Given the Following All-Integer Linear Programming Problem
Given the following all-integer linear programming problem: Max 3x1 + 10x2 s.t.2x1 + x2 < 5 x1 + 6x2 < 9 x1 - x2 > 2 x1,x2 > 0 and integer a.Solve the problem graphically as a linear program. b.Show that there is only one integer point and that it is optimal. c.Suppose the third constraint was changed to x1 - x2 > 2.1.What is the new optimal solution to the LP? To the ILP?
Kloos Industries has projected the availability of capital over each of the next three years to be $850,000,$1,000,000,and $1,200,000,respectively.It is considering four options for the disposition of the capital: a.Research and development of a promising new product b.Plant expansion c.Modernization of its current facilities d.Investment in a valuable piece of nearby real estate Monies not invested in these projects in a given year will NOT be available for the following year's investment in the projects.The expected benefits three years hence from each of the four projects and the yearly capital outlays of the four options are summarized in the table below in millions of dollars. In addition,Kloos has decided to undertake exactly two of the projects.If plant expansion is selected,it will also modernize its current facilities. Formulate and solve this problem as a binary programming problem.
Given the following all-integer linear program: Max 15x1 + 2x2 s.t. 7x1 + x2 < 23 3x1 - x2 < 5 x1,x2 > 0 and integer a.Solve the problem as an LP,ignoring the integer constraints. b.What solution is obtained by rounding up fractions greater than or equal to 1/2? Is this the optimal integer solution? c.What solution is obtained by rounding down all fractions? Is this the optimal integer solution? Explain. d.Show that the optimal objective function value for the ILP (integer linear programming)is lower than that for the optimal LP. e.Why is the optimal objective function value for the ILP problem always less than or equal to the corresponding LP's optimal objective function value? When would they be equal? Comment on the optimal objective function of the MILP (mixed-integer linear programming)compared to the corresponding LP and ILP.
A business manager for a grain distributor is asked to decide how many containers of each of two grains to purchase to fill its 1,600-pound capacity warehouse.The table below summarizes the container size,availability,and expected profit per container upon distribution. a.Formulate as a linear program with the decision variables representing the number of containers purchased of each grain.Solve for the optimal solution. b.What would be the optimal solution if you were not allowed to purchase fractional containers? c.There are three possible results from rounding an LP solution to obtain an integer solution: (1)The rounded optimal LP solution will be the optimal IP solution. (2)The rounded optimal LP solution gives a feasible,but not optimal IP solution. (3)The rounded optimal LP solution is an infeasible IP solution. For this problem,(i)round down all fractions; (ii)round up all fractions; and (iii)round off (to the nearest integer)all fractions (Note: Two of these are equivalent.)Which result above (1,2,or 3)occurred under each rounding method?