Please help me solve this linear programming problem (d) and (e). I have solved (a)(b)(c) and had some thoughts on (d) but not so sure. I attached my solution to (a)(b)(c)(d). Please help me check (d) and give me the solution of (e). Very urgent. Thank you!
Poo we have V We pick the smaller value and get the allowable range for X, Therefore, the range of the optimal value would be 5201 2 - 600-015X12 ~ 5/7,78 Therefore, the allowable range of P, is [571, 78 , 600] (e) keep(a ) Ducil Min 400 4 + 400 4/z s . t . : 3 y , t by > > 4 491 + 24 2 2 6 8 4 1+ 5y, 2 10 by, + 842 79 4, , 1 270 CS conditions for primal : ( 3 X 1 + 4 x 2+ 8 X3 + 6xx-400)y, =0 ( 6 X 1 + 2 X > + 5x3 +8X4-400) 1/2 =0 CS conditions for dual ( 34 1 + 6 4 2 - 4 ) X , = 0 (491+ 242 - 6 ) X 2 20 (8 4 1 + 54 2 - 10 ) 23 = 0 ( by , + 84 2 - 9) X4 = 0 (b ) There are four variables and two constraints in the primal LP. thus there must be two basis variables and two non-basis variables . Off B= Ex, ,x 23. N= Sx3, xx]. then x170, X270 According to CS conditions , two constraints ane binding 53417 64 2- 4= 0 sy1 = ] L 49, 7 24/ 2-6 = 0 so this solution is not feasibleThen the objective value of dual is too x z (c ) In summery , we can see for the three feasible corner , the objective value of dual based on B =Ex 2. x4 is the smallest. Therefore, the optimal solution to the dual is sy x - 2 ( c ) From part ( b) , we get the optimal value should be 400. ( y * + 42* ) = 600 If X3 = 15 , then we can look at the opened, Z= 600 -0.5x1 - 23 - libs, The reduced cast of X3 is -2 . Therefore , if As increases from 0 to 15 , the change of optimal value would be Jobj = - 2 x 15 = 130 That is , the investor lose $3000 in potential profit. ( d ) keep X3 = X4 = SI zo , then the optimal dictionary X2 = 100- 0175x1 20 52 = 200 - 4:5%, 20 8 - 600 - 0,5x , because X, and Ss should stay feasible. Then2. (14 points) An investor has a choice of four potential investments, named P1, P2, P3, and P4. The returns on these investments are, respectively, {4,6, 10, 9) percent respectively. Each of these investments, however, causes some harm to society, and the investor would like to pick an investment tha.this harm to a reasonable amount. One difculty is that the \"cost\" to society is either the vector (0.03, 0.04, 0080.06) or the vector (0.06, 0.02, 0.05, 0.08), depending on the realization of some uncertainty. The investor is conscious of tmike to limit it to at most $4, regardless of the scenario that realizes. Letting :2,- denote the amount (i-f dollars) invested in asset i, the investor solves the following linear programming problem: Max 4:161 63:2 10m3+9x4 subject to: 3E1 43:2 8333 5334 g 400 6331 2332 5333 8334 S 400 m1.m2,m3.m4 Z 0 Introducing slack variables 31 and 32 for the two constraints, and solving using the simplex method, we end up with the following optimal dictionary: 1'12 2 100 0.75331 2:333 1.5334 0.2531 200 4.51:1 333 5174 + 0.581 600 0.51:1 2563 1.531 M 32 Z W Write down the dual of the given linear programming problem and the CS conditions. 3% sing the CS conditions or otherwise, write down an optimal solution to the dual. If management insists that $1500 be invested in P3, how much does the investor lose in potential prot? \\5 (d) What is the range of the prot margin of P1 for which the given basis{$2,32}remains optimal? fe) What is the range of the prot margin of P2 for which the given basis remains optimal
