Instituto Superior Técnico. Theme 1: Transport networks design and evaluation Case study presentation and synthesis

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1 Instituto Superior Técnico REGIÕES E REDES (REGIONS AND NETWORKS) Theme 1: Transport networks design and evaluation Case study presentation and synthesis Luis Martinez 1

2 OUTLINE Case study: Soybean transportation in Brazil (Fajardo, A. P., Uma Contribuição ao Estudo do Transporte Intermodal -Otimização da Expansão Dinâmica das Redes Intermodais de Soja Produzida no Estado de Mato Grosso. Dissertation submitted in Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro: 170) Description Present situation Base Optimization model (initial solution) Network expansion Optimization model Analysis of the results Synthesis of the topic 2

3 IN BRAZIL - DESCRIPTION Soy produced in the State of Mato Grosso tons around 8 production centers Almost all the production of soybean is exported (mainly to Asia) The soybean production must reach the ports located at the South and North of Brazil Capacity, cost Soy transportation problem 3

4 IN BRAZIL PRESENT SITUATION Ports available: Santos (São Paulo) Paranaguá (Paraná) Vila do Conde (Belém) Itaqui (São Luís) Santos handles nearly 50% of the country s soybean exports Existing transportation options of soybean production 4

5 IN BRAZIL BASE OPTIMIZATION MODEL (I) Mathematical modeling (Min-cost-flow): Each node representing a production center (centroid) - Source Final port Sink Super-source + Super-sink Intermodal transfer nodes Each intermodal node split into: Transport arcs constrained by transport capacity Processing arcs interchange processing capacity 5

6 CASE STUDY: SOYBEAN TRANSPORTATION IN BRAZIL BASE OPTIMIZATION MODEL (II) Variables: For arcs: A : set X c U j j j of : all arcs flow passingin arc( : unitcost of : capacity (max flowin arc( flow)of arc( For nodes: B i : Load balanceof nodei 6

7 IN BRAZIL BASE OPTIMIZATION MODEL (III) Objective function (Cost): Constraints: ( A c( X ( Node equilibrium X ( X ( j, i) j: ( A j:( j, i) A Arcs capacity B( i)... i N 0 X ( U(... ( A Cost: R$ ,57 Optimal solution with currently available transportation options. Expensive, but capacity is enough 7

8 IN BRAZIL NETWORK EXPANSION MODEL (I) The purpose is to invest in existing or new arcs, with the aim of reducing operating costs Two models runs were made: Free investments (no amortization) to compute minimum possible operating costs Real investments (amortization considered over a 15 year period) to get a realistic optimal solution Initially, all apparently interesting investments (road, rail, rivers, ports, interchange stations) identified and budgeted Some investments for capacity (to take advantage of lower costs along a favorable path which was strangled by a bottleneck) Some investments for replacement (lower costs, no additional capacity) In some cases, successive investments considered at same place 8

9 CASE STUDY: SOYBEAN TRANSPORTATION IN BRAZIL NETWORK EXPANSION MODEL (II) All investments modeled for arcs (even when physically in nodes) Each node with possible investments modeled as a local network No two arcs ( may exist in the network (Staged) Capacity increase in node C New arcs or arcs replacement 2X 3X Y j : decision variable of investmentin arc ( Two stages of capacity increase Y( E2) Y( E3) Arc replacement Y( A, C1) Y( A, C2) 1 9

10 CASE STUDY: SOYBEAN TRANSPORTATION IN BRAZIL NETWORK EXPANSION MODEL (III) New variable: Objective function: Constraints: K j : Annualizedcost of c( X ( ( A ( Y( A i investmenton arc ( K( Node equilibrium X ( X ( j, i) B( i)... i N j: ( A j:( j, i) A Arcs capacity 0 X( j ) U( j )... ( j ) A 0 0 X ( Y( * U(... ( ( Abs Ais Aia ) Binary Expansion variables Y ( j ) { 01, } j A / 10

11 CASE STUDY: SOYBEAN TRANSPORTATION IN BRAZIL NETWORK EXPANSION MODEL (IV) Constraints (cont d): Sequential investments Y ( j1) Y( j2) Y( j3)... ( j n ) A ia Arcs replacement / upgrade Identical use of both part of real arcs (with auxiliary node introduced) Y( m) Y( m,... ( m),( m, ( A bs Ais ) Alternate use due to arc replacement Y( m) Y( n) 1... ( m),( n) ( A bs Ais ) 11

12 IN BRAZIL ANALYSIS OF RESULTS (I) Results for free investments (all K=0): Transport costs: R$ ,87 Only 35% of current operating costs Results with real investments (K>0): Annualized Investment costs: R$ ,66 Transport costs: R$ ,87 Total Costs: R$ ,53 Only 62% of current operating costs Optimal solution with network expanded and unconstrained budget 12

13 IN BRAZIL DYNAMIC NETWORK EXPANSION MODEL (I) Optimal network expansion plan computed without considering Possible capital restrictions Time needed to make several (possibly many) investments operational, all across Brazil So, a dynamic network expansion problem was defined Adding a constraint of available capital at z% of the full optimal package With z varying from 1% to 100% ( j ) Y( A i j )*K( j ) TOTINV * it 1 it

14 Custos anuais Milhões REGIONS AND NETWORKS IN BRAZIL ANALYSIS OF RESULTS (II) Incremental investment analysis (thresholds of investment for total cost reduction) Transport costs reduction % of Investment Resulting cost w.r.t. current cost 0,4% 81,8% 2,5% 76,4% 11,8% 76,04% 31,2% 71,55% 40,5% 71,2% 42,2% 68,21% 44,3% 62,8% 53,6% 62,46% Redução dos Custos com a Intensidade de Investimento Custo total (bilhões reais) Custo Transporte Invest (Custo Anualizado) em % 100% 62,19% Total costs and transport cost reduction with growing investment 14

15 Investment arcs in the optimal solution Arcos de Investimento no Opção Ótima REGIONS AND NETWORKS IN BRAZIL ANALYSIS OF RESULTS (III) Incremental investment analysis (types of investment arcs) a) Arcs that became part of the optimal solution and remain until the total investment b) Arcs that became part of the optimal solution, exit and reenter again until the total investment c) Arcs that became part of the optimal solution and exit and do not reenter again until the total investment d) Arcs that became part of the optimal solution, exit, reenter, exit again and are not part of the total investment solution e) Arcs that are only part of the optimal solution with the total investment % % % % % % % % 8100% Porcentagem de Investimento da Solução Ótima Percentage of Investment of the optimal solution Identification of different types of investment arcs 15

16 Investment arcs in the optimal solution Arcos de Investimento no Opção Ótima REGIONS AND NETWORKS IN BRAZIL ANALYSIS OF RESULTS (III) Preparing a staged investment without regret (aiming at full investment) Incremental investment analysis (removal of arcs type c and d ) Solutions with these arcs Increment in total cost w.r.t. optimal dynamic path 0%=< Inv<0.4% 0,00% 0.36%<=Inv<2.5% 0,00% 2.5%<=Inv<11.8% 6,60% 11.8%<=Inv<31.2% 0,00% 31.2%<=Inv<40.5% 5,91% 40.5%<=Inv<42.2% 0,00% 42.2%<=Inv<44.3% 4,19% 44.3%<=Inv<53.6% 11,77% 53.6%<=Inv<59.9% 12,27% 59.9%<=Inv<71.3% 12,11% 71.3%<=Inv<100% 4,37% Inv = 100% 0,00% % % % % % 5100% Percentagem de Investimento da Solução Ótima Percentage of Investment of the optimal solution Identification of different types of investment arcs 16

17 Investment arcs in the optimal solution Arcos de Investimento no Opção Ótima REGIONS AND NETWORKS IN BRAZIL ANALYSIS OF RESULTS (IV) Preparing a staged investment without regret (possibly stopping without at full investment) Incremental investment analysis (network only with arcs type a ) Solutions with these arcs Increment in total cost w.r.t. optimal dynamic path 0%=< Inv<0.4% 0,00% 0.36%<=Inv<2.5% 0,00% 2.5%<=Inv<11.8% 6,60% 11.8%<=Inv<31.2% 0,00% 31.2%<=Inv<40.5% 5,91% 40.5%<=Inv<42.2% 0,00% 42.2%<=Inv<44.3% 4,19% 44.3%<=Inv<53.6% 11,77% 53.6%<=Inv<59.9% 12,27% 59.9%<=Inv<71.3% 0,00% 71.3%<=Inv<100% 0,00% Inv = 100% 0,00% % 0, % 1, % 2, % 3, Percentagem de Investimento da Solução Ótima Identification of different types of investment arcs Percentage of Investment of the optimal solution 17

18 IN BRAZIL ANALYSIS OF RESULTS (V) Incremental investment analysis (Final solutions) Investment = 1% Investment = 12% 18

19 IN BRAZIL ANALYSIS OF RESULTS (V) Incremental investment analysis (Final solutions) Investment = 41% Investment = 100% 19

20 SYNTHESIS OF THE TOPIC (I) Some of the main transport design problems result from the combination or adaptation of some basic NP-hard or NP-complete problems (e.g. Min-cost-flow) All these models use linear programming optimization tools with: Linear Objective Function Linear Constraints Where the decision variables present commonly binary or integer form, which do not have a trivial linear programming solution method But fortunately there are sophisticated algorithms to obtain (quasi) optimal solutions in acceptable time for many of them 20

21 SYNTHESIS OF THE TOPIC (II) The objective function and constraints of the model should contain the main strategic goals of the decision makers the network design: Coverage all settlements must be accessible Accessibility Travel distances, times, operating costs must be reasonable Public Economy Avoid excessive investment cost Level of Service Low levels of delay Redundancy / Low level of vulnerability Preserve functionality in case of some breakdown Induce efficient and sustainable land-use patterns promote compact settlements, low need for motorized travel 21

22 SYNTHESIS OF THE TOPIC (III) The consideration of all the goals simultaneously in the optimization process can turn the problem unfeasible or take to long to find n optimal solution, due to the conflict between some of them Some of these goals should be left of the optimization process and assessed after to find out if the obtained solution satisfies all the established goals The result of this evaluation can be a multicriteria process were several optimal solutions are evaluated and the final solution will result from the multicriteria evaluation considering the decision maker s preferences 22