Consider the following 0-1 knapsack problem, the items, their weights and their profits appear in table below. If the knapsack has capacity 11 select the items that appear in the solution? i.e. select answers 1, 2,and 3 if those are the items in the optimal napsack. Item 1 4 6 Weight 4 5 6 Profit 2 6 8 10 13 15 J item 1 I item 2 J item 3 I item 4 item 5 irem 6

Portfolio manager Max Gaines needs to develop an investment portfolio for his clients who are willing to accept a moderate amount of risk. His task is to determine the proportion of the portfolio to invest in each of the five mutual funds listed below so that the portfolio maximizes the expected return but provides an annual return of no less than 3%. for each of the following scenarios. Annual Returns (Planning Scenarios): mutual fund yr 1 yr 2 yr 3 yr 4 international stock 22.37 26.73 6.46 -3.19 low-cap blend 14.88 18.61 10.52 5.25 mid-cap blend 19.45 18.04 5.91 -1.94 small-cap blend 13.79 11.33 -2.07 6.85 intermediate bond 7.29 8.05 9.18 3.92 Formulate the appropriate linear program for this situation. (state the objective function, the decision variables, and the constraints)

The following LP formulation represents a transportation problem where raw material (in tons] from Supplier i, i={1,2,3} to Plant j, j={1,2,3). The problem is solved and the optimal objective function value is 5200. Using the sensitivity report shown below, what would be the optimal total cost if the cost of transporting 1 ton from Supplier 1 to Plant 1 becomes 2 and the cost of transporting 1 ton from Supplier 3 to Plant 3 becomes 2.5? Min Total cost = 1x11 + 3x12 + 5x13 +3.5x21 +4x22+4.8x23 +3.5x31+3.6x32+3.2x33 Subject to x31+x32+x33 = 0, for i= {1,2,3} &j= {1, 2, 3} Final Reduced Objective Allowable Allowable Value Cost Coefficient Increase Decrease Name 500 1 2.5 1E+30 700 3 0.6 1E+30 200 1E+30 0.2 0 2.5 0 1E+30 1E+30 0.2 1E+30 1 1E+30 200 2.5 0 1E+30 0.6 1.8 1E+30 Variables X11 X12 X13 X21 X22 X23 X31 X32 X33 10 200 000000000 5 3.5 4 4.8 3.5 3.6 3.2

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