ColumnGeneration-Exercise-Set

Mathematical Programming Exercise set 2 - Column generation Exercise 1 In this exercise we study the facility location problem . Consider m facilities and n customers. The cost to service customer j ∈ { 1 , 2 , . . . , n } from facility i ∈ { 1 , 2 , . . . , m } is c ij . The cost for opening facility i is equal to f i , for i ∈ { 1 , 2 , . . . , m } . At most W facilities may be opened and each customer must be serviced by a facility. The problem is to decide (1) which facilities should be opened and (2) which open facility should service which customer such that the total costs are minimized. We define a facility plan as a pair ( i ( p ) , S ( p )), where i ( p ) ∈ { 1 , . . . , m } is a facility that is opened, and the subset of customers S ( p ) ⊆ { 1 , . . . , n } is serviced from i ( p ). Let P be the set of all facility plans. a. Formulate this problem as a binary programming problem. Make use of the variables x p , indicating whether facility plan p ∈ P is used. Clearly define the corresponding objective coefficients, and clearly explain your formulation. b. Suppose that we want to solve the LP relaxation of the binary programming formulation using column generation. For given values of the dual variables, describe the pricing problem and an algorithm to solve it. c. Suppose we perform column management as follows. In each iteration of the column gen- eration algorithm, we remove all columns that have not been part of an optimal solution to the RMP for at least 10 consecutive iterations. Explain why this could speed up the column generation algorithm. Also explain why this could slow down the column generation algorithm. ( The take-away is that testing is required to see if it column management helps. ) Answer of Exercise 1 a. The cost coefficient corresponding to x p is C p = f i ( p ) + ∑ j ∈ S ( p ) c ij . Let the parameter a jp be 1 if j ∈ S ( p ) and 0 otherwise. Consider the following binary programming formulation. min summationdisplay p ∈ P C p x p (1) summationdisplay p ∈ P a jp x p = 1 ∀ j ∈ { 1 , . . . , n } (2) summationdisplay p ∈ P x p ≤ W (3) x p ∈ { 0 , 1 } ∀ p ∈ P (4) 1 Downloaded by Taghi Khaniyev (tagikhaniyev@gmail.com) lOMoARcPSD|24058970

The objective (1) is to minimize the total costs. Constraints (2) ensure that every customer is serviced. Constraint (3) ensure that not more than W facilities are opened. Finally, the binarity conditions are given by (4). b. Let λ j , for j ∈ { 1 , . . . , n } , be the dual variables corresponding to (2). Furthermore, let μ be the dual variable corresponding to (3). The reduced cost of a variable x p is RC ( x p ) = C p - n summationdisplay j =1 a jp λ j - μ = f i ( p ) - μ + summationdisplay j ∈ S ( p ) ( c i ( p ) j - λ j ) The pricing problem is to find a facility plan that minimizes the reduced cost. This pricing problem is not very difficult to solve, and a pricing algorithm is the following. Step1: For all facilities i ∈ { 0 , . . . , m } , we select the set S of customers for which c ij - λ j < 0. Step 2: For each facility i and corresponding subset S found in Step 1, compute f i - μ + ∑ j ∈ S ( c ij - λ j ). The lowest cost is the optimal solution value, and corresponds to the optimal solution of the pricing problem. c. Having less columns in the RMP speeds up solving the RMP. On the other hand, keeping track of columns to remove, and removing them takes (a tiny bit of) computation time. Moreover, columns that are removed might be useful in a later iteration of the column generation algorithm, and then need to be generated again. This would slow down the algorithm. Exercise 2 In this exercise we study the bin packing problem . Consider n items each with known volume, indicated by v i for item i ∈ { 1 , . . . , n } , which have to be packed into a number of bins, each having the same volume V . Items packed into the same bin should have a total volume not exceeding the volume of the bin and the objective is to pack all items using a minimal number of bins. We define a packing as a subset of items that do not exceed the bin capacity. Let P be the set of all packings. a. Formulate this problem as a binary programming problem. Make use of the packing vari- ables x p , indicating whether packing p ∈ P is used. Clearly define the corresponding objective coefficients, and clearly explain your formulation. b. Suppose that we want to solve the LP relaxation of the binary programming formulation using column generation. For given values of the dual variables, provide the pricing problem. Do you recognize the pricing problem? c. Explain why it is not always necessary to solve the pricing problem to optimality when generating additional columns. d. Now suppose we use a branch-and-price algorithm to solve the bin packing problem. Provide a disadvantage of using a branching rule that branches on a fractional packing variable. Answer of Exercise 2 2 Downloaded by Taghi Khaniyev (tagikhaniyev@gmail.com) lOMoARcPSD|24058970