GE330_lect8 - Lecture 8 Algebraic Sensitivity Analysis...

Info icon This preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon
Lecture 8 Algebraic Sensitivity Analysis February 17, 2009
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Algebraical Sensitivity Analysis TOYCO Model: TOYCO assembles three types of toys–trains, trucks and cars–using three operations. The daily limits on the available times for the three operations are 430,460, and 420 minutes, respectively, and the revenues per unit of toy train, truck, and car are $3, $2, and $5, respectively. The assembly times per train at the three operations are 1, 3, 1 minutes, re- spectively. The corresponding times per train and per car are (2,0,4) and (1,2,0) minutes (a zero time indicates that the op- eration is not used.) LP Model: Let x 1 , x 2 , and x 3 be the daily number of units assembled of trains, trucks, and cars, respectively. max z = 3 x 1 + 2 x 2 + 5 x 3 s.t. 2 x 1 + x 2 + x 3 430 (Operation 1) x 1 + 2 x 3 460 (Operation 2) x 1 + 4 x 2 420 (Operation 3) x 1 , x 2 , x 3 0 9
Image of page 2
TOYCO Model: Optimal Tableau Introduce slack variables x 4 , x 5 , and x 6 , the initial tableau is: Basic x 1 x 2 x 3 x 4 x 5 x 6 Solution z -3 -2 -5 0 0 0 0 x 4 1 2 1 1 0 0 430 x 5 3 0 2 0 1 0 460 x 6 1 4 0 0 0 1 420 The optimal tableau is: Basic x 1 x 2 x 3 x 4 x 5 x 6 Solution z 4 0 0 1 2 0 1350 x 2 - 1 4 1 0 1 2 - 1 4 0 100 x 3 3 2 0 1 0 1 2 0 230 x 6 2 0 0 -2 1 1 20 10
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Changes in the Right-Hand Side After adding slack variables, rewrite the constraints as follows: x 1 + 2 x 2 + x 3 = 430 - x 4 (Operation 1) 3 x 1 + 2 x 3 = 460 - x 5 (Operation 2) x 1 + 4 x 2 = 420 - x 6 (Operation 3) We can say that a one-unit decrease in the slack variables is equivalent to a one-unit increase in the resource (operations time). On the other hand, from the optimal tableau we know that: z + 4 x 1 + x 4 + 2 x 5 + 0 x 6 = 1350 , which is equivalent to z = 1350 - 4 x 1 + 1 × (increase in operation 1 time) + 2 × (increase in operation 2 time) + 0 × (increase in operation 3 time) Note: 1. In this case, the shadow price for a resource is actually the corresponding z -row coefficient in the optimal tableau! 2. The shadow price of operation 3 is zero, this is reasonable because this resource is already abundant (the slack is positive). 11
Image of page 4
Determine the Feasibility Ranges The idea: Find the range of the changes in the right-hand side so that the basis of the optimal solution remains the same, i.e., x 2 , x 3 , and x 6 are still the basic variables in the optimal tableau after the changes.
Image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 6
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern