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2. Consider the linear programm (a) Fill in the initial tableau below in order to start the Big-M Method tableau by performing one pivot operation. (6) The first tableau below is the tableau just before the optimal tableau, and the second one oorresponds to the optimal tableau. Fill in the missing entries for the second one. 1 7 56 M15 25 01 3/2 2 0 0 1/2 0 15/2 #310 0 5/2-1 o 1-1/2 0133/2 a1 a rhs
(i) Exhibit an optimal solution to the dual problem. i) Write the dual problem, and show that the solution in (i) really is a feasible solution to (ii) Explain to a skeptical boss why the solution in (i) is an optimal solution to the dual it. problem.
.)-coefficient of excess variable in BV row o If we know BV, B , and the or part of the simplex tablcan for is the the currees store on any pivot is (1)-(7) can be used to on ised siemplex algorth (e) Suppose that two new variables, r4. rs, are added to the original linear program. The variable ra has the objective coefficient 100 and tableau in part (b) so that it includes the new variables z4 and rs. Is the resulting tableau optimal? If not, how do you restore the column vector (2.1.3). Similarly, The rs has the objective coefficient 50 and the column vector (1,5.2). Update the optimal restore optimality, but indicate the first step in this procedure) optimality? (Do not go through the actual steps to rhs 3666/5 267/5 2 13/5 1/5 3/5 1/5 87/5 33/5 (i) Suppose that ri in the original linear program is urs. How does the dual problem you found in (c) change? (i) Suppose that the original linear program is a minimization problem instead of maximiza- tion. What is the second constraint in the dual problem now? (ii) Suppose that we add the constraint 1-2r50 to the original linear program. What is new first constraint in the dual problem now?
M +cayB ucofficient of artificial variable a, in BV row (in a max problem) cavB-b right-hand side of BV row 0 e any know BV B , and the original tableau, formulas (1)-(7) enable us to lex tableau for any set of basic variables -form the simple BV. This means that if a com- x algo rithm, then all the computer hasic variables, B-, and the initial tableau. Then implex tableau. This idea is the basis (i) Calculate the allowable increase and decrease of each of the rhs values. (ii) Calculate the allowable increase and decrease of the objective coeficients of ri. 72 and 73. ve coefficients of x1, 2 and
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