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Resource Allocation under the Contingency Planning D. Keselman, Los Alamos National Laboratory

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Problem Description The problem is to minimize over the set of possible events and possible storage locations the average summary cost of resources, their storage, and transportation to the pre-defined locations, as well as additional penalty cost of not supplying quantities of resources needed at the locations. General formulation is due to W.B. Daniel.

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Objective function objFun = min Wn { min Qn {∑ l (p(e l ) * ∑ k (∫ Ω ((p w (l, k, x 1k, …, x nk )*∏ i dx ik ) *∑ Pm (p s (l, k, π) * min F {C W,X }))))}} C W,X = ∑ Ic (c(w i ) + ∑ k (c k *y ik ) + ∑ k ∑ Jc (d(w i, s j )*c tk * y ik )) + ∑ k ∑ Jck (max{0, (P(s j ) - y ijk / q mink )*c pk }) ∑ k V ik <= V(w i ); V ik <= C k (w i ); ∑ Jc y ijk <= y ik * x ik

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Notation N – number of candidate buildings to storages. n – number of storages to be selected. m – number of consumption points. D(s j ) – demand size of the jth consumption point. V(w i ) – total volumetric capacity of the ith storage. V ik – limit of volumetric capacity for the kth item in the ith storage. d(w i, s j ) – distance between ith storage and jth consumption point. C k (w i ) – capacity of ith storage for the item k, C k (s j ) – max demand in the jth demand point.

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Notation, cont. c(w i ) – cost of maintenance of the corresponding facility. c k – cost of unit of item k. c tk – transportation cost of one unit of item k over a distance unit. s – number of the events under consideration. e l – lth event. p(e l ) – probability of event e l in a given time span. p w (l, k, x 1k, …, x nk ) – combined probability distribution (density function or discrete probability) that after the event l, amount of item k in the warehouses the useable amount will be reduced by the factor x ik (0<= x ik <=1, i = 1,…, n).

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Notation, cont. p s (l, k, π) – probability of a need in item k in the consumption points after the event l, where π represents a vector (α 1, …, α m ) of 0’s and 1’s with 0 in j position signifies that the jth consumption point has zero demand in the kth item, and 1 represents the opposite. We’ll denote the set of all 2 m such vectors by P m. c pk – penalty cost coefficient in the objective function incurred by the shortage in item k for one person. W – set of all N possible warehouses. Wn – set of n-element subsets of W.

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Network Flow Subproblem Source Cp 3 Cp 2 Cp 1 Wh 1 Wh 2 Sink objFun = mean(cost(stored items) + cost(storage maintenance) + cost(delivery) + cost(unsatisfied demand)), objFun –> min

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Capacities and Costs For source outgoing arcs: capacities – stored item units in a corresponding warehouse, costs – 0. For Wh outgoing arcs: capacities – infinite, costs – trasportation cost of one item unit to a corresponding Cp. For sink incoming arcs: capacities – consumption demands, costs – 0.

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Algorithm Structure Preprocessing – Probability combination enumeration – Edge cost sorting – Warehouse combination enumeration Non-linear optimization – MinCost MaxFlow Batch Linear Programming

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Probability Enumeration p1p2p3 q1q2q3 Assumption: all unusability factor probability distributions are discrete and independent for different warehouses

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Min Cost Max Flow Batch Exact – Min Cost Max Flow solution (all combinations) uses Edmonds-Karp algorithm Approximation – Min Cost Max Flow approximate solution (all combinations) with periodic updates uses a proposition in the next slide and cost edge sorting Monte-Carlo – Sampling combined distribution

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Approximation Algorithm Proposition. Let e = (s, w) be a s arc with capacity cap(e) = c, and a maximum flow of minimum cost fMax is equal to f on e: fMax(e) = f. If the capacity of e is changed to c1 with the rest of the network unchanged. Then, c >f and c1≥f implies that fMax is also a maximum flow of minium cost solution in the changed network.

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Non-Linear Optimization Simulated annealing Newton-Raphson method Gradient descend Combinatorial brute force

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Linear Programming Formulation ∑ j w ijkl (comb) - w ik * x ikl (comb) <= 0, c jkl (comb) + ∑ j w ijkl (comb) / q mink >= P(s j ), w ik <= C k (w i ), ∑ k V ik * w ik <= V(w i ), objFun = ∑ jklcomb c jkl (comb) * p klcomb ) * p(e l ) * c pk + ∑ ik w ik * (c k + c(w i )) + ∑ ijklcomb prob ijkl (comb) * p(e l ) * w ijkl (comb) * d(w i, s j ) *c tk * sFactor ijkl –> min.

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Some Computational Results Num. Wh34567 Time (Exact)0.290.842.274.3814.97 Obj. fun.13.814.214.314.614.7 Time (Appr)0.160.260.380.650.98 Obj. fun.13.914.315.0 15.4 Time (MC)1.360.49112.011.921.73 Obj. fun.13.214.014.114.414.8 Time (LP)0.290.842.274.3841.65 Obj. fun.13.714.014.214.414.8 Number of Consumption Points = 15

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Improvements, directions Improve sampling – Find optimal sampling size as a function of distribution complexity – Use MCMC for higher dimensional problems Employ multi-commodity flows – Apply LP approach for solving batch flow problems – Expand formulation to mutually interchangeable items

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