5.1.7 (Duality Gap of the Knapsack Problem)wwwGiven objects i = 1,... , n with positive weights w; and values vi, we want toassemble a subset of the objects so that the sum of the weights of the subset doesnot exceed a given A > 0, and the sum of the values of the subset is maximized.%3D(a) Show that this problem can be written asmaximize >i=1subject to > Wit; < A,Σxi E {0,1}, i = 1,..., n.i=1(b) Use a graphical procedure to solve the dual problem where a Lagrangemultiplier is assigned to the constraint , w;X; < A.(c) Let f* and q* be the optimal values of the primal and dual problems,respectively. Show that0 < q* – f* < max vi.|i=1,...,n(d) Consider the problem where A is multiplied by a positive integer k andeach object is replaced by k replicas of itself, while the object weights andvalues stay the same. Let f* (k) and q*(k) be the corresponding optimalprimal and dual values. Show thatq*(k) - f*(k)f* (k)1 maxi=1,...,n Vif*kso that the relative value of the duality gap tends to 0 as k o.

SOLUTION: Use CAS performs following steps by implementing the method of Lagrange multipliers by…

Answered: 5.1.7 (Duality Gap of the Knapsack…

Practical Management Science

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