Discrete Covering. Location. Problems. Louis. Luangkesorn. Housekeeping. Dijkstra s Shortest Path. Discrete. Covering. Models.

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1 Network Design Network Design Network Design Network Design Office Hours Wednesday IE 079/079 Logistics and Supply Chain Office is closed Wednesday for building renovation work. I will be on campus (or close). if you want to see me. University of Pittsburgh Homeworks will be Wednesday to Wednesday Department of Industrial Engineering June 9, 009 GLPK on Windows Testing 3 Circles code Setup Complete package, except sources Unittest using a problem you know the answer Install Setup.exe C:/Program Files/GnuWin3/bin (-, ), (, ), (, -) Shift (-, 6), (, 6), (, 3) Create batch file in working directory glpsol.bat Where are people in the homeworks? C:/Program Files/GnuWin3/bin/glpsol % % %3 %4 %5 %6 %7 %8

2 Network Design Network Design Network Design Network Design Problem Solving Many of the discrete location problems that we will cover require that the distance between two nodes is known. Most descriptions of networks describe lengths in terms of the lengths of individual arcs. Problem: given a network with costs associated with each of the links, find the shortest path from a specified node (s) to another node (t). The homework asks you to solve this through a linear programming formulation. This works, but it is inefficient. 3 The solution to a shortest path algorithm contains two pieces of information Cost of shortest path from origin s to destination t Sequence of links needed to get from s to t Define a node label that includes cost of getting from s to the node, and the node that we came from in getting to here from s. Node label for any node j is [Vj,Pj] s Algorithm Initialize Label node s [0, -]. I.e. Vs = 0 Label other nodes [inf, ]. I.e. initialize all other nodes to infinity. Set node s as scanned Label Updates Call the last scanned node, node m. For all links (m,j) such thaat node j is not scanned, computer Tj = Vm + cmj If Tj < Vj, relabel node j with [Tj,m] 3 Scan a node Find the unscanned node with the smallest label Vj. In the event of a tie, arbitrarily choose one of the nodes with the smallest label Vj. Scan this node. 4 Termination Check Are all nodes scanned? YES - Stop NO - go to Label Updates Initialize node s as [0, ] s is scanned From s, scan A and C [Inf, -] 3 A [0, -] s 3 C [Inf, -] [, s] 3 A [0, -] s 3 C [, s] Shortest path example [Inf, -] B E 3 [Inf, -] D [Inf, -] [Inf, -] B t 3 [Inf, -] D [Inf, -]

3 Network Design Network Design Network Design Network Design Shortest path example Shortest path example Node C has the minimum Vj, so set C scanned Scan A and D. Note that A remains unchanged Node D has the minimum Vj, so set D scanned From A and D have the minimum Vj, so pick one, A and scan B [, s] [Inf, -] 3 A B [0, -] s 3 t 3 [Inf, -] C D [, s] [, C] Scan B and t. Note that B can be improved, so relabel B Done! [, s] [4, D] 3 A B [0, -] s 3 t 3 [5, D] C D [, s] [, C] [, s] A [0, -] s 3 C 3 [5, A] B t 3 [Inf, -] D Note: In real life you would use a math library. Boost Graph Library [, s] [, C] Motivation of location research Examples of decision contexts decisions are made at all levels of human organization from individuals to governments and international agencies. Decisions are strategic in nature. They involve large sums of captial resources and their effects are long term: ability of a Airlines and airports Garages firm to compete, efficiency of services, ability to attrack other Service centers activity. Emergency services/public safety 3 The frequently impose economic externalities: pollution, congestion, economic development. Health facilities Truck terminals 4 Often extremely difficult to solve. Even basic models are intractable for large problem instances. Require the use of Warehouses digital computers. 5 Application specific. Their structure is determined by the particular location problem under study.

4 Network Design Network Design Network Design Network Design Basic Facility Set covering Maximal covering p-center p-dispersion p-median Fixed charge Set covering location model Maximal covering location model p-center problem p-dispersion problem Hub Maxisum Set covering location model Set Formulation Objective: locate the minimum number of facilities required to cover all of the demand nodes. set I = The set of demand nodes. set J = The set of candidate facility locations dij = distance between demand node i and candidate site j Obj :Min xj () s.t. xj i I () j Ni DC = Coverage distance Ni = j dij Dc xj {0,} j J (3) Alternative : aijxj i I (4) = set of candidate lcoations that can cover demand point i Alternate: aij = if locate at i is covered by j, 0 otherwise. Decision variable xj = if site located at j, 0 otherwise Alternative (4) in place of () is to create a parameter aij that is set to if a demand at i can be covered by facility at j

5 Network Design Network Design Network Design Network Design Maximal Problem Maximal Formulation What if it is not possible to cover all demand nodes? Locate a predetermined number of facilities, p, in such a way as to maximize the demand that is covered. hi = demand at node i p = number of facilities to locate zi = if demand node i is covered, 0 otherwise Obj :Max hizi (5) i I s.t. xj zi 0 i I (6) j Ni xj = p (7) xj {0,} j J (8) zi {0,} i I (9) Objective maximizes total demand covered. Constraint (6) ensures that node i is not considered covered unless at least one facility covers i. You can vary p to find the marginal benefit of one additional facility. p-center Problem p-center Components What if Dc was a goal or target instead of fixed? Minimize the maximum distance that demand is from the closest facility given we are siting p facilities. Vertex problem: facilities can only be at the nodes of a network Absolute problem: facilities can be anywhere along arcs of a network Additional decision variables W = the maximum distance between a demand node and the facility to which it is assigned yij = if demand node i is assigned to facility at node j, 0 otherwise Weighted or unweighted

6 Network Design Network Design Network Design Network Design p-center Formulation p-dispersion Problem Min :W (0) s.t. xj = p () yij = i I () yij xj 0 i I,j J (3) W hidijyij 0 i I (4) xj {0,} j J (5) yij {0,} i I,j J (6) Only concerned with the siting of new facilities Objective is to maximize the minimum distance between any pairs of facilities. M = a large constant, maxi I,dij D = minimum separation distance between any pair of facilities. (4) sets a lower bound on maximum demand weighted distance W Note that (6) can be replaced by yij 0 because y is already forced to be by (3). Includes correction to objective function from p. 88 p-dispersion Formulation Total or MaxD (7) s.t. xj = p (8) p-median problem D+ (M dij)xi + (M dij)xj M dij i,j J,i < j (9) xj {0,} j J (0) Fixed Charge Problem Hub location problem Maxisum problem (9) defines the minimum separation between any pair of open facilities. If a facility xi or xj is 0 (not open) the constraint is not binding.

7 Network Design Network Design Network Design Network Design p-median Problem p-median Formulation Objective: locate p facilities to minimize the demand-weighted total distance between demand nodes and the facilities to which they are assigned. Min i I s.g. hidijyij () xj = p () yij i I (3) yij xj 0 i I,j J (4) xj {0,} j J (5) yij {0,} i I,j J (6) Fixed Charge Problem Fixed Charge Problem Formulation Minimize total facility and transportation costs Does not assume each site has the same fixed costs Does not assume that sites are capacitated Does not assume that there is a set number of facilities p that should be opened fj = fixed cost of locating a facility at candidate site j Cj - capacity of a facility at candidate site j α cost per unit demand per unit distance Min fjxj + α iini s.t. hidijyij (7) yij = i I (8) yij xj 0 i I,j J (9) hiyij Cjxj 0 i I (30) xj {0,} j J (3) yij {0,} i I,j J (3) (3) implies that each demand is assigned to a single facility. (30) implies that facilities are capacitated.

8 Network Design Network Design Network Design Network Design Hub Problem Hub Formulation Minimize the total cost (as a function of distance) of transporation between hubs, facilities and demands hij = number of units of flow between nodes i and j cij = unit cost of transporattion between nodes i and j α = discount factor for transport between hubs Min i N j N ( hij k N cikyij + m N cjmyjm + α k N m N ckmyikyjm (33) s.t. xj = p (34) j N yij = ) i I set N = all nodes xj = if a hub is located at node j, 0 otherwise yij = if demands from node i are assigned to a hub located at node j, 0 otherwise yij xj 0 xj {0,} (35) i I,j J (36) (37) j J yij {0,} i I,j J (38) Note: (33) is corrected here from p. 93 Maxisum Problem Maxisum Formulation Objective: Maximize the total demand weighted distance between demand nodes and the facilities to which they are assigned. Note: Demands are assigned to nearest facility. Max i I s.t. hidijyij (39) x + j = p (40) yij = i I (4) yij xj 0 i I,j J (4) m y i[k]i x [m]i 0 i I,m =,...,N (43) k= xj {0,} j J (44) yij {0,} i I,j J (45)

9 Network Design Network Design Network Design Network Design Maxisum notes Routing models where demands are served by vehicles that are routed from a central facility. (43) forces demands to be assigned to the nearest facility [k]i is the k th farthest candidate location from demand node i (43) if mth closest facilty to demand node i is opened, the facility must be assigned to that facility, or a closer one. Where to locate facilities Allocating customers to facilities Routing vehicles to serve customers Not covered in this course (vehicle routing problem) Identify additional constraints Capacity constrained vehicle routing problems cost constrained vehicle routing problems cost constrained location - routing problems Facility Network Design models Strategic issues lead to multiple, competing objectives Both costs of locating facilities as well as costs of arcs Two approaches Hub location models Airlines, utilities, computer networks Generating techniques: Identify Pareto optimal siting configurations Preference techniques: Rank the objectives, find solution that optimizes ranking

10 Network Design Network Design Network Design Network Design examples approaches Ranking objectives Cost Preference weights on objective function: optimize a convex Risk Equity combination of the weighted objective. Lixicographical ordering of objectives: Optimize the primary, then from alternative optima, optimize a secondary objective etc. Dynamic Stochastic When a single solution has to optimize over time Environment changes over time Allow for facilities to open or close (with some cost) Demand, travel time/cost, facility availability, fixed and Parameters are not known with certainty. Demand, travel time, facility costs, distances, facility availability, number of facilities to be sited. variable costs, number of facilities to be opened.

11 Network Design Network Design Network Design Network Design Approaches for Stochastic Scenario Planning Approximate the uncertainty with a deterministic surrogate. Cost uncertainty into the model, but still use a deterministic estimate (e.g. using a lower discount rate for future costs to add a cost of uncertainty) Develop chance constrained model. Allow for a probability that a facility is busy. Account for queuing interactions that occur in a spatially distributed queueing system with facilities at multiple locations in a network. Scenario planning: Optimize the solution over a set of possible scenarios. Allow for phases of the solution (before actual scenario known, after scenario is realized) Optimize over all scenarios Minimize regret as an objective Set a maximum regret Scenario planning Solution Approaches for Set Formulation Mixed integer linear programs: Branch-and-bound or cutting planes Problem sizes get large fast, and become impractical to solve Greedy heuristics - Add facility with greatest improvement on objective, or remove the facility with the smallest lost Improvement heuristics Neighborhood search - Start with a feasible solution. Define neighborhoods by determining the nearest facility for each demand. Within the neighborhood, solve a single facility problem Genetic algorithms - Tabu search (will be covered later in course) Obj :Min xj (46) s.t. xj i I (47) j Ni xj {0,} j J (48) Alternative : aijxj i I (49) Lagrangean relaxation - solve relaxed problem, with a constraint replaced by a lagrangean variable that is in the objective.

12 Network Design Network Design Network Design GMPL Set Cover sets and parameters GMPL Set Cover Constraints and objectives set F within D; /* candidate sites*/ param coverdistance; /* distance over which a facility F can cover a demand */ param distance{d, F}; /* distance between D and F */ param cost{f}; /* cost to locate a facility at F */ param a{d in D, f in F} := if distance[d,f] <= coverdistance then else 0; /* if candidate site F can cover demands at node D */ /* if we locate a production facility at site f */ s.t. coverdemand{d in D}: sum{f in F} a[d,f] * x[f] >= ; /* for each demand location, covered by at least one eligible facility*/ minimize covercost: sumf in F cost[f] * x[f]; /* minimize total cost */ end; GMPL Set Cover Data set D := A B C D E F; param coverdistance := 9; param distance default e7 : A B C D E F := A B C D E F ;