Solve the questions 6-7. For 7th question use example 5

6 For an LP in standard form with constraints Ax=b and x0, show that d is a direction of unboundedness if and 4. 4 Why Does an LP Hare an Optitial bis? 139 only if Ad=0 and d0 7 Recall that Example 5 of Chapier 3 is an unbounded L. Find a direction of unboundedness along which we can move for which the objective function becomes arbitranily large. Graphically solve the following LP: naxz=2r1x2s.t.x1x212x1+x26x1,x20 From Figure 7, we see that (19) is satisfied by all points on or above AB(AB is the line x1x2=1). Also, (20) is satisfied by all points on or above CD(CD is 2r1+x2=6). Thas, the feasible region for Example 5 is tbe (ahaded) unbounded region in Figure 7 , which is bounded only by the x2 axis, line segment DE, and the part of line AB beginning at E. To find the optimal solution, we draw the isoprofit line passing through (2, 0). Thi isoprofit lioe has z=2r1x2=2(2)0=4. The direction of incereasing t is to the southeast (this nakes x1 larger and x2 snaller). Moving parallel to z=2r1=x2 in a southeast direction, we see that any ispprofit line we draw will intersect the feasible re: gion. (This is becouse any isoprofit line is steeper than the line x1x2=1 1) Thus, there are points in the feasble region that have arbitrarily large s-valoes. For ex. ample, if we wanted to find a poiat in the feasible region that had z1,000,000, we could choose any point in the feasible restion that is soutbeast of the isoprofit line 281,000,000. From the dicession in the iast two secbons, we see that every L with two sariables must fall islo oec of the following four cases: Case 1 The LP has a unique optional solution. Case 2 The L.P has alternative of meltiple optimal solutions: Two or more estrethe points are eptimul, and the L. will have an infinite number of optimal solutions Case 3 The L.P is infasible. The feasible region contains no points. Case 4 The LP is unbounded There are points in the feasible region with arbutrarily farge 2yalues (max problem) or arbitrarily small zyalues (min problcm) In Chaper 4, wo show that every L.P (not just L.Ps, with two varables) mait fall into one of Canes I 4 6 For an LP in standard form with constraints Ax=b and x0, show that d is a direction of unboundedness if and 4. 4 Why Does an LP Hare an Optitial bis? 139 only if Ad=0 and d0 7 Recall that Example 5 of Chapier 3 is an unbounded L. Find a direction of unboundedness along which we can move for which the objective function becomes arbitranily large. Graphically solve the following LP: naxz=2r1x2s.t.x1x212x1+x26x1,x20 From Figure 7, we see that (19) is satisfied by all points on or above AB(AB is the line x1x2=1). Also, (20) is satisfied by all points on or above CD(CD is 2r1+x2=6). Thas, the feasible region for Example 5 is tbe (ahaded) unbounded region in Figure 7 , which is bounded only by the x2 axis, line segment DE, and the part of line AB beginning at E. To find the optimal solution, we draw the isoprofit line passing through (2, 0). Thi isoprofit lioe has z=2r1x2=2(2)0=4. The direction of incereasing t is to the southeast (this nakes x1 larger and x2 snaller). Moving parallel to z=2r1=x2 in a southeast direction, we see that any ispprofit line we draw will intersect the feasible re: gion. (This is becouse any isoprofit line is steeper than the line x1x2=1 1) Thus, there are points in the feasble region that have arbitrarily large s-valoes. For ex. ample, if we wanted to find a poiat in the feasible region that had z1,000,000, we could choose any point in the feasible restion that is soutbeast of the isoprofit line 281,000,000. From the dicession in the iast two secbons, we see that every L with two sariables must fall islo oec of the following four cases: Case 1 The LP has a unique optional solution. Case 2 The L.P has alternative of meltiple optimal solutions: Two or more estrethe points are eptimul, and the L. will have an infinite number of optimal solutions Case 3 The L.P is infasible. The feasible region contains no points. Case 4 The LP is unbounded There are points in the feasible region with arbutrarily farge 2yalues (max problem) or arbitrarily small zyalues (min problcm) In Chaper 4, wo show that every L.P (not just L.Ps, with two varables) mait fall into one of Canes I 4