7 The following questions refer to the Giapetto problem (Section 3.1). Giapetto's LP was max z = 3x, + 2x2 2x, + x2 < 100 X, + x2 < 80 < 40 (Finishing constraint) (Carpentry constraint) (Limited demand for soldiers) st. (x, = soldiers and x2 = trains). After adding slack variables S1, 82, and s3, the optimal tableau is as shown in Table 12. Use this optimal tableau to answer the following questions: a Show that as long as soldiers (x1) contribute between $2 and $4 to profit, the current basis remains optimal. If soldiers contribute $3.50 to profit, find the new optimal solution to the Giapetto problem. b Show that as long as trains (x2) contribute between $1.50 and $3.00 to profit, the current basis remains optimal. c Show that if between 80 and 120 finishing hours are available, the current basis remains optimal. Find the new optimal solution to the Giapetto problem if 90 fin- ishing hours are available. d Show that as long as the demand for soldiers is at least 20, the current basis remains optimal. e Giapetto is considering manufacturing toy boats. A toy boat uses 2 carpentry hours and 1 finishing hour. Demand for toy boats is unlimited. If a toy boat contributes $3.50 to profit, should Giapetto manufacture any toy boats? TABLE 12 Basic Variable rhs 1 1 180 z = 180 1 -1 20 X1 = 20 1 -1 60 X2 = 60 -1 20 S3 = 20

6.6. Consider the following linear programming problem. maximize z = 3x + 10x₂ + 5x3 + 11x4 + 6x5 + 14x6 subject to x₁ + 7x₂ + 3x3 + 4x4 + 2x5 + 5x6 (a) Write the dual problem. (b) Solve the dual problem by inspection. = 42 x; ≥ 0, for all j

QUESTION 15 Calculate the minimum value of 3x + 6y subject to the following constraints: x+y22 x + 2y ≤ 5 x+y≤6 x ≥ 0 y≥0

6.2.2