Location Models in Logistics Management Vincent F. Yu Department of Industrial Management National Taiwan University of Science and Technology Oct. 21, 2010 Vincent F. Yu (IM, NTUST) Location Models Oct. 21, 2010 1 / 24 Outline 1 Location Models 2 Applications in Logistics Management: Location Routing Problem and Its Variants 3 References 4 Q&A Vincent F. Yu (IM, NTUST) Location Models Oct. 21, 2010 2 / 24 Location Models Location Models Location! Location!! Location!!! Location decisions arris in a variety of public and private sector problems. Public Sector: Hospitals, Fire Stations, Police Departments, Landfills for Hazardous Waste, Public Schools, Bus/Train Stations, ..., etc. Poor decisions lead to .... Increased likelihood of property damage, lost of life, ... Private Sector: Offices, Plants, Retail Stores, Distribution Centers, Hubs, Call Centers, ..., etc. Poor decisions lead to .... Increased cost, decreased efficiency/competitiveness, ... The success or failure of both public and private sector facilities depends in part on the locations chosen for those facilities. Vincent F. Yu (IM, NTUST) Location Models Oct. 21, 2010 3 / 24 Location Models Key Questions 1 How many facilities? 2 Where should each facility located? 3 How large should each facility be? 4 How should demand for the facilities’ services be allocated to the facilities? Vincent F. Yu (IM, NTUST) Location Models Oct. 21, 2010 4 / 24 Location Models Prototype Problems 1 Covering Problems. 2 Center Problems. 3 Median Problems. 4 Fixed Charge Facility Location Problems. 5 Extensions of Location Models: Location Routing Problems. Hub Location Problems. Dispersion Models. Location of Undesirable Facilities. Hierarchical Location Models. Multiobjective Problems, ... Vincent F. Yu (IM, NTUST) Location Models Oct. 21, 2010 5 / 24 Location Models Application: Ambulance Location Set Covering Model - minimize the number of ambulances needed so that all demand nodes are within a given number of minutes. Maximum Covering Model - maximize number of demands that can be covered within a specified service standard using a given number of vehicles. P-center Problem - minimize the maximum response time using a given number of vehicles. P-medium Problem - minimize the average response time using a given number of vehicles. Other Considerations: stochastic demands, workload balance, service type/characteristics, temporal variation (relocatable ambulances v.s. fixed sites). Vincent F. Yu (IM, NTUST) Location Models Oct. 21, 2010 6 / 24 Location Models Application: Siting Landfills for Hazardous Wastes P-medium Problem - minimize the average or total shipping distance. Maxisum or Maximin Model - locate a given number of facilities to maximize the (weighted) distance between population centers and the nearest sites. Fixed Charge Facility Location Problem - balance the initial capital investment and ongoing operating costs. Other Considerations: reduce the inequalities across communities (spread the risk or disbenefit), routing, ... Vincent F. Yu (IM, NTUST) Location Models Oct. 21, 2010 7 / 24 Location Routing Problem and Its Variants Location Routing Problem Traveling Salesman Problem (TSP) Vehicle Routing Problem (VRP) Location Routing Problem (LRP): includes two key decision problems in strategic logistics management Determination of Facility Location Planning of Vehicle Routes These decisions are often interdependent. Therefore it is beneficial to consider them simultaneously. Vincent F. Yu (IM, NTUST) Location Models Oct. 21, 2010 8 / 24 Location Routing Problem and Its Variants Applications of LRP Food and drink distribution Blood bank location Newspapers delivery Waste collection/Reverse logistics Bill delivery Military application Parcel delivery Various consumer goods distribution Vincent F. Yu (IM, NTUST) Location Models Oct. 21, 2010 9 / 24 Location Routing Problem and Its Variants Solution Strategy The LRP belongs to the class of NP-hard problems. As the problem size grows larger, heuristic approaches become the only viable alternative. Vincent F. Yu (IM, NTUST) Location Models Oct. 21, 2010 10 / 24 Location Routing Problem and Its Variants Problem Definition G = (V , E , c) be a complete, weighted, and undirected network. V : set of nodes comprised of a subset I of m potential depot sites and a subset J = V \I of n customers. cij : traveling cost (distance) between each pair of nodes i, j. Wi , Oi : capacity and opening cost associated with each potential depot site i. Each customer j ∈ J has a demand dj which must be fulfilled be a single vehicle. A set K of identical vehicles with capacity Q is available. Each vehicle, when used, incurs a fixed cost Fi and performs a single route. Vincent F. Yu (IM, NTUST) Location Models Oct. 21, 2010 11 / 24 Location Routing Problem and Its Variants Problem Definition (cont.) Each route must start and terminate at the same depot. Each route’s total load can not exceed vehicle capacity. The total loads of the routes assigned to a depot can not exceed the capacity of the depot. The objective is to determine which depots should be opened and which routes should be constructed to minimize the total traveling cost. Vincent F. Yu (IM, NTUST) Location Models Oct. 21, 2010 12 / 24 Location Routing Problem and Its Variants Mathematical Model Define binary variables: yi =1 iff depot i is opened; = 0 otherwise; fij = 1 iff customer j is assigned to depot i; = 0 otherwise; xjlk = 1 iff edge (j, l) is traversed from j to l in the route performed by vehicle k. Then the problem can be formulated as a binary integer program. Vincent F. Yu (IM, NTUST) Location Models Oct. 21, 2010 13 / 24 Location Routing Problem and Its Variants Mathematical Model (cont.) min z = ∑ Oi yi + ∑∑∑ cij xijk + ∑∑∑ Fi xijk i∈I i∈V j∈V k∈K subject to ∑∑ x = 1 ∀j ∈ J k∈K i∈V ijk j∈J i∈V j ijk ∑∑ d x ∑d j∈J (3) f ≤ Wi yi ∀i ∈ I (4) j ij ijk ∑∑ x j∈V jik (1) (2) ≤ Q ∀k ∈ K ∑ x −∑ x j∈V k∈K i∈I j∈J = 0 ∀i ∈ V , k ∈ K (5) i∈I j∈J ijk ≤ 1 ∀k ∈ K (6) i∈S j∈S ijk ≤| S | −1 ∀S ⊆ J , k ∈ K (7) ∑∑ x ∑x u∈J iuk + ∑ u∈V \{ j } xujk ≤ 1 + fij ∀i ∈ I , j ∈ J , k ∈ K xijk ∈ {0,1} ∀i, j ∈ V , k ∈ K (9) yi ∈ {0,1} ∀i ∈ I (10) fij ∈ {0,1} ∀i ∈ I , j ∈ V Vincent F. Yu (IM, NTUST) (8) (11) Location Models Oct. 21, 2010 14 / 24 Location Routing Problem and Its Variants Simulated Annealing Heuristic for LRP Solution Representation Initial Solution Simulated Annealing Heuristic Vincent F. Yu (IM, NTUST) Location Models Oct. 21, 2010 15 / 24 Location Routing Problem and Its Variants Solution Representation 24 0 0 25 10 9 17 2 23 14 15 16 19 0 8 11 6 22 0 0 4 1 12 18 20 13 5 7 3 21 Figure 1. An example of solution representation. Figure 2. Visual illustration of the example solution given in Figure 1. Vincent F. Yu (IM, NTUST) Location Models Oct. 21, 2010 16 / 24 Location Routing Problem and Its Variants Construction of Initial Solution Step 1. Let U be the set of unused depots. For each depot i ∈ U, let cc(i) be the number of unassigned customers whose closest depot in U is depot i. Choose the depot in U with the highest cc value. If there is a tie, select the depot with the highest capacity. Step 2. For all unassigned customers, assign them to the chosen depot one by one in the increasing order of the distance between the customer and the chosen depot. Stop when the capacity of the depot is violated. Step 3. Construct a TSP route which starts from and ends at the depot using the Lin and Kernighan’s heuristic (1973). Step 4. Split the TSP route into several routes so that the route capacity constraint is not violated. Step 5. If there are still unassigned customers, go to Step 1; otherwise, terminate the procedure and encode the current solution. Vincent F. Yu (IM, NTUST) Location Models Oct. 21, 2010 17 / 24 Location Routing Problem and Its Variants SA Procedure SALRP(T0, TF, α, K, P, Nnon-improving, Iiter) ⎡ Step 1: Let Ndummy = ⎢ ⎢ ∑Q i di ⎤ ⎥. k ⎥ Generate the initial solution X by the greedy heuristic. Step 2: Let T=T0; I=0; N=0; Fbest=obj(X, P); Xbest=X; Step 3: I=I+1; Step 4: (Generate a solution Y based on X) Step 4.1: Generate r = random (0, 1); Step 4.2: Case r≤1/3: Generate a new solution Y from X by random swap operation; Case 1/3<r≤2/3: Generate a new solution Y from X by random insertion operation; Case 2/3<r≤1: Generate a new solution Y from X by random 2-opt operation; Step 5: If Δ=obj(Y, P) - obj(X, P) ≤0 {Let X=Y;} Else { Generate r = random (0,1); If r<exp(-Δ/KT) {Let X=Y;} Vincent F. Yu (IM, NTUST) Location Models Oct. 21, 2010 18 / 24 Location Routing Problem and Its Variants ≤ SA Procedure (cont.) ≤ Δ ≤ Generate r = random (0,1); If r<exp(-Δ/KT) {Let X=Y;} } Step 6: If (obj(X, P)<Fbest and X is feasible){Xbest=X; Fbest=obj(X, P);N=0;} Step 7: If I=Iiter { T=αT; I=0;N=N+1; Perform Local search based on swap operation on Xbest; Perform Local search based on insertion operation on Xbest; } Else {Go to Step 3;} Step 8: If T<TF or N=Nnon-improving {Terminate the SA heuristic;} Else {Go to Step 3;} Vincent F. Yu (IM, NTUST) Location Models Oct. 21, 2010 19 / 24 Location Routing Problem and Its Variants Computational Study Benchmark instances Barreto (2004): 19 instances. Prins et al. (2004): 30 instances. Tuzun and Burke (1999): 36 instances. Parameter setting Computational Results Barreto (2004): 18/19 best solutions (best parameters; new: 8; gap: 0.40%); 19/19 best solutions (parameter analysis; new: 9) Prins et al. (2004): 16/30 best solutions (best parameters; new: 7; gap: 0.38%); 30/30 best solutions (parameter analysis; new: 18) Tuzun and Burke (1999): : 7/36 best solutions (best parameters; new: 5; gap: 1.1%); 29/36 best solutions (parameter analysis; new: 25) Vincent F. Yu (IM, NTUST) Location Models Oct. 21, 2010 20 / 24 Location Routing Problem and Its Variants Conclusions and Future Directions Special solution encoding method Better results 41/85 best solutions using best parameters; 20 of them are new. 78/85 best solutions found during parameter analysis; 52 of them are new. Longer computational time Extensions Vincent F. Yu (IM, NTUST) Location Models Oct. 21, 2010 21 / 24 Location Routing Problem and Its Variants LRP Variants LRP variants with applications in reverse logistics and integrated logistics: Multiple echelon LRP LRP with simultaneous pickup and delivery ... Vincent F. Yu (IM, NTUST) Location Models Oct. 21, 2010 22 / 24 References Reference Mark S. Daskin (1995), Network and Discrete Location - Models, Algorithms, and Applications, John Wiley & Sons. Zvi Drezner and Horst W. Hamacher (2001), Facility Location Applications and Theory, Springer. A simulated annealing heuristic for the capacitated location routing problem, Computers & Industrial Engineering, 58 (2010), pp. 288-299. ( Co-authors: Prof. Shih-Wei Lin (CGU), Prof. Wenyih Lee (CGU), and Prof. Ching-Jung Ting (YZU)) Vincent F. Yu (IM, NTUST) Location Models Oct. 21, 2010 23 / 24 Q&A Q&A Vincent F. Yu Department of Industrial Management National Taiwan University of Science and Technology 43, Sec. 4, Keelung Rd., Taipei 106, Taiwan Tel: 02-2737-6333 E-mail: vincent@mail.ntust.edu.tw Web: web.ntust.edu/∼vincent Vincent F. Yu (IM, NTUST) Location Models Oct. 21, 2010 24 / 24