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GENERAL ASSIGNMENT PROBLEM via Branch and Price JOHN AND LEI Outline • Review the column generation in Generalized Assignment Problem (GAP) 2 • GAP Examples in Branch and Price Assignment Problem • The assignment problem is to find a maximum profit assignment of n tasks to n machines such that each task (i=1,2,…,n) is assigned to exactly one machine (j=1,2,…n). 3 • To find a maximum weight matching in a weighted bipartite graph. • One of the fundamental combinatorial optimization problem Generalized Assignment Problem (GAP) • m tasks vs. n machines (m>=n) • Each machine is allowed to be assigned to more than one task • • • • 4 • z: total profit pij: profit of task i on machine j yij: association between task i and machine j wij: resouce consumption of task i on machine j dj: total available resource on machine j Generalized Assignment Problem (GAP) • GAP: 5 • If there is only one machine • Knapsack Problem (KP): Generalized Assignment Problem (GAP) • GAP KP1 KP2 . . . KPn 6 • (NOT means the solution to the subproblems KPs can be combined to that of the GAP) • It motivates us to re-represent the set of points satisfying constraint (13.10) To use B & P Review the B & P Approach DANTZIG-WOLFE DECOMPOSITION Conventional Problem reformulation To choose the entering basic variable: Pricing Problems (Sub-Problems) 7 Master Problem Column Generation Restricted Master Problem Master Problem Formulation Consider the set of points satisfying: S is just a finite set of points: where Conventional Problem “Kj is the number of feasible solutions for j” ---- textbook Kj is the number of feasible assignment for the machine j Z1j, Z2j, …….. ZKjj are the feasible solutions of KPj 8 Master Problem Formulation Consider the set of points satisfying: Every point y can be represented by S is just a finite set of points: where Conventional Problem “Kj is the number of feasible solutions for j” ---- textbook Kj is the number of feasible assignment for the machine j Z1j, Z2j, …….. ZKjj are the feasible solutions of KPj 9 Master Problem Formulation Every point y can be represented by Master Problem Conventional Problem Substitute y The master problem of GAP is formulated in terms of columns representing feasible assignments of tasks to machines. 10 Structure of the formulation Example: m=3, n=2 Master Problem Symbol 0/1 in the entries represents 0 or 1 value of z (assignment) The task lines indicate the assignment constraint The machine lines indicate the convexity constraint 11 The master problem of GAP is formulated in terms of columns representing feasible assignments of tasks to machines. Issues with the formulation Conventional Problem • Fundamental difference between these two: replaced by IP solution y in S A finite set of points Usually a very huge number (exponential) • Issue 1: Too much columns in the reformulation problem. (So we choose to work with its restricted version instead of solving directly) • Issue 2: Choosing entering variables (Solve the sub-problem/pricing problem) 12 Reformulated Master Problem Why bothering to re-formulate? Conventional Problem • A compact IP formulation (e.g. conventional GAP) has a weak LP relaxation. IP solution y in S • A finite set of points The Master Problem formulation is tighter than the conventional one. Reformulated Master Problem Intuitive explanation for GAP Fractional solutions in LP that are not convex combinations of the solution (z) to the knapsack problems, are not feasible to the master problem • 13 The decomposition formulation is not to speed up but to tight the bound. Sub-problem/Pricing Problem • Issue 2: Choosing entering variables • (u,v) is the optimum dual solution to the LP relaxation of the restricted master problem • KP has m+n columns, which is usually much smaller than the total number • The relaxation of the restricted master problem is solved by the revised simplex method. • The pivot column associated with the entering variable is generated by solving the pricing problem (Solve the sub-problem/pricing problem) sub-problem/pricing problem: Then new decomposition 14 Sub-problem/Pricing Problem sub-problem/pricing problem: 15 • If the optimum value of any pricing problem is positive, then the column with positive reduced cost can be added to the restricted master problem. • If the maximal reduced costs of all the KPs are non-positive, then the LP solution obtained is also maximal for the relaxation of the UN-RESTRICTED master problem. GAP Example m=3, n=2 (d1,d2)=(11,18) Conventional Formulation: 16 GAP Example Conventional Formulation: 17 GAP Example Conventional Formulation: Decomposition Re-formulation: 18 GAP Example Initial feasible solution: (Only a subset of all variables) (13.18) (13.19) (13.20) (13.21) (13.22) Two-phase method Decomposition Re-formulation: 19 GAP Example Update the tableau Passing the dual value u and v to the sub-problems, we have Decomposition Re-formulation: 20 GAP Example We do this until … No negative objective value of sub-problems is found Decomposition Re-formulation: Z1=(1,0,0) 21 Z2=(0,1,1) Branch-and-Price Algorithm Original Problem Formulation Master Problem Restricted Master Problem (RMP) Solve LPR of RMP Add such columns to RMP Generate Duals from RMP Solve Sub-problem by passing duals Are there any columns generated with positive reduced cost? N Solution Integral ? N 22 Branch Y Stop Y www.liu.se