(a)Maximize Zx3.(b)Maximize Zx12x3.5.1-20.Consider the three-variable linear programming problemshown in Fig. 5.2.(a)Explain in geometric terms why the set of solutions satisfyingany individual constraint is a convex set, as defined in Ap-pendix 2.(b)Use the conclusion in part (a) to explain why the entire feasi-ble region (the set of solutions that simultaneously satisfiesevery constraint) is a convex set.5.1-21.Suppose that the three-variable linear programming prob-lem given in Fig. 5.2 has the objective functionMaximizeZ3x14x23x3.Without using the algebra of the simplex method, apply just itsgeometric reasoning (including choosing the edge giving the max-imum rate of increase of Z) to determine and explain the path itwould follow in Fig. 5.2 from the origin to the optimal solution.5.1-22.Consider the three-variable linear programming problemshown in Fig. 5.2.(a)Construct a table like Table 5.4, giving the indicating variablefor each constraint boundary equation and original constraint.(b)For the CPF solution (2, 4, 3) and its three adjacent CPF so-lutions (4, 2, 4), (0, 4, 2), and (2, 4, 0), construct a table likeTable 5.5, showing the corresponding defining equations, BFsolution, and nonbasic variables.(c)Use the sets of defining equations from part (b) to demonstratethat (4, 2, 4), (0, 4, 2), and (2, 4, 0) are indeed adjacent to (2, 4, 3), but that none of these three CPF solutions are adja-cent to each other. Then use the sets of nonbasic variables frompart (b) to demonstrate the same thing.5.1-23.The formula for the line passing through (2, 4, 3) and (4, 2, 4) in Fig. 5.2 can be written as(2, 4, 3)[(4, 2, 4)(2, 4, 3)](2, 4, 3)(2,2, 1),where 01 for just the line segment between these points.After augmenting with the slack variables x4,x5,x6,x7for the re-spective functional constraints, this formula becomes(2, 4, 3, 2, 0, 0, 0)(2,2, 1,2, 2, 0, 0).Use this formula directly to answer each of the following ques-tions, and thereby relate the algebra and geometry of the simplex2245THE THEORY OF THE SIMPLEX METHODx11012342345(2, 5)(4, 5)x25.2-1.Consider the following problem.MaximizeZ8x14x26x33x49x5,subject tox12x23x33x4x5180(resource 1)4x13x22x3x4x5270(resource 2)x13x22x3x43x5180(resource 3)
andx10,x20,x30.Let x4,x5, and x6denote the slack variables for the respective con-straints. After you apply the simplex method, a portion of the fi-nal simplex tableau is as follows:andxj0,j1, . . . , 5.You are given the facts that the basic variables in the optimal so-lution are x3,x1, and x5and that1217.(a)Use the given information to identify the optimal solution.(b)Use the given information to identify the shadow prices for thethree resources.I5.2-2.*Work through the revised simplex method step by stepto solve the following problem.