(a) Graph the constraints for this problem. Use dots to indicate all feasible integer solutions. X₂ X₂ 8 6 6 (b) Solve the LP Relaxation of this problem. at (x₁, x₂) = (c) Find the optimal integer solution. at (x₁, x₂) = 8 4 6 8 DO X₂ DO 8 6 8 G
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- For the products A, B, C, and D, which of the following could be a linear programming objective function? Select one: a. Z = 1A + 2BC + 3D b. Z = 1A + 2AB + 3ABC + 4ABCD c. Z = 1A + 2B + 3C + 4D d. Z = 1A + 2B/C + 3DUse the simplex method to find the optimal solution. Assume all variables are nonnegative. Minimize f = 7x + 9y subject to the following constraints. x2 5 y ≤ 19 -x + y 2 2 X = y = f= Need Help? Submit Answer xxx Read It3. SOLVE USING TWO-PHASE METHOD Minimize x₁ - 3x3 subject to x₁ + 2x₂x3 ≤ 6 - x₁ - x₂ + 3x3 = X1, X2, X3 20
- Simplex Method Write in normal form and solve by the simplex method, assuming x, to be nonnegative. 1. The owner of a shop producing automobile trailers wishes to determine the best mix for his three products: flat-bed trailers, economy trailers, and luxury trailers. His shop is limited to working 24 days/month on metal- working and 60 days/month on woodworking for these products. The following table indicates production data for the trailers. Usage per unit of trailer Flat-bed Economy Luxury Available resources 1 2 1 24 Metal work days Wood work days Contribution (Rx100) 60 4 6 2 14 13For minimization linear programming problem, the simplex method is terminated when all values of: a. Z (NB) s0 Ob. None of them Z (NB) = 0 Z (NB) 2 0Suppose we are solving a maximization problem andthe variable xr is about to leave the basis.a What is the coefficient of xr in the current row 0?b Show that after the current pivot is performed, thecoefficient of xr in row 0 cannot be less than zero.c Explain why a variable that has left the basis on agiven pivot cannot re-enter the basis on the next pivot.
- For the remaining questions, consider the following problem description: An oil company is considering exploring new well sites S₁, S2, ..., S10 with respective costs C1, C2, C10. And in particular they want to find the least-cost selection of 5 out of the 10 possible sites. The binary decision variables x₁,x2,..., X10 denote the decision to explore the corresponding site.Which is the correct relationship between IP and its LP relaxation? a. The optimal objective value of LP relaxation is better than that of IP b. The optimal objective value of LP relaxation is worse than that of IP c. IP is easier to solve d. IP can be solved by rounding the optimal solution of the LP relaxationWhich of the following is the optimal value of the objective function for the LP model in this problem? Minimize z=4x + 8y Subject to: x+1.6y>=45 2x+0.5y>=40 X+y<=40 XY>=0 a.220 b.160 c.240 d.200
- Innis Investments manages funds for a number of companies and wealthy clients. The investment strategy is tailored to each client's needs. For a new client, Innis has been authorized to invest up to $1.2 million in two investment funds: a stock fund and a money market fund. Each unit of the stock fund costs $50 and provides an annual rate of return of 10%; each unit of the money market fund costs $100 and provides an annual rate of return of 4%. The client wants to minimize risk subject to the requirement that the annual income from the investment be at least $60,000. According to Innis' risk measurement system, each unit invested in the stock fund has a risk index of 8, and each unit invested in the money market fund has a risk index of 3. The higher risk index associated with the stock fund simply indicates that it is the riskier investment. Innis's client also specified that at least $300,000 be invested in the money market fund. Refer to the computer solution shown below. Optimal…Consider the following Pareto maximization problem with decision variables x and y: vmax (x2 + x, - 2y) s.t. 0< x< 10 0< y< 5. What is the unique efficient point for this problem? (0,0) (10,5) (10,0) (110,0)2(a) The tableau is not optimal for either maximization or a minimization problem. Thus, when a non-basic variable enters the solution it can either increase or decrease Z or leave it unchanged, depending on the parameters of the entering non basic variable. Basic x1 X2 X3 X4 X5 X6 X7 X8 solution -5 4 -1 -10 620 X8 3 -2 -3 -1 1 12 X3 1 1 3 1 3 X1 1 -1 -4 (i)Categorize the variables as basic and non-basic and provide the current values of all the variables. (ii)Assuming that the problem is of the maximization type, identify the non- basic variables that have the potential to improve the value of Z. If each such 3 variable enters the basic solution, determine the associated leaving variable, if any, and the associated change in Z. (iii)Repeat part (b) assuming that the problem is of the minimization type. (iv)Which non basic variable(s) will not cause a change in the value of Z when selected to enter the solution? (b) Let xij be the amount shipped from source i to destination j in a 5X5…