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# 数学代考|运筹学代写Operations Research代考|OPR561 illustRation of duality

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## 数学代考|运筹学代写Operations Research代考|illustRation of duality

Maximize $\mathrm{Z}=2 \mathrm{x}{1}+6 \mathrm{x}{2}+5 \mathrm{x}{3}$ Subject to: $$\begin{gathered} \mathrm{x}{1}+2 \mathrm{x}{2}+2 \mathrm{x}{3} \leq 10 \ 3 \mathrm{x}{1}+3 \mathrm{x}{2}+2 \mathrm{x}{3} \leq 12 \text { with } \mathrm{x}{1}, \mathrm{x}{2} \geq 0 \end{gathered}$$ i. Construct the dual form of given primal problem. ii. Solve only primal by simplex method and identify basic solution at each iteration. Also, identify basic and non-basic variables of dual problem. Solution: i. Dual form would be: Minimize $\mathrm{Z}^{*}=10 \mathrm{y}{1}+12 \mathrm{y}{2}$ Subject to: $$\begin{gathered} 1 \mathrm{y}{1}+3 \mathrm{y}{2} \geq 2 \ 2 \mathrm{y}{1}+3 \mathrm{y}{2} \geq 6 \ 2 \mathrm{y}{1}+2 \mathrm{y}{2} \geq 5 \text { with } \mathrm{y}{1}, \mathrm{y}{2} \geq 0 \end{gathered}$$ ii. Following would be initial models of both primal and dual: Maximize $\mathrm{Z}=2 \mathrm{x}{1}+6 \mathrm{x}{2}+5 \mathrm{x}{3}+0 \mathrm{~s}{1}+0 \mathrm{~s}{2}$
Subject to:
$$\begin{gathered} x_{1}+2 x_{2}+2 x_{3}+1 s_{1}+0 s_{2}=10 \ 3 x_{1}+3 x_{2}+2 x_{3}+0 s_{1}+1 s_{2}=12 \text { with } x_{1}, x_{2} \geq 0 \end{gathered}$$
where $s_{1}, s_{2}$ are slack variables.

## 数学代考|运筹学代写Operations Research代考|SUMMARY

This chapter discussed concepts of sensitivity analysis and duality by illustrating linear programming problems. It was inferred that coefficients and parameters used in formulation of a linear programming model are only estimates, and it is important to apply sensitivity analysis to identify which of these are sensitive and which are insensitive to change. Sensitivity analysis was performed by finding an allowable feasible range of RHS values, which indicate resource availability and also a range of optimality was identified for objective function coefficients. It is important to note that in this chapter such deduction was done by changing only one coefficient, keeping all others as unchanged, though limited analysis is possible by adopting multiple changes.

Another section was devoted to understanding the concept of duality that was helpful in implementing and interpreting sensitivity analysis. A mirror image called dual can be constructed of original problem that is complementary to primal. Solving any one would provide the solution for the other. The benefit of creating and solving dual is that any variation in solution identifies economic contribution by variable that has caused such variation. The formulation of dual problem and its relationship with primal was discussed in detail.

## 数学代考|运筹学代写Operations Research代考|illustRation of duality

$$\mathrm{x} 1+2 \mathrm{x} 2+2 \mathrm{x} 3 \leq 103 \mathrm{x} 1+3 \mathrm{x} 2+2 \mathrm{x} 3 \leq 12 \text { with } \mathrm{x} 1, \mathrm{x} 2 \geq 0$$

$$1 \mathrm{y} 1+3 \mathrm{y} 2 \geq 22 \mathrm{y} 1+3 \mathrm{y} 2 \geq 62 \mathrm{y} 1+2 \mathrm{y} 2 \geq 5 \text { with } \mathrm{y} 1, \mathrm{y} 2 \geq 0$$
ii. 以下是原始模型和对偶模型的初始模型: 最大化 $\mathrm{Z}=2 \mathrm{x} 1+6 \mathrm{x} 2+5 \mathrm{x} 3+0 \mathrm{~s} 1+0 \mathrm{~s} 2$ 受制于:
$$x_{1}+2 x_{2}+2 x_{3}+1 s_{1}+0 s_{2}=103 x_{1}+3 x_{2}+2 x_{3}+0 s_{1}+1 s_{2}=12 \text { with } x_{1}, x_{2} \geq 0$$

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