Question

Consider the following problem. Maximize Z= cixi + c2x2 + C3X3 subject to x1 + 2x2 + x3 = b 2x1 + x2 + 3x3 = 2b and x 20, X220, X2 > 0. Note that values have not been assigned to the coefficients in the objective function (C1, C2, C3). and that the only specification for the right-hand side of the functional constraints is that the second one (2b) be twice as large as the first (b). Now suppose that your boss has inserted her best estimate of the values of C1, C2, C3, and b without informing you and then has run the simplex method. You are given the resulting final simplex tableau below (where x4 and X5 are the slack variables for the respective functional constraints), but you are unable to read the value of Z". Coefficient of: Basic Variable Right Side Eq. X1 X2 X3 X4 X5 @ NUMA @ @ (a) Use the fundamental insight presented in Sec. 5.3 to identify the value of (C1, C2, C3) that was used. (b) Use the fundamental insight presented in Sec. 5.3 to identify the value of b that was used. (c) Calculate the value of Z* in two ways, where one way uses your results from part (a) and the other way uses your result from part (b). Show your two methods for finding Z*.

This is question 5.3-5 from Introduction to Operations Research (Hillier). Relevant text:

Answer 1

solution a) we know that 2 - CB Yg - co where Cg = cost asociated with bagpo vorable co - cost of vooriable in function Yo - Ith columo in simplex table Now for Sale 2 - I - CB4, -C, 163] [ 15 ]-ci 1 - 2 + 3 C3-C, 26₂ +6C3-106, -7 - (1) for or 3=4 au 3] L -cy Cụ - 5g - . 322 - C₂ -3 - (2) for 3-526-4-[(4 53 ] [ 2016 ]-cs 4 1 2 12 + 9 = 23 263 - C₂ = 4 - (3)

From (2) 6 and (3) - 263 = 6 az - Ca = 4 Lt 5₂ = 10 using in (2) Cz- 3(2) -3 -3. using C, C in 4) 21a) + 613) - 60C, 27 10c, = 4+18 7 =15 c = 3/2 Hence, c, 23, Ca - 2, C3 23 we know that XB & Bb 3 3154157 L-15 215 -1915 1957b7 L-lls 215]Lab Lb+46

c) By pont a с. - 3), C = 8, Ca -3 WKT Рcxp) - СвХв 3* - [ 3 30 31 349 =]] WKT according to fundmental insight ах - Ухь : Вы (я з 1 1 (2 31181 1 3+8 2 | Hence we get at in two ways Thank you