110年 - 110 國立臺灣大學_碩士班招生考試_工業工程學研究所系統工程組:作業研究#101830

(a) (5 points) Let us introduce the slack variables x₃, x₄, x₅, x₆ to constraints 1, 2, 3, and 4, respectively so X1,1220 that the objective function and equations, which are called dictionary, can be written as

We let x₃,x₄,x₅,X₆ in the above dictionary be basic variables, and x₁and x₂ be the nonbasic variables. What is the initial feasible solution (x₁, X₂,X₃,X₄.X₅, X₆.Z) based on the above dictionary in (a) if both nonbasic variables are set as zero?

(c) (5 points) In (a), the current basis is composed of x₃,x₄,X₅, X₆. Suppose x1 enters the basis and x₃ lcaves the basis. What is the dictionary at the next iteration? In other words, please complete the followings.
  

(e) (5 points) What is the optimal dictionary of this probiem? In other words, please complete the followings in the final dictionary.

(b) (10 points) Someone tells us that the optimal solution is . Use the complementary slackness to obtain the value of dual variables.

4.(10 points) When one person is infected with c-virus and interacts with another one who is healthy, there is a probabilityp = 0.2 to get the healthy one infected. Two people in the community are selected randomly at any single period of time and interact with each other. Let x_n be the number of infective people at time n and there are five people in the community. Given x₀ = 1, what is the mean time to infect everyone in the community?

(b) (10 points) Find and demonstrate a better ordering policy than the one Marie uses, ie., the policy with a lower cost.