141 Scientific Journal of Maritime Research 30 (2016) 141-150 © Faculty of Maritime Studies Rijeka, 2016 Multidisciplinary Multidisciplinarni SCIENTIFIC JOURNAL OF znanstveni časopis MARITIME RESEARCH POMORSTVO 1 1 Decision making background2 for the location of inland terminals Tomislav1 Rožić , Dario Ogrizović , Marinela Galić 2 University of Zagreb, Faculty of Transport and Traffic Sciences, Vukelićeva 4, 10000 Zagreb, Croatia, e-mail: [email protected] University of Rijeka, Faculty of Maritime Studies Rijeka, Studentska ulica 2, 51000 Rijeka, Croatia ABSTRACT ARTICLE INFO Inland terminals are becoming important objects (knots) in today’s supply chain. This implies new Original scientific paper methods to determine the location of inland terminal. Appropriate inland terminal location (ITL) Received 20 September 2016 decision depends on analyzing demand, employee’s availability, investment capabilities and other KeyAccepted words: 25 October 2016 factors. In this paper we present gravity center method and median method as quantitative ones and Analytic Hierarchical Process (AHP), ELECTRE and PROMETHEE as complex decision methods to determine inland terminal location. Given methods are implemented on the example of inland Inland terminals terminal location determination in Republic of Croatia. AHP method Gravity center method Median method ELECTRE method PROMETHEE method 1 Introduction which is referred to herein as the alternatives and five cri- teria which are evaluated. Inland terminals represent objects that enable port In classical gravitation method the optimal inland ter- capacities disburdening and the expansion of the port minal location (ITL) is determinedN as a point on the map gravitational areas. This makes them vital part in today’s where[11 the] total transport costs that are induced by trans- supply chain management. One of the most important porting goods to and from different customers are mini- Ri , Vi di i – trends in the inland terminal system are methods to de- mal . Because the costs depend on unit transport costs termine location of inland terminals by satisfying different demand volume and the shortest distance from requirements. th customer to the optimal location as the center of gravity: m Choosing the location of inland terminals has to be conducted with care, because it can cause irreversible con- ∙ (1) sequences in urban planning and can create bottlenecks that lead to increase of the price of logistics services [1]. In median method the ITL is obtained by –the –calcu- According to Sorensen et al. [2] the best approach to de- Xi Yi i N X Y lation of the cumulative weight. Cartesian coordinates termine the location of inland terminal is the application [ ] 0 0 of network models and the use of multi-criteria analysis. ( , ), = 1, ..., are used for customers, and ( , ) are coordinates for ITL 6 . In the minimization of objective In this paper we used the gravity center method Vi Ri wi and median method as quantitative ones and Analytic function, we use the Manhattan or taxi – cab metric pon- dered by weights � = . Hierarchical Process (AHP), ELECTRE and PROMETHEE as | | + | | complex decision methods to determine inland terminal location. The analysis contains five cities across Croatia: ∙ (2) Slavonski Brod, Karlovac, Koprivnica, Osijek and Split, T. Rožić et al. 142 / Scientific Journal of Maritime Research 30 (2016) 141-150 = 0 and = 0 Complex decision by AHP method is to rank criteria [8] (7) and alternatives in purpose to obtain optimal solution. Qualitative criteria are compared in pairs . Preference TC After finding the derivative of the transport cost func- is used in the process of finding optimal choice, iin jiour tion , it is possible to express the unknown coordinates jcase the location of inland terminal.j It is usual to mark the explicitly: weak preferencei relation by symbol �. In our case � � = = represents that the location has weak preference over aij ∑ ∑ location . i j (8) We calculateA each coefficient which represent the – – ∑ X Y ∑ value preference of criterion ahead of criterion and con- X Y V R struct matrix shown in (3). These coefficientsa presenta i i 0 i 0 i ji ij Initial average coordinates , are the mean values ranking preference from the same over weak, strong, dem- … of , respectively, pondered by � according to the fol- onstrated and absolute preference. In that case, = 1/ . … lowing= formula: = … ∑ ∑ (9) ∑ ∑ ⋮ ⋮ ⋮ (3) The unknown –values– are found by iterative method w X Y Xi Criteria are obtained as the components of eigenvec- Ythroughi theN following steps: αmax A i tor and then can be ranked. In the procedure we also get 1. The initial– – ( , ) is the arithmetic mean of given ( , X Y di maximal eigenvalue of the matrix using the power ), = 1, ..., TC – – method. a b aSb aPb a 2. With ( , X), theY distance can be calculated by (6) In Elimination Et Choice Translating Reality (ELECTRE) – – b a b 3. Calculate the TC by (1) X Y method alternatives and are ranked by or if is 4. The new ( , ) is now obtained by (8) at least good as or if is much better[4 than] . Alternatives 5. Calculate the by (1) with new ( , ) are ranked according the criteria, by counting criteria that 6. Repeat steps 2.-5. till the costs become the same un- support one of given alternative pair . der some tolerance X Y Preference Ranking Organization METHod for i i S [ ] Problem described in (2) is solved using median meth- Enrichment Evaluation (PROMETHEE), for complex deci- [ ] Max f a f a od by findingX a mean coordinate within and separately, sion in a set1 of alternativesk , represent a problem 3 : i considering their weights 6 . After aligning the coordi- f a j ( ( ), ..., ( )) i (4) nates increasingly, we calculate the cumulative values for related sequence of weights. If – th cumulative value where criterion functions ( ) are defined discretely by Xj X amounts a half or a more than a half of the last cumulative sequences of arbitrary ordinal numbers assigned to each – j alternative.F S S Preference function value, then = . Formally, this method is written in fol- lowing inequality< conditionsand for – th coordinate: : × � �0,1� F a,b (5) a b F a,b a b ∑ ∑ ∑ Y ≥ ∑Y (10) is defined such thatF a,b( ) =P 0f meansa f weakb preferenceP d Pofx k k – over and ( ) = 1 means strong preference of over . Analogue solution method for = is written in fol- If we assumed thatf a ( f )b = ( ( ) – ( )) = ( ), then ( ) lowing inequality conditions for – th coordinate: is some non-decreasing function which takes value zero < and 2 2 for negative = ( ) – ( ). Problem could be defined vice versa, but the preference function must be defined first. In ≥ –(11)– our paper we defined criterion functions first. X Y 2 Solution methods Xj Yk Finally, optimal location is obtained as a pair ( , ) = λmax ( , ). w A R V i i In AHP[ method,] the maximal eigenvalue and its ei- Given values[ 11in] the classic gravitation problem (1) are genvector for the matrix from (3) satisfies following , and Cartesian coordinates for each customer in kil- A w λ w relation 10max: ometer’s units . If we use the shortest, Euclidian, dis- tance, then (1) is transforming into � = � (12) ( ) + ( ) A Each eigenvector’s component is obtained, in the first K ∙ (6) witeration, by summing related row of matrix . Vector in i the first iterationA A is hardly close to the wanted eigenvector where is strong unit weight value in coordinate system. A . InA theA. second iteration, elements in rows are summing Unknown ITL coordinates are obtained from necessary for matrix � . In the third iteration summing is done for conditions for extrema: � � Process stops when difference between compo- T. Rožić et al. / Scientific Journal of Maritime Research 30 (2016) 141-150 143 = nents of two successivew vectors are less then (small) value given in advance. The vector obtained in the last iteration (19) is wanted eigenvector from (12).[ ] Solving the complex decision problem by ELECTRE … 6. Calculate domination in concordance matrix by the method presumes given matrix 13 … formula: = ℎ ℎ … ℎ ℎ ℎ … ℎ (13) (20) ⋮ ⋮ ⋮ E ekl ℎ ℎhij f g ℎ kl kl 7. Format Total DominanceF Matrixf = � � withG eleg - where elements are numbers from an arbitrary or- kl kl hij ments obtained by multiplying related elements � fkl ckl c gkl dinary scalei such that every column describesj order from the obtained matrix = � � and matrix = � �. gkl dkl d of given alternatives. So every is a ranking estima- Elements are given by condition that if � then = gkl tion for – th alternative according to the – th criteri- 1. Elements are given by condition that if � then on. Furthermore, the vector for ranking criteria is also B b b b = 1. Otherwise, both elements equal zero. necessary:1 n 8. The order of alternatives is defined by counting val- = � 2 ... � (14) ues 1 in related row for every alternative. Absolutely dom- A B inated alternative has the highest number of values 1 in its where components are elements of arbitrary ordinary correspondence row. f scale. Considering and are given, one must Rproceedr i ij PROMETHEE methodπ giveni by (4)k is based on assump- through the next steps. a i ij tion that decision maker defined criterion[ ]functions and 1. Calculate the normalized decision matrix = ( ) its importance factors , = 1, ..., . For example, prefer- = with elements 2calculated by from (12): ence (function) can( be) defined( ) by formula( ) = 3 : ℎ ∑ ℎ (15) (21) Vij bi 2. Calculate the0 normalized… weighted decision matrix… 0 Final result is obtained from the preference function by values from (13):… … 0 0 values matrix with rows defined by every possible pair of … k k … alternatives, and columns defined by criteria. Every row as 0 0 – dimensional vector in dot product with – dimensional ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 0 … 0 … vector of scalar weights gives decision intensities for each alternative pair. 0 … 0 … (16) i Scalars obtained above are elements in matrix with al- … … – j – i – ternatives in rowsj – and in the columns.