b. Report clearly Question 3 Consider the following linear program: Max Z = 3X1 + 2x2 S.T. 5x1 +5x2 5 400 X1 - X22-10 X1 + 3x22 90 4x1 + 2x2 2 180 X2 35 X1, X220 a. Using the graphical solution method, identify clearly the feasible region. b. Show the direction of improvement for the objective function. C. Identify the optimal solution and the corresponding optimal Z value. d. Are there any redundant constraints? If yes, which one(s)? e. What are the values of the slack and surplus variables at the optimal solution? Tyne here to search cion 2 Consider the following linear program (LP): Maximize z = 3x1 + 2x2 Subject to 2x1 + x2 = 2 3x1 - 4x2 5-12 X120 fint 534 PM 10/27/2020 a. Construct the initial tableau for the Big-M method and determine whether the initial basic feasible solution of the resulting artificial problem is optimal. If not, proceed to the optimal solution through the simplex iterations, and report the optimal solution to the original LP given above. b. Report clearly the optimal solution obtained. Cold Question 3 Bic d. Are there any redundant constraints? If yes, which one(s)? e. What are the values of the slack and surplus variables at the optimal solution? Question 4 The table below corresponds to one of the simplex iterations reached while solving a linear program. Basic variable X1 X2 X3 X4 X5 X6 X7 Right hand side -5 4 -1 0 0 280 z -2 0 X6 2 0 3 -2 1 3 15 X2 1 1 - 1 3 -2 0 5 0 Given that the problem is of maximization type: a. Which of the current nonbasic variables has the potential to improve the objective function value? b. For each of these nonbasic variables identified in part (a), if such variable enters the basic solution (i.e. becomes basic), d leaving variable, if any, and the associated change in Z. Do not use the elementary row operations, c. Which nonbasic variables will not change the value of Z once selected to enter the basic solution? Why is this the case? Tuesday, October 27, 2020 5:34:20 PM GST BI c C (0 Type here to search Question 1 An oil refinery uses four ingredients in the production of three types of gasoline: Special, Super and SuperPlus. The cost per barrel for each ingredient and the maximum amount available of each is shown in the following table: Cost/barrel Ingredient ($) 1 19 Barrels available/day 4200 3500 5000 12300 12 nd Support 3 4 8 11 17 Each type of gasoline has certain requirements concerning its chemical composition, which is defined by the percentages of the various ingredients. Those restrictions as well as the selling price per barrel are shown in the following table: Gasoline type Composition restriction Selling price/barrel ($) Not less than 30% of ingredient 1 Special Not more than 20% of ingredient 2 13 Not less than 40% of ingredient 3 Not less than 40% of ingredient 2 Super 16 Not more than 20% of ingredient 3 Not less than 50% of ingredient 1 SuperPlus Not more than 20% of ingredient 2 Not less than 20% of ingredient 4 20 The refinery would like to produce at least 4000, 3500 and 2500 barrels of the Special, Super and SuperPlus gasoline, respectively, each day. The management would like to plan the production for the three types of gasoline such that the profit is maximized. Formulate this problem as a linear program (clearly define the decision variables)