Optimisation of a Distribution Network

Transcription

1 Optimisation of a Distribution Network by MARTIN DE WET Submitted in partial fulfillment of the requirements for the degree of Bachelors of Industrial and Systems Engineering in the Faculty of Engineering, Built Environment and Information Technology University of Pretoria October 2010

2 Executive Summary Presented in the document is a method of optimising a distribution network. The distribution network chosen is known as a Vehicle Routing Problem and this particular Vehicle Routing Problem is part of Smart Publishing s (magazine distributor in Pretoria) distribution network. Due to external factors the VRP selected can transform into a Capacitated Vehicle Routing Problem. A Capacitated Vehicle Routing Problem is considered NP-hard and requires a metaheuristic to solve. The different types of metaheuristics are listed and their operating procedure explained. A possible metaheuristic that can be used to solve a CVRP is a simulated annealing approach. Improving Smart Publishing s CVRP and building the simulated annealing metaheuristic will be the main purpose of the project. Results of the new route compared with the old route are presented and a shorter route has been found using a simulated annealing approach. As result, the company will save on fuel usage, maintenance cost and time. The quality of a heuristic can be tested through special measures and a good quality heuristic will provide good approximations near the optimal solution on a regular basis in reasonable time. i

3 Contents 1 Introduction Problem statement Problem description Complexity theory Research question Research design Research Methodology Document Structure Literature Review Background The company Vehicle routing problem Capacitated vehicle routing problem Algorithms Exact algorithms Branch-and-bound approach Heuristics and metaheuristics Nearest-neighbour-heuristic Genetic algoritms Tabu search Comparing heuristics Quality in heuristics Performance guarantees Probalistic analysis Empirical analysis Similar cases Evaluation of heuristics Conclusion Simulated annealing algorithm Background Procedure Example The model Pseudocode ii

4 4 Results Input requirements Parameter requirements Sensitivity analysis Starting temperature effect Maximum transitions at each temperature Maximum accepted transitions at each temperature Effect of temperature reduction Maximum iterations Change in delta Stopping temperature effect Output Smart Publishing s distribution network Smart Publishing s distribution network in Pretoria Single route for basket distribution Effect of changing vehicle capacity Actual capacitated route for basket distribution Simulated annealing capacitated route 1 for basket distribution Simulated annealing capacitated route 2 for basket distribution Simulated annealing capacitated route 3 for basket distribution Conclusion Future Work 35 Acronyms 36 Bibliography 38 iii

5 List of Tables 1.1 Complexity Growth of Polynomial Algorithms Complexity Growth Rate of Exponential Algorithms Types of NP Class Problems The volumes of the primary magazines distributed by Smart Publishing Distribution schedule Metaheuristics Compared Heuristic Neighbourhood Searches Possible Heuristics Initial input parameters Final input parameters Results compared with bechmark problems Capacaty effect on distance iv

6 List of Figures 1.1 NP Problem Topology Research Procedure Smart Publshing s Distribution Network Smart Publshing s basket distribution in Pretoria VRP relationships Complete Graph Starting temperature effect Maximum transitions at each temperature Maximum accepted transitions at each temperature Effect of temperature reduction Maximum iterations Change in delta Stopping temperature effect Smart Publshing s Distribution Network Distribution Network in Pretoria Compared single routes to service Pretoria Effect of capacity on total distance travelled Actual capacitated route to service Pretoria day capacitated route to service Pretoria multi baskets day capacitated route to service Pretoria multi baskets day capacitated route to service Pretoria multi baskets v

7 Chapter 1 Introduction The world we know consist of a multitude of variable and determinable elements. Elements like nature, technology and people. These elements have at least one characteristic in common; they produce problems. Real world problems are often represented as mathematical problems because it is easier to find solutions for mathematical problems than finding solutions for the same problems using anything but math. Operational Research is a typical area where mathematical problems can be found and an example of these problems are the amount of births and deaths in a country, growth rate of certain bacteria and determining the shortest route between a number of customers for a delivering company, the last named will be explained thoroughly in this document. Solutions to most mathematical models can be easily determined and sometimes special methods are required to solve more complex problems. 1.1 Problem statement Problem description Lets take the delivery company for instance. A vehicle visits a network of clients and each client require some kind of service. Clients have to be visited once a day and at the end of the day the vehicle returns to where it started. This type of problem is considered a Vehicle Routing Problem (VRP) and is similar to a Traveling Salesman s Problem (TSP) where a salesman travels from customer to customer and back home again. According to Schrijver (2005), TSP is the simplest routing problem and known to have been formulated as a mathematical model in 1930 already (the TSP was found in a German manual depicting the dangers of woman in and outside the business environment) and the first mathematician to write about TSP was Karl Menger in VRP also requires the minimization of the distance traveled like the TSP and can also include vehicle constraints of capacity, fuel usage and running cost. The VRP problem was first introduced by Dantzig and Ramser in 1959 depicted by Bräysy et al. (2009). TSP and VRP are types of OR models and are nothing more than real world problems expressed mathematically. Route finding, route allocation and servicing of networks in these systems is a science and finding solutions of optimal nature is no easy task. 1

8 1.1.2 Complexity theory Algorithms are normally used to solve mathematical problems. An algorithm is a set of steps that can be followed to find a solution to a mathematical problems that can t be found using conventional methods. With complex or large problems these algorithms can take an enormous amount of computation time to find or suggest a solution. For example, take a VRP with 3 clients. If a vehicle visits client 1, then there are 2 clients left. The combination or order these clients can be visited in equal 2 3. For 5 clients, this equals to or 5! and totals to 120 different combinations or tours in which all clients can be visisted. A tour is a single route in which all clients are visisted exactly once. The problems grows exponentially as the number of clients increase. The computational time and problem class are directly proportional. If a problem grows exponentially, then the computational time will grow exponentially and polynomial problems behave in the same manner. As the VRP problems increase in size, the number of different tours increase and thus the time taken to find a near optimal solution (tour) increases. This phenomenon is called Complexity Growth described by Winston (2003). The tables below presents the number of tours for each problem class. Table 1.1: Complexity Growth of Polynomial Algorithms Size N log(n) N ,000 N ,000 1,000,000 N 5 100,000 3,200, ,500,000 10,000,000,000 N 10 10,000,000, E E+16 1E+20 According to Winston (2003) if the algorithm is bounded by a N 10 polynomial the amount of time it will take to solve will be excessive. Some problems are bounded by exponential expressions and their growth rates are as follows: Table 1.2: Complexity Growth Rate of Exponential Algorithms Size N 1,024 1,048, E E+30 N logn 2, , ,879,201, E+13 N! 3,628, E E E N 9,765, E E E N 10,000,000,000 1E+20 1E+50 1E+100 Thus the concept of nondeterministic NP class problems was developed. A mathematical problem belongs to the NP class if there is a algorithm that can confirm a 2

9 certificate to the problem. Certificates is more or less like a solution to the problem, but not optimal. Refer to Winston (2003) for more information. Problems that can t be solved within polynomial time are considered hard and there is no polynomial algorithm to solve them with. These hard problems are known as NP-complete and NP-hard problems and NP-complete are the hardest type in the NP space according to Winston (2003). Table 1.3: Types of NP Class Problems Types NP-Complete Nondeterministic P Description Cannot be solved by known polynomial time algorithms Where polynomial time algorithms are possible Can be solved in polynomial time Figure 1.1: NP Problem Topology As the size of NP class problems grow (move up in the topology) they require special methods in order to find near optimal solutions. These special methods are known as heuristics and they provide approximate answers to NP class problems. 1.2 Research question We know that a VRP consist of a network of nodes and routes. The solution to the problem would be the shortest route that services all the nodes or customers. Effective 3

10 servicing of customers depend on the external demand of the customers and carry capacity of the vehicle. A VRP with these constraints is known as a Capacitated Vehicle Routing Problem. The solution to the CVRP would be the shortest route that services all the customers demand using the vehicle capacity. CVRP is known to be a NP-Hard as explained by Idsardi (2006), and Lin et al. (2009). We now know heuristics are used to find solutions to NP class problems and that brings us to the following question: How can the shortest route to a capacitated vehicle routing problem be found under the specific model constraints using metaheuristic? 1.3 Research design In this paper a mathematical algorithm will be identified for a Vehicle Routing Problem. There will also be a primary investigation as to the possibility of building a simulated annealing algorithm to solve a CVRP. Thus trying to find the optimal route in the network concerning the vehicle capacity and obtain a near optimal solution by implementing a working mathematical model. 1.4 Research Methodology Figure 1.2: Research Procedure The hierarchy above provides a step orientated procedure for analysing the VRP and consists of 4 steps. The aim is finding a suitable metaheuristic to solve a VRP with all its constraints and translate the findings into a structured solution. To find a solution to the problem, the problem must first be understood and all the deliverables known. By researching all possible variations of VRP and comparing the different heuristics associated with them, an applicable heuristic can be found for the VRP. Documenting the findings and using the appropriate heuristic will be the process. The VRP 4

11 algorithm can then be used as backbone for the CVRP algorithm. When an accepted CVRP metaheuristic is found, it needs to be analysed and presented. After presenting the solution, a conclusion will be given explaining possible future use and improvement of the given model and problem. 1.5 Document Structure This chapter concludes the cause and types of problems found in OR. Because of Complexity Theory, VRP and TSP belongs to the NP class problems. If a problem belongs to the NP-space, special algorithms are used to solve them. These algorithms are known as heuristics and are presented in the following chapter. Chapter 1 presented the problem and the remainder of the report will be divided into 4 chapters. Chapter 2 consists of a detailed explanation of the VRP and CVRP, a literature study about the different algorithms applicable and it demonstrates the metaheuristics to be used and why. Chapter 3 presents the detailed model (metaheuristic) to solve the VRP. In chapter 4 the computational results and the interpretation of the solution are illustrated. Chapter 5 explains future work regarding possible improvements of the metaheuristic. 5

12 Chapter 2 Literature Review 2.1 Background The company Smart Publishing is located in Pretoria near Hatfield Square. They specialize in the free magazine market, designing and circulating magazines locally. Currently they are one of the biggest magazine developing and distribution companies in South Africa. The distribution network has been in place for almost 13 years. The network is monitored by Smart Distribution, a division of Smart Publishing. They currently distribute 5 different magazines and their 3 main magazines together with their volumes include: Table 2.1: The volumes of the primary magazines distributed by Smart Publishing Magazine Volume Techsmart 100,000 ProAgri 100,000 Retreat 60,000 They distribute closely to 300,000 magazines a month. The main location of distribution takes place within Gauteng. ProAgri is a national magazine and circulated as such. Compared to other magazine companies (which distribute roughly between 10,000 and 50,000 copies nationally) a total of 100,000 magazines are distributed only in Gauteng by Smart Distribution. The only magazine that has a higher volume than this is the Huisgenoot and You with a total of 300,000 copies. Techsmart and Retreat is published monthly and bi-monthly respectively and their main focus areas are the LSM 5-8 (Living Standards Measure) groups. This LSM groups indicates middle to upper income every month. A higher income means the ability to afford more luxuries and Techsmart and Retreat advertises a large number of these luxuries. The magazines are placed in stands or multi baskets and the multi baskets are located at the entrance of shopping malls, retail outlets and business estates. A large number of the Techsmart and Retreat magazines are also placed within schools and businesses around Pretoria and Johannesburg. There are currently 850 distribution points located over Gauteng 6

13 and the circulation network has not been optimized for better effectiveness and efficiency. The red dots in the following figure are the business & school locations and the green dots the multi baskets. Figure 2.1: Smart Publshing s Distribution Network. The CVRP in connection with Smart Publishing consists of 6 different networks integrated in one big distribution system with 850 customers. The network is serviced according to attribution and done weekly each month. Multi baskets are visited bi-weekly and schools and businesses monthly in each area. Thus, a tipical schedule would be: Table 2.2: Distribution schedule Area Vehicle Week 1 Week 2 Week 3 Pretoria KIA K2700 Baskets Businesses & Schools Baskets Johannesburg Ford Bantam Baskets Businesses & Schools Baskets For purpose of this document only the part of the distribution network consisting of 121 customers (baskets) located in Pretoria will be addressed. Distribution of the magazines in Pretoria (multi basket network) takes 4 days, using a different route every day. No time limit exists between servicing of customers and the only time constraint will be distributing as much commodity as possible every working day. The current vehicle has 7

14 enough capacity for magazines and the magazines come in packets of 50. Each basket has a demand of between 50 and 1500 magazines. The multi basket demand totals to magazines. An average of books can be distributed daily. An alternative vehicle is also available if required and has a much smaller capacity (only one vehicle is used at any given stage). The shortest route found to service the baskets located in Pretoria will be presented and an investigation into writing a CVRP algorithm, in order to improve the capacitated route, will be explored. The figure represents Smart Publishing s stand distribution in Pretoria and the red dot is the location of Smart Publishing head office. Figure 2.2: Smart Publshing s basket distribution in Pretoria Vehicle routing problem VRP or TSP is the simplest form of the vehicle routing problems. Meaning that TSP is the starting point or most basic. If a VRP has an extra constraint, the VRP is known by name of the constraint. A CVRP is a VRP which uses a vehicle with a certain capacity to service the network whilst CMVRP stands for capacitated multi vehicle routing problem and consist of more than one vehicle with capacity constraints. HVRP is a heterogeneous VRP and consists of a fixed amount of non identical vehicles. A LSVRP is a large scale VRP (the following diagram represents the hierarchy of the VRP system and how they relate to each other from Paolucci (2002) and Toth and Vigo (2001)). 8

15 Backhauls TSP Time windows TSPB Multi MTSP TSPTW Vehicle capacity Backhauls CVRP Route length VRPB Time windows DCVRP VRPTW Mixed services VRPPD VRPBTW VRPPDTW Figure 2.3: VRP relationships There is a broad number of VRP s and each requires a different set of heuristic to solve. Thus it is required to find an appropriate heuristic to use with the CVRP. The VRP and TSP are very similar when the VRP constraints are minimal. A heuristic is a set of mathematical steps or procedure to solve the TSP or VRP. Characteristics and Components According to Paolucci (2002) a VRP will consist of a graph G = (V, A). Where V is known as the vertices. These vertices are also known as the depots, customers, nodes or road crossings. The A in the equation is known as the arcs, roads, links or edges connecting each vertice or node. If there is N number of nodes, then there will be N(N 1)/2 edges in the graph for a complete graph. Meaning that all nodes in the graph must connect to all other nodes in the graph in order for it to be complete. The next figure shows a complete graph with 7 nodes. Figure 2.4: Complete Graph 9

16 For the purpose of this project an undirected graph will be used. Thus the edges has no specific orientation and this means that going from one node to another will use the same amount of resources when moving in either direction. In summary a minimal constrained VRP or otherwise a TSP will consist of the following: A complete graph G = (V, A) G = (V, A ). Where V is your customers including your depot and A is the network of links connecting all customers and depot. The cost (i, j) A, c ij 0 will be the minimum cost or distance assosiated with moving from point i to point j according to Paolucci (2002). To represent the differences when modeling for a VRP and CVRP the mathematical models are presented. Following is the mathematical representation of the TSP or VRP from Jayaswal (2000): Customers = 1,2,3,...,N U ij is the accumulated deliveries at each node c ij = distance from city i to j { 1 if the solution goes from i to j x ij = 0 otherwise min Z = N i N j c ijx ij (1.1) s.t. N i=1 x ij = 1 ( j i) (1.2) N j=1 x ij = 1 ( i j) (1.3) U i = 1 (1.4) 2 U i N ( i = 1) (1.5) U i U j + 1 (N 1)(1 x ij ) ( i = 1, j = 1) (1.6) U ij 0 ( i) (1.7) x ij {0, 1} ( i, j) (1.8) Constraints (1.4) to (1.7) are the subtour elimination constraints. When these constraints are left out, the problem represents a normal assignment problem and can be solved using a linear programming method Capacitated vehicle routing problem Characteristics and Components Toth and Vigo (2001) state that CVRP requires the following elements. These elements are more or less the same as for the VRP except that extra vehicle and capacity constraints are applicable: A complete graph G = (V, A) N customer nodes, (i = 1,..., N) 10

17 A single depot (i = 0) Demand of each customer: q i > 0, (i = 1,..., N), (q 0 = 0 for the depot). q(s) = i S q i, S V (the sum of demands of a subset in V or other words a subtour). M amount of vehicles with a capacity Q A distance matrix: d ij, (i,j) A (with d ii = and d ij = d ji, a symmetric matrix). Using the elements above, the mathematical representation of a normal and mixed integer CVRP model can now be explained respectively: CVRP: Customers = 1,2,3,...,N { 1 if the solution goes from i to j x ij = 0 otherwise c ij = distance from city i to j { 1 if point serviced by the k th truck y ik = 0 otherwise q i = Demand at point i Q = Vehicle capacity min Z = i j c ijx ij (2.1) s.t. N i=1 q iy ik Q (k = 1, 2,..., m) (2.2) i=1,j S x ij S 1 (S {2,..., n}) (2.3) m k=1 y ik = { m i = 1 1 i = 2,..., n n i=1 x ij = { n m j = 1 i=1 x ji 1 j = 2,..., n (2.4) (2.5) x ij {0, 1} (i = 1,..., n; j = 1,..., n) (2.6) y ik {0, 1} (i = 1,..., n; k = 1,..., n) (2.7) Equations (2.1) is your objective function and minimizes your total distance. (2.2) is your capacity constraint, (2.3) is the node allocation and (2.4) is the vehicle allocation constraint. (2.5) Completes the symmetric matrix and (2.6) and (2.7) is your integer allocation constraints. Mixed integer CVRP from Lin et al. (2009): 11

18 c ij = distance from city i to j K = Vehicles N = Customers S i = Service time of customer i Q k = Capacity of vehicle k T k = Maximum distance vehicle k may travel d i = Demand of customer i { 1 if traveled from i to j Xij K = 0 otherwise min Z = N N N i=0 j=0 K=1 c ijx k ij (3.1) s.t. N i=0 N j=0 xk ijd i Q k (1 k K) (3.2) N i=0 N j=0 xk ij(c ij + S i ) T k (1 k K) (3.3) N j=1 x ijk = N j=1 x jik 1 (for i = 0, k {1,..., k}) (3.4) k=1 j=1 x ijk K (for i = 0) (3.5) Equations (3.1) is your objective function and minimizes your total distance. (3.2) is your capacity constraint, (3.3) is the vehicle maximum distance and (3.4) completes the symmetric matrix. Equation (3.5) is the vehicle allocation constraint. 2.2 Algorithms Different approaches to solving optimisation problems exist. Algorithms are normally used and 2 types of algorithms will be encountered when dealing with a distribution problem; exact and approximation algorithms. Exact algorithms will find the optimal solution to a problem if the problem is within its specific range or complexity. Meaning that the size of the distribution network and constraints are not too complex. An approximation algorithm will try to find the closest solution to the optimal solution regarding the size and constraints applicable. The approximate algorithm or heuristic is developed in such a way to accomodate these qualities. According to Zeng et al. (2007) heuristics belongs to 3 basic categories. These categories are: local search improvement methods, constructive methods and metaheuristic methods. Metaheuristic are higher heuristic and the most promising in finding solutions for VRP explained by Zeng et al. (2007). Types of metaheuristics include tabu search, simulated annealing and genetic algorithms. Heuristics can consist of only one type of method or could include a combination of different heuristic methods. Winston (2003) state that a heuristic that can solve multiple problems hasn t been found until the introduction of Artificial Intelligence (AI). AI is also known as metaheuristics and are superior in solving problems based on clever search techniques. One major problem with heuristics are that they sometimes get trapped in local solutions and fail to find global solutions. All metaheuristics are formulated to prevent this from happening. To explain some of the following heuristics a VRP with 5 customers is used. 12

19 2.3 Exact algorithms Branch-and-bound approach Branch-and-bound (BnB) approach uses a systematic approach in disregarding one possibility after another. For a VRP the BnB approach will determine the shortest route between the customers through an initial assignment problem approach. Refer to Winston (2003) for more details. Lets say the solution to the problems was Notice that the tour starts at 1 and ends at 1. If the BnB first solution is and note that there is no tour and that the solution consists of a subtour The BnB will then make one of the subtours of zero to achieve a tour. This either making 2 4 or 4 2 zero. After this is done the BnB approach can find a new solution which also contains a subtour. The steps above will then be repeated until a feasible tour is found. Winston (2003) state that the problem with this method is that with larger problems its more complicated getting rid of subtours. The following formulation is an example of an BnB approach as Integer Programming (IP) fromwinston (2003): cities = 1,2,3,...,N for i j c ij = distance from city i to j c ii = M, where M is a very large number { 1 if the solution goes from i to j x ij = 0 otherwise min z = i j c ijx ij s.t. i=n i=1 x ij (for j = 1,2,...,N) j=n j=1 x ij (for i = 1,2,...,N) u i u j + Nx ij N 1 ( for i j; i = 2, 3,..., N; j = 2, 3,..., N) All x ij = 0 or 1 All u j Heuristics and metaheuristics Nearest-neighbour-heuristic The nearest-neighbour-heuristic (NNH) also known as the Greedy heuristic will take the first customer and find the next closest customer to it. Then it will find the nearest customer to the most recent one found. This is repeated until a tour is completed. The solution is noted and the second customer is used as starting point. This is repeated for all five customers and the best solution is presented. The NNH is not a very good heuristic for large scale VRP according to Winston (2003) 13

20 2.4.2 Genetic algoritms Winston (2003) state that a Genetic Algorithms (GA) is based on the natural selection of 2 breeding pairs. This metaheuristic uses 6 steps to complete a cycle. The step are: Start with an answer Evaluate Select Reproduction Mutation Repeat. GA takes 2 solutions or tours in this case. It then changes parts of these 2 solutions to get 2 other solutions. The 2 new solutions stil contains large parts of the first 2. As the algorithm goes on, the better solutions have an improved chance of being selected again. Similar to Simulated Annealing, the GA uses a random probability to accept or reject certain values to avoid local solutions. The following model code is from Rardin (1998): Method: Step 0 = Initialization. Choose a network size p, an initial solution x = [1,..., p], an iteration limit t max, and subdivisions for the population where p e = elites, p c = crossovers and p i = immigrants. Set t 0. Step 1 = Stopping. When t = t max, end and present the best answer of the current network as an estimated optimum. Step 2 = Elite. Initialize the network of generation t + 1 with copies of the p e best answers in the current generation. Step 3 = Immigrants. Choose p i new solutions at random. Include them in the t + 1 network. Step 4 = Crossovers. Take p c /2 nonoverlapping pairs of answers from generation t network. Proceed with crossover of two randomly chosen points to complete the generation t + 1 network. Step 5 = Increment. Increment t t + 1, return to Step Tabu search Tabu Search (TS) uses short and long term memory for approximating its solutions and TS declares some instances forbidden or tabu. It does not imitate physical or biological processes. It uses memory exclusively. Intensification, diversification and strategic oscillations are used with the long term memory. Long term memory is used to keep searching in the most promising neighbourhoods. The short term memory is used not to stay in an area too long. All the above are depicted by Winston (2003) With TS you start by choosing a random starting point and call it x. This point is generally not random but more a smart calculated one. Take other tours and call them ry. Where a tour is y and a predetermined set of r = 1 : 5. These 5y are chosen from a certain sequence and compare with the first one. Which one of these 5y change the greatest or less when compared with the initial point? Then use this as the new top solution but not the optimal one. Then choosing other y from another sequence or the same as the one above and compare again with x or the new top solution. Because the difference in the value of your searches can be positive or negative the TS can move up or down and thus not converge at a local optimum. The following model code is from Rardin (1998): 14

21 Method: Step 0 = Initialization. Choose any feasible beginning solution x (0), and iteration limit t max. Set solution ˆx x (0) and solution index t 0. No moves are tabu. Step 1 = Stopping. If no non-tabu move x in move set M leads to a feasible neighbour of current solution x (t), or if t = t max then stop. Solution ˆx is a estimate optimum. Step 2 = Move. Take non-tabu feasible move x M as x (t+1). Step 3 = Step. Update x (t+1) x (t) + x (t+1) Step 4 = Incumbent Solution. If the objective function of x (t+1) is better than ˆx, then replace ˆx x (t+1). Step 5 = Tabu List. Remove any moves that have been used for a sufficient number of iterations. Add moves that includes any returning immediately from x (t+1) to x (t). Step 6 = Increment. Increment t t + 1 and return to Step Comparing heuristics The following tables present the main metaheuristics and compares them. Each algorithm has a different solution approach and search technique. The algorithms are not problem specific and none is superior to another. There is speculation however that certain metaheuristics will out perform another under certain conditions, but it is possible that inferior algorithms would by chance find a more optimal solution. Contained in the table is the simulated annealing algorithm not explained previously. This is because the specific metaheuristic was used as approximation algorithm to this particular optimisaton problem and discussed in detail in chapter 3. Table 2.3: Metaheuristics Compared Heuristic Memory Solution Type Search Simulated annealing No Memory One Random Genetic search No Memory Population Random Tabu search Memory-based One/Population Systematic Table 2.4: Heuristic Neighbourhood Searches Heuristic Tabu search and Neural Networks Genetic search and simulated annealing Search Area Neighbourhood Random neighbourhood 15

22 2.5 Quality in heuristics Giving accurate approximate solutions makes a heuristic more reliable. A reliable heuristic can be used to make managerial decisions. The only way a heuristic can really be tested is with time. If it keeps a stable output within the known best solutions then you probably have a good quality heuristic. Determining which heuristic to use with what type of routing problem will be the greatest challenge. Not all heuristics are compatible with all routing problems. A time test will reveal which problems a heuristics has good qualities for and for which not. The following measures of quality in heuristics have been presented by Winston (2003) Performance guarantees A heuristic should be able to prove mathematically that it can approximate a solution within a minimal range of the optimal solution Probalistic analysis Here heuristics are tested by a ratio of compliance. Ratio = Expected length of the path found by the heuristic Expected length of the optimal tour The closer this ratio equals to 1, the better the heuristic s computational powers Empirical analysis Performance and efficiency can be measured through comparing the algorithms answers with existing problems for which the optimal solutions are known. These have been prepared by B. Golden (1998) to compare heuristics and determine which should be used with what type of problem. 2.6 Similar cases A case similar to this one is: A decision support system for a real vehicle routing problem by Ruiz et al. (2004). The general concept of Ruiz et al. (2004) s routing problem is the same as the problem presented in this document. The only difference is Ruiz et al. (2004) have more constraints in their model. Multiple methods were addressed in the mentioned article and a graphical user interface (GUI) was build to use these models. Modeling problems they experienced with traditional modeling techniques is that the model didn t take all the variables into consideration and the computing time was unknown. Ruiz et al. (2004) tried to divide the problem into subsections and find solutions individually. For example, they took 10 points and grouped them under one area, call it N. The optimal route was found for N and completed for N + 1. The problem was that these different area s were calculated independently and a solution couldn t be found as a whole. Other techniques included a two phased approach for solving the problem. In the first phase, an enumeration 16

23 algorithm generates and finds the best routes and for the second phase, an integer programming model chooses the best routes from the first phase. The number of routes chosen by the enumerator was excessively large and thus the integer programming model was used to narrow the search by implementing new constraints like maximum route distance, etc. The 2 phased method proved adequate and was used as a solution to Ruiz et al. (2004) problem. Heuristic approaches can find single solutions but metaheuristic approaches can approximate better solutions for CVRP. Types of metaheuristic approaches include the tabu search (TS), simulated annealing (SA) and genetic algorithm (GA) as presented in Lin et al. (2009). Another example is the heterogeneous vehicle routing problem (HVRP). This is a variation of the CVRP. The difference between the two is that you have a fixed amount of vehicles with different capacities for HVRP and only one vehicle for the CVRP. The fleet size and mix vehicle routing problem (FSMVRP) is the same as the HVRP except that with the FSMVRP the fleet has to be determined. An algorithm associated with HVRP is presented by Lima et al. (2004). The solutions found in the article Liu et al. (2009) were calculated by using a genetic algorithm based heuristic. The solutions were compared with current best solutions and competitive results were noted. Golden et al. (1997) state that there is an alternative method which can be used alongside TS which is called Adaptive Memory Procedure (AMP) for the capacitated vehicle routing problem with multiple use of vehicles CVRPM. A CVRPM allows the vehicles to use routes they have already used. This procedure allows for a more diverse search and thus allow for superior solutions. Lastly, a very large-scale VRP is considered (LSVRP). In this model you have a very large customer database requiring services (Large VRP considered 500 customers). This article considers a simulated annealing approach called a record-to-record traveling algorithm (RTR) for finding its solutions. This was first introduced by Golden in 1998 described by Lia et al. (2005). 2.7 Evaluation of heuristics After comparing the articels above it was found that the best heuristics to use with the CVRP would be the two-stage-exact approach from Ruiz et al. (2004), the hybrid genetic algorithm from Lima et al. (2004) and simulated annealing algorithm from Lin et al. (2009). An algorithm for the CVRP can thus be designed from one of these models. Reference will be made to Winston (2003) for additional assistance. The next table presents heuristic usage with VRP type. 2.8 Conclusion In chapter 2 a background on VRP was given and the different variations there off. Also explaining that different heuristics are needed to solve VRP and that metaheuristics are required to solve complex VRP. The different types of heuristics and their mathematical mechanisms are listed. It was found that a simulated annealing approach can be used to solve the CVRP. The SA model will be explained in detail in the following chapter. The quality measures are a good indication if a heuristic is useful or not. Refer to Lin et al. 17

24 Table 2.5: Possible Heuristics Developer Method Problem Type Ruiz et al. (2004) Two-Stage Exact Approach CMVRP Lin et al. (2009) Simulated Annealing CVRP Lima et al. (2004) Hybrid Genetic Algorithms HVRP Liu et al. (2009) Genetic Algorithms FSMVRP Golden et al. (1997) Adaptive Memory Procedure CVRM Lia et al. (2005) Record-to-Record Traveling Algorithm LSVRP (2007) for more detail on quality and heuristics. A SPC study can also be completed on the deviation the heuristics gives from the known answers of Golden and using the quality measures. 18

25 Chapter 3 Simulated annealing algorithm 3.1 Background According to Breedam (1993) the similarities of the simulated annealing approach and physical annealing of metals are obvious. Physical annealing tries to achieve a solid in its lowest energy form. A metal for example is melted and then slowly cooled until the lowest energy state is found that represents a solid metal. The cooling of the metal until this energy state is found, should be done very slowly. This lowest energy form or state is similar to the objective function of the distribution model. By rearranging the molecules or atoms in the metal, different energy states can be found for the same conditions. When equilibrium is reached, the metal s composition or structure is distributed or formed according to Boltzmann distribution. In the simulated annealing process a solution to the distribution problem is a certain configuration of the nodes to visit, same as the different configuration of the molecules in the metal. By rearranging the nodes, different solutions can be found. For the metal to reach equilibrium, the temperature must stay constant for certain duration of time. Similar for simulated annealing, the number of solutions found at a certain temperature must be large enough to reach equilibrium or be feasible. A new solution is generated from the old solution and the new solution differs slightly in node placement. This is the same as changing the atoms structure in the metal to find a new energy state. The objective function value of each of these solutions are calculated and compared. The difference between two of each of these solutions is defined as δ. This is the same as the energy change from molecule reconstruction in the physical annealing approach. If δ < 0, the solution becomes the current solution and if δ > 0, the solution generated is accepted through comparing it with Boltzmann function. e δ/k BT is the Boltzmann function with k B the Boltzmann constant and T the temperature. A traditional descent algorithm only accepts the solutions that improves the objective function, but with simulated annealing, solutions that deteriorates the objective function is also accepted. This allows for the algorithm not to get trapped in a local optimum. A problem concerning simulated annealing is determining the cooling schedule. The annealing (cooling schedule) consist of an initial temperature, cooling rate and a stopping criteria. For the initial temperature a general rule can be used. This rule of thumb specifies that the initial temperature should be equal to: (Worst case objective function value) (Best case objective function value) and the cooling schedule should not be too fast. The iterations at each temperature 19

26 needs to be enough to find equilibrium but if too many solutions are chosen at each temperature, the algorithm will take inordinately long to execute. Your stopping criteria can be either a stopping temperature, a fixed number of temperature decreases, a minimum acceptance ratio or max number of iterations. The acceptance ratio for each temperature is: Acceptance Ratio = 3.2 Procedure Number of feasible solutions Total number of feasible solutions The following model code is from Rardin (1998) and explains the general procedure for executing the algorithm: Method: Step 0 = Initialization. Choose any feasible beginning solution x (0), and iteration limit t max and a large initial temperature q > 0. Set solution ˆx x (0) and solution index t 0. Step 1 = Stopping. If no x in move set M leads to a feasible neighbour of current solution x (t), or if t = t max then stop. Solution ˆx is a estimate optimum. Step 2 = Provisional Move. Randomly choose a feasible move x M as a provisional x (t+1), and compute the objective function improvement for moving from x (t) to (x (t) + x (t+1) ) Step 3 = Acceptance. If x (t+1) improves or with probability e obj/q if obj 0, accept x (t+1) and update x (t+1) x (t) + x (t+1) Step 4 = Incumbent Solution. If the objective function of x (t+1) is better than ˆx, then replace ˆx x (t+1). Step 5 = Temperature Reduction. If a sufficient number of iterations have passed since last temperature change, reduce temperature q. Step 6 = Increment. Increment t t + 1 and return to Step Example According to Winston (2003) SA is also a probabilistic process and lower-ranking solutions can be recognised based on Boltzmann s function. This memoryless technique optimises without knowledge of the previous solution. Firstly, 2 random customers are chosen through a probability function. SA changes these 2 customers order in the original tour. Say the starting tour is with a tour value Z (distance) and the 2 random customers are 3 and 5. Then the tour will become The Z new of the tour is used to calculate a new probability say X. X is checked with another random probability to see if it should be rejected or not. The Z new can be rejected or accepted based on probability X. Meaning that the value (objective function) can thus be penalised or improved and the technique moves on. After all the customers have been complete or listed, the counter checks how many iterations it has done. If it is less than the predetermined amount, the technique carries 20

27 on until the amount has been reached. SA prevents the convergence to a local solution through this method. Simulated annealing is based on the physical process of annealing a metal to achieve a better molecular structure at lower temperatures. 3.3 The model Pseudocode From Winston (2003) and Jayaswal (2000) the following simulated annealing algorithm is used to solve the VRP: x (0) = Initial solution (Greedy algorithm is used). T (0) = Initial temperature. i = Number of iterations. δ = Cooling temperature ratio T cur = T (0) x (cur) = x (0) x (best) = x (cur) z (best) = z (x(cur) ) for count = 1 : i x (count) = Randomly generated solution where 2 randomly selected nodes are changed. If z (x(count) ) < z (x(cur) ) then x (cur) = x (count) δc/t (cur) else C = z (x(count) ) z (x(cur) ) and set x (cur) = z (x(count) ) with probability e If z (best) > z (x(cur) ) then x (best) = x (cur) and z (best) = z ()cur set count = count + 1. else T (cur) = T (cur) δ If T < (chosen temperature) go to step 2 with T (cur) else z (best) = z (best) and x (best) = x (best) 21

28 Chapter 4 Results The values in this document represent the actual data recorded from the distribution vehicles. 4.1 Input requirements In MS Excel the longitude and latitude of every network node was converted to a decimal value using a Visual Basic programming code. There are 850 total network nodes to process. This decimal value was used with the Great Circle Distance (shortest distance between two points) or Pythagoras s distance equations to determine the average distance. For simplification purposes the normal distance formulae was used and expressed as one degree of latitude is equal to 111 km and 57.3 convert the latitude 1 to a radial value. The distance formula is: Distance Latitude = 111km Latitude2 Latitude1 Distance Longitude = 111km Latitude2 Latitude1 cos(latitude1/57.3) Distance = ((Distance Latitude) 2 + (Distance Longitude) 2 ) 0.5 Because the city distance is not the same as the straight distance between any 2 points, an increasing factor is needed. But because this factor applies to all points to the same ratio, it is left out of calculation. The same answers will be generated even with the increasing factor. When comparing the results this will be clarified. Using the distance formula above, a distance matrix was designed and used as input for the algorithm. The following inputs are required for the model to work: Coordinates of each node Distance between each node in the network Demand of each node 4.2 Parameter requirements The simulated annealing algorithm requires additional parameters in order to function properly. The results of the algorithm depend on the value of these parameters. If they are insufficiently used, the algorithm will be insufficient. The list of the input parameters follows: 22

29 Initial temperature: This is the starting temperature of the algorithm. The temperature is chosen through intelligent selection or rule of thumb. Maximum transitions at each temperature: Number of objective function values compared to see if the value will be accepted or rejected according to the Boltzmann function and a cooling factor is invoked if reached. Maximum accepted transitions at each temperature: After this number of objective function values have been accepted the cooling factor is invoked. Temperature cooling factor: Reduces the temperature with a certain ratio after it was invoked. Max iterations: Number of total iterations allowed to run. Change in delta: This input changes the way the objective function value is accepted according to the probabilities. Meaning that it will allow a bigger or smaller difference in the objective function value found to be accepted or not. The bigger this value, the larger the change in the objective function value will be accepted. Termination values: If a certain temperature is reached or the objective function value didn t change for a set of iterations then the program terminates and presents the solution found. Stopping criteria should be carefully chosen as the relationship of your starting temperature, stoping temperature, temperature reduction and maximum iterations all influence each other. Meaning that the stopping temperature could be reached before your maximum number of iterations was achieved. 4.3 Sensitivity analysis A sensitivity analysis is conducted to find the optimal input parameters that improves the simulated annealing algorithm performance. Using default preset input values (non-changing), 10 different combinations for a single input (chosen input not part of default values) were chosen and run for 5 instances. The 2 values recorded were the average objective function value and the mean deviation from the average objective function. The average objective function value show the algorithms potential to achieve a best optimal solution. The MAD or mean average deviation from the objective function show the algorithms potential conformance for the chosen inputs. The inputs will allow the algorithm to converge quicker, more accurate and more often to a near optimal solution. A primary analysis was completed to find initial input parameters that converged to a near optimal solution. A second analysis was completed to optimise the initial inputs. After completing the first analysis the following inputs were recorded: 23

30 Table 4.1: Initial input parameters Input Value Input Value Starting temperature 0.5 Temperature reduction 0.7 Stop temperature 1E 200 Maximum iterations 500 Tried transitions 400 Accepted transitions 350 Change in delta 0.6 Then the inputs above were used to determine the final inputs that would optimise the SA algorithms performance. The following results were found: Starting temperature effect Figure 4.1: Starting temperature effect Using the rule of thumb procedure for choosing the starting temperature resulted in long computational time. After reducing the initial temperature, better results were recorded in shorter time. From the graph the lowest MAD and objective function value was achieved at

31 4.3.2 Maximum transitions at each temperature Figure 4.2: Maximum transitions at each temperature Changing this value allows for greater chance of finding a better solution but this greatly increases your computational time. From the graph the best value is 400. It was chosen because it allowed a good answer in reasonable time and produced the lowest MAD and objective value Maximum accepted transitions at each temperature Figure 4.3: Maximum accepted transitions at each temperature As with the number of transitions tried, the number of transitions accepted also increases your computational time but improves your chance of finding an improved solution. When considering the above mentioned, of the 400 transitions tried, 350 accepted transitions delivered the best results compared with other input values. 25

32 4.3.4 Effect of temperature reduction Figure 4.4: Effect of temperature reduction This input greatly affects the probability function in the algorithm. Thus, the higher the temperature the higher the chance of choosing a larger objective function value. The lower the temperature the greater the chance of choosing an improved objective value but if the temperature is too low, the algorithm would not improve the current solution. Thus, the temperature reduction should not be too slow or too quick. From the analysis it can be seen that a reduction of 99% delivers the best results Maximum iterations Figure 4.5: Maximum iterations The number of iterations the algorithm should perform is an indication of how long it will run. If this number is too big, the computation time will be very long and if too small the chances of finding a good solution will be slim. The best value for finding an average good solution was calculated at 500 iterations. 26

33 4.3.6 Change in delta Figure 4.6: Change in delta Experimenting in Excel it was found that changing the delta value (difference in two consecutive objective function values) the acceptance of these values could be changed in the probability function. It was decided that a possibility existed that this change in delta could improve the algorithm s ability to find an improved solution more rapidly. According to the analysis, 0.6 proved to be the most effective Stopping temperature effect Figure 4.7: Stopping temperature effect Stopping temperature is a criteria used by the SA algorithm as a stopping criteria. The value that delivered the best objective function values considering the starting temperature, temperature reduction and maximum iterations was 1E 6. 27