Banasthali University P.O.- Banasthali Vidyapith-00, Rajasthan, India pusde@redimail.om This paper desribes two dierent methods to deal with the uzzy shortest path problems. The irst one is to study uzzy shortest path in a network by Bellman dynami programming approah and the seond one is to study same problems by multi objetive linear programming (MOLP) tehnique. It is onsidered that the edge weights o the network as unertain. To analyze this idea o unertainty our eamples have been taken with two dierent network where edge weights have been presented by triangular uzzy numbers and trapezoidal uzzy numbers respetively. Both the problems have been solved by the above two methods. In the irst method sign distane ranking proedure has been applied to get real value o the uzzy edge weights where as in the seond method MOLP, only 0- variables have been onsidered to get integer solution without using the Branh and Bound tehnique. It is observed that the length o the shortest paths in uzzy sense as obtained by both the methods are same/almost same and the shortest path orresponds to the atual path in the network. It is obvious that uzzy shortest path is an etension o the risp problem. AMS Mathematis Subjet Classiiation (MSC 000) : 0E7, 90B0, 9D0 Key words and phrases: Shortest-path, triangular uzzy number, trapezoidal uzzy number, multi objetive linear programming.. INTRODUCTION The shortest path problem is one o the most important problems in network optimization. The problem o shortest path in a network has attrated many previous researhers sine it is important to many transportation problems, routing, ommuniations, eonomial, and other appliations. Graphs/Networks emerge naturally as mathematial model o the observed real world system. The weights o edges an epress geographial distanes, transportation ost or time between two verties onneted by the edge. While geographial distanes an be stated deterministially, ost or time an lutuate with trai onditions, pay load and so on. Thereore unertainty arises to epress edge weights o a network through ost or time. A typial way o epressing these unertainties in the edge weights is to utilize uzzy numbers. Also in a network uzziness an be introdued in a variety o ways through edge weights, edge apaities, verte restrition, ar lengths [,7,,, 9]. The onept o uzzy deision making problems with maimizing deision was irst proposed by Bellman and Zadeh []. Zimmerman [] presented a uzzy approah to multi objetive linear programming problems. He also studied the duality relations in uzzy linear programming. Fuzzy linear programming problem with uzzy oeiients was ormulated by Negoita is alled robust programming. Dubois and Prade [] investigated linear uzzy onstraints. Liu and Kao s [9] developed an algorithm or inding non-dominated shortest path. In this approah they transorm all the uzzy ars into risp ars by applying Yoger s ranking method and then solve the uzzy shortest path problem with risp ars by using 0- variables. In this paper, we disussed two dierent methods or the same uzzy shortest path problems. Both the methods have been eeuted with the same numerial eamples. Two dierent

2 0 NSP Vol. () (0), 9 networks have been onsidered whose edge weights are uzzy numbers- triangular uzzy numbers and trapezoidal uzzy numbers. The irst method is Bellman dynami programming approah. When solving by this method the uzzy shortest path problem is irst onverted into an equivalent risp problem and then reursion proedure is applied. The seond method we use here is the multi objetive linear programming method as suggested by []. Enough literatures about the uzzy shortest path an be ound, e.g., [,,,,8,]. Dubios and Prade [] irst in 980 treated the uzzy shortest path problems. It is obvious that uzzy shortest path an be determined but in reality it may not orrespond to an atual path o the network. To analyze this problem Klein [0] proposed a dynami programming reursionbased uzzy algorithm that speiied eah ar length within an integer value rom one to a ied number. In [], Okada and soper introdued an order relation between uzzy numbers based on uzzy min onept and that a uzzy non dominated path or Pareto optimal path rom any node to a speiied node o a network. To deal with impreision parameters in mathematial programming problems, uzzy set theory has been applied to real word deision making problems. Fuzzy linear programming models and uzzy multi objetive programming problems are designated or suh a purpose in uzzy set theory, a orresponding membership untion is usually employed to quantiy the uzzy objetives and onstraints, using the linear membership untion, Zimmermann proposed the min operator models to the multi objetive linear programming. Okada and Soper also restrited eah uzzy number o ar length to L-R uzzy number, thus the our objetive untions are required in their approah []. Apparently this problem an be ormulated as a linear multi objetive programming problem without using any 0- variable beause it belongs to the problem type o network linear programming []. The weights (lengths or osts or times) o edges an epress geographial distanes o the orresponding verties or transportation osts epended (or times spent) to move between their and verties. Whole geographial distanes an be stated deterministially, ost or times an lutuate with trai onditions, pay load and so on. In the last two ases (ost or time), deterministi values or representing the edge weights an not be used. A typial way o epressing these unertainties in the edge weights is to utilize uzzy numbers based on uzzy set theory. In the ase we must deine an order relation between uzzy numbers, beause the uzzy variant o the problem evaluates uzzy min operations. As many approahes or the omparison o uzzy numbers do not guarantee that uzzy number are totally ordered, they lead to a number o non dominated paths (or pareto optimal paths). In this paper our approah is based on Cheng s uzzy ranking method. Soure Fig - Destination Aording to bellman s equation [BL] a Dynami programming ormulation or the shortest path problem an be given as ollows, onsider a network with an ayli direted graph G ( V,E) with n verties number rom to n suh that is the soure node and n is the destination node. Then, by using orward alulation we have,

3 0 NSP Where weight o the direted edge i, j () 0 ( j) min ( i) l< j { + } () i length o the shortest-path rom the soure verte. Vol. () (0), 9 (B). DEFINATION AND PREREQUISITES Deinition : A Fuzzy set is a set whose boundary is not lear, whose elements are haraterized by a membership untion. Let X be a universal set. A uzzy set à deine on X. A set o order pair o element whose irst element X, seond element µ ~ ( Α ) is the membership value o element in the set Ã.It is denoted by à or A, and it deined by A ~, µ A X Where µ ( ) K {( ( )) } A Signed distane ranking method or uzzy numbers In the given network the edge weights are represented as triangular uzzy numbers. A triangular uzzy number ~, ~ ( a,b,) is a uzzy set. The signed distane o ~ measured rom 0 ~ is deined by d ( ~ ~, 0 ) ( a + b + ) whih will map the uzzy number ~ on the real line R. Prop- The sum o two uzzy numbers A ~ ( u,v,w ) and B ~ ( p, q, r) then the binary operation is given by A ~ B ~ ( u + p, v + q, r + w) ~ ~ and d ( A ~ B ~, 0 ) d( A ~, 0 ) + d( B ~, ~ 0 ) Prop- The ranking o uzzy number A ~ ( u, v, w) and B ~ ( p, q, r) is deined by ~ ~ A ~ B ~ i d ( A ~, 0 ) ( B ~ < d, 0 ) A ~ B ~ i d ( A ~, ~ 0 ) ( B ~, ~ d 0 ) Fuzzy numbers Fuzzy number is epressed as uzzy set deining in the interval o real number R.Sine the boundary o this interval is ambiguous thus interval is also a uzzy set. There are three types o uzzy number:. Interval Fuzzy Number. Triangular Fuzzy Number. Trapezoidal Fuzzy Number Triangular Fuzzy Number: It is a uzzy number represented with three points as ollows A ( a, b, )

4 0 NSP Vol. () (0), 9 µ A ( ) 0 a b Fig..Traingular uzzy number. Computation o shortest path base on uzzy numbers Doubis and Prade [] irst shown how to determine shortest path in a network in uzzy environment. Fuzziness an be introdued in a network in a variety o ways, e.g., through edge apaities, edge weights, or verte restritions. In real lie situations, some unepeted events may our so that the edge weight in the network may hanges slightly. I ~ (,, ) then ( + + ) > 0 d ( ~ ~, 0 ) 0 ~ is a positive distane measured rom to ~ and also number rom 0.. Thereore, we alulate that is also a positive Eample- We onsider the ollowing ayli network G ( V, E) with topologial ordering whose edge weights are given triangular uzzy numbers. (,, 8) (,, 0) (,, ) (7,, ) (, 0, 9) (8, 9, 0) (,, 8) Fig - (7,, ) I ~ (,, ) then ( + + ) > 0 d ( ~ ~, 0 ) 0 ~ is a positive distane measured rom to ~ and also number rom 0. (,, ). Thereore, we alulate that is also a positive

5 0 NSP Vol. () (0), 9 Using distane ranking method we ind the real values o the triangular uzzy numbers o edge weights as ollows Similarly + + ( + + 8) ( ) 9 0 The solution o the Bellman dynami programming an be derived as ollows, min { } { } { } min{ ( ) +, ( ) + } () 0, ( ) ( ) ( ) min < ( i) + i min { 0 +, + } 9 0 Fig - Similarly we get { + } { + } + { } min{ ( ) +, ( ) +, ( ) + } ( ) min ( ) () min () i + i min{ +, +, + 9} 8 + ( ) min{ ( i) + } min{ ( ) +, ( ) + } min { +, 8 + 0} Eample- We onsider the ollowing ayli network G ( V, E) with topologial ordering whose edge weights are given trapezoidal uzzy numbers. 7

6 0 NSP Vol. () (0), 9 (,, 0, ) (, 7,, ) (0,,, ) (9, 9,, ) (7, 8,, ) (8, 0,, ) (, 0,, ) 7 (0, 0, 0, 0) (9,,, ) (70, 80, 0, 0) (, 0,, ) Fig - G is onsists o 7 nodes and edges with topologial ordering o nodes. The edge weights have been onsider as L-R trapezoidal uzzy numbers ~ and are represented as,, α, β The seond network ( V,E) LR α β _, then we an transorm the numbers rom triangular type to L-R trapezoidal type suh as l, m, n m, n, m-l, n-m I the edge weights are triangular uzzy numbers, like ( l m, n) ( ) ( ) LR by using Yager s [Bel A proedure or ordering ] enter o gravity or entroid deuzziiation method the L-R trapezoidal numbers an be transormed into real numbers or utility values as ollows,, α, β + + α + ( + β ) _ _ _ Thereore (,, 0, ) [ + + ( - 0 ) + ( + ) ]. ( 8, 0,, ) [ ( 8 - ) + ( 0 + ) ] 9. Similarly the other numbers are ( 0, 0,0,0) 0 ( 0,,, ) 9,,,. (, 0,, ) 8 ( ) Fig- 8

7 0 NSP Vol. () (0), 9 (, 7,, ) ( 9, 9,,) 9 (, 0,, ) 7. ( 7, 8,,) 8.7 ( 70, 80, 0, 0) 7 7 with this real values the above network an be rewritten as Fig - Now we use Bellman method o dynami programming we have () 0 min { } { } { } ( ) ( ) < () min{ () i + i} min ( ) +, ( ) + min { ,. + } 9. ( ) min{ ( i) + i} min ( ) +, ( ) + min { 0 + 0, 9. +.} 0 () min{ () i + i} min ( ) +, ( ) + min{. +, } 8. + ( ) min{ ( i) + i} min ( ) +, ( ) + min{ 0 + 8, } + + ( 7) min ( i) + min +, + { } { } { } { i7} { ( ) 7 ( ) 7} 7 7 min{ , + 7} Computation o shortest path by using multi objetive linear programming In this setion we purpose a simple multi objetive linear programming to deal with the uzzy shortest path problem. In this approah 0- variables are not required to obtain shortest path in a network [Yu &Wei; solving the F. S. path M. O. P].Yu and Wei obtained a ompromising non-dominated integer optimal solution o uzzy shortest path without adding etra onstraints. 9

8 0 NSP Vol. () (0), 9 Weighted Additive Method to solve Multi Objetive Linear Programming Chen et al have used weighted average in Fuzzy goal programming with dierent importane and properties. Tiwari, Dharmar and Rao have mentioned an additive model in uzzy goal programming whih inorporates eah goal s weight U K into the orresponding objetive untion ie m Z W K U Where K K W K denotes the m th K uzzy goal and K U. In the additive model weights show the relative importane o the goals. Now or simpliity the importanes o these objetives (goals) are assumed as the same. Hene all the objetive untion s an be reormulated as a single objetive untion without adding the onstraints. min Z U suh that ( ) + U W ( ) + U W ( ) + U ( ) W W K n j - n j ji, -, i i n 0, otherwise or i,,,..., n and j,,,..., n n K U K The desription o the network LPs onstraints properly is as ollows A linear programming is said to be a network. I eept or simple upper and lower bound onstraints (suh as ), eah variable appears in at most two onstraints.. I eah variable appear in two onstraints, its oeiients in the two ar + and -. I the variable appears in one onstraint, its oeiient is either + or - Multi objetive linear programming ormulation o Eample- The edge weights o the network in eample- are triangular uzzy number. Thereore, the orresponding uzzy shortest path problem an be ormulated with three objetives as ollows min Z W U + WU + WU we assume equal weights U U U, U i i ( ) + min Z ( ) ( ) Where, W W

9 0 NSP Vol. () (0), 9 W suh that The optimal solution is Z 7. W 7, W 9, W 7 and,, 0, 0, 0 0, 0, 0 The shortest path is shown in igure-7 { } (7, 9, 7) Multi objetive linear programming solution o Eample- and its uzzy shortest path length is The edge weights o the network in eample- are L-R trapezoidal uzzy number. So, the orresponding multi objetive uzzy shortest path problem an be ormulated with our objetives as ollows min Z [ W + W + W + W ], when weight are equal U U U U ( ) + ( ) ( ) + ( ) suh that + +, + + 0, , , 7 7 the optimal solution is Z 0.,,, 0, 0, 0, 0, 0, 0, 0, Fig 7 7 W, W, W, W 7 0,

10 0 NSP Vol. () (0), 9 7 Fig-8 Fuzzy shortest path and the orresponding uzzy shortest path length is (,, 9, 8). The optimal value o Z is 0., that mean the shortest path length is 0. whih is the same as obtained by Bellman dynami programming method. The shortest path is shown in ig-8 as { 7} Eamples Bellman Dynami Programming Method S. P. length in E- Length 8 Path : S. P. length in E- Length 0. Path : 7 Multi Objetive Linear Programming Method Optimal value 7. S. Path { } F.S.P. length is (7, 9, 7) Optimal value 0. S. Path { 7} F.S.P. length is (,,9, 8). CONCLUSIONS In this paper some uzzy problems in network have been presented through two dierent methods irst one is Bellman Dynami Programming method and the seond one is Multi Objetive Linear Programming method. In the seond method non-dominated integer optimal solution is obtained without using 0- variables. Also this method redues the ompleity o solving shortest path in a network. Two dierent networks have been onsidered whose edge weight triangular uzzy numbers and trapezoidal uzzy numbers. Both the networks have been solved by eah o the above two methods. It is observed that shortest path length in Bellman method is same/nearly same with the optimal value in the Multi Objetive Linear Programming and also the orresponding uzzy numbers also shown. REFERENCES. [Bellman 970] Bellman.R.E, and Zadeh.L.A, Deision making in a uzzy environment management siene,7 pp.-. [Zimmermann 97] Zimmermann,H.J. Desription and optimization uzzy systems,international journal o general system, 09-. [Zimmermann 778] Zimmermann,H-J, Fuzzy programming and linear programming with several objetive untions,,-

11 0 NSP Vol. () (0), 9. [Dubios 980] Dubois and H. Prade, Theory and Appliations : Fuzzy Sets and Systems, Aademi. Aademi Press, New York. [Yager, 98] Yager, R.R., A proedure or ordering uzzy subsets o unit interval, Inormation Sienes,, -, 98. [Chanas 98] Chanas, S and W. Kolodziejzyk, Maimum low in a network with uzzy ar apaities, Fuzzy Sets and Systems, Vol. 8, p--7, [Mares, 98] Mares, M. and J. Horak., Fuzzy quantities in networks, Fuzzy Sets and Systems, Vol. 0, p- -, [Zimmermann 98] Zimmermann, H.J, Fuzzy mathematial programming Compute and Ops. Res. Volume 0 no. pages [Twari 987] Tiwari, R. N., S. Dharmar and J. R. Rao, Fuzzy goal programming an additive model, Fuzzy Sets and Systems,, [klein 99] Klein, C.M., Fuzzy Shortest Paths, Fuzzy Sets and Systems, Vol. 9, E, pp.7-.. [Zimmer 99 ] Zimmermann, H.J., Fuzzy Set Theory and its Appliations, nd ed, Klwer Aademi Publishers, London, 99. [Okada 99] S.Okada, M.Gen, Fuzzy shortest path problem. Computers and Industrial Engineering;7:-8.. [Lin 99] Lin, K. and Chen, M., The uzzy shortest path problem and its most vital ars, Fuzzy Sets and Systems, Vol. 8, pp.-,99. Fortemps, 99] Fortems, P. and M. Roubens, Ranking and Deuzziiation Methods Based on Area Compensation, Fuzzy Sets and Systems, Vol. 8, p-9-0, 99.. [Cheng 998] Cheng, C.H., A new Approah or ranking uzzy numbers by distane method, Fuzzy Sets and Systems, Vol.9, p-07-7, [Okada 000] Okada, S and Soper, T, A shortest path problem on a network with uzzy are lengths, Fuzzy Sets and Systems, Vol. 09, pp. 9-0, [Yao, 000] Yao, J.S. and K.M.Wu, Ranking uzzy numbers based on deomposition priniple and signed distane, Fuzzy Sets and Systems, Vol., p-7-88, [ Yao, 00] Yao, J. S. and Feng-Tse Lin, Fuzzy Shortest-Path Network Problems With Unertain Edge Weights, Journal o Inormation Siene and Engineering, 9, 9-, [Liu 00] Liu, S. T. and C. Kao, Network low problems with uzzy ar lengths, IEEE Transations on Systems, Man, and Cybernetis: Part B, : [Hillier 00] Hillier, F. S. and G. J. Lieberman. Introdution to Operations Researh, MGraw-Hill, New York P. K. De has obtained his M.S and B.Ed degrees rom Kalyani University and reeived his M. Phil and Ph. D. degrees rom Indian Shool o Mines University, Dhanbad. He was employed in many institutions like National Aerospae Laboratories (C-MMACS), Bangalore, Delhi College o Engineering, KIET Ghaziabad (U.P.Teh. University) and Banasthali University as a Senior Researh Fellow, Leturer, Senior Leturer, Reader and Assoiate Proessor. Presently, Dr.De is working as an Assoiate Proessor in Mathematis in the National Institute o Tehnology, Silhar. His researh areas inlude Fuzzy Optimization, Operations Researh, Fuzzy Logi and Belie Theory, Elastodynamis, Finite Element Modelling, Mathematial Modelling and History o Mathematis. Amita Bhinhar is pursuing her Ph.D degree rom Banasthali University. She has obtained her B.S with Mathematis Honours rom Banasthali University and also reeived her M.S in Mathematial Sienes with speialization in Operations Researh rom the same university. Presently she is working in the area o Fuzzy Optimization. Now she is employed as a leturer in Mathematis at Rajdhani Institute o Tehnology and Management, JAIPUR-090, (Rajasthan) India.