数学代写|运筹学代写Operations Research代考|KMA255 SIMPLEX METHOD SOLVES BOTH THE PRIMAL AND THE DUAL
数学代写|运筹学代写Operations Research代考|SIMPLEX METHOD SOLVES BOTH THE PRIMAL AND THE DUAL
Let us explain this using the formulation in Illustration 1.1. The simplex table is shown in the usual notation in Table 3.3.
In the optimal tableau, let us observe the $C_j-Z_j$ values. These are $0,0,-3$ and $-1$ for variables $X_1, X_2, u_1$ and $u_2$. We also know (from complimentary slackness conditions) that there is a relationship between $X_j$ and $v_j$ and between $u_j$ and $Y_j$.
The values of $C_j-Z_j$ corresponding to $X_j$ are the negatives of the values of dual slack variables $v_j$ and the values of $C_j-Z_j$ corresponding to $u_j$ are the negatives of the values of dual decision variables $Y_j$. (We will see the algebraic explanation of this subsequently in this chapter.) Therefore, $Y_1^=3, Y_2^=1, v_1^=0, v_2^=0$. We also know from the complimentary slackness conditions that if $X_j$ is basic then $v_j=0$. This is also true because when $X_j$ is basic, $C_j-Z_j$ is zero. When $u_j$ is non-basic and has zero value, its $C_j-Z_j$ is negative at optimum indicating a nonnegative value of the basic variable $Y_j$.
The optimum solution to the dual can be read from the optimum tableau of the primal in the simplex algorithm. We need not solve the dual explicitly.
Let us look at an intermediate iteration (say with basic variable $u_1$ and $X_1$ ). The basic feasible solution is $u_1=1, X_1=4$ with non-basic variables $X_2=0$ and $u_2=0$. When we apply the above rule (and complimentary slackness conditions), we get the corresponding dual solution to be $Y_1=0, Y_2$ $=2, v_1=0, v_2=-1$ with $W=24$ (same value of $Z$ ).
This solution is infeasible to the dual because variable $v_2$ takes a negative value. This means that the second dual constraint $Y_1+2 Y_2 \geq 5$ is not feasible making $v_2=-1$. The value of $v_2$ is the extent of infeasibility of the dual, which is the rate at which the objective function can increase by entering the corresponding primal variable. A non-optimal basic feasible solution to the primal results in an infeasible dual when complimentary slackness conditions are applied. At the optimum, when complimentary slackness conditions are applied, the resultant solution is also feasible and hence optimal.
Consider the linear programming problem given by ILLUSTRATION $3.4$ Minimize $Z=4 X_1+7 X_2$ Subject to $$ \begin{array}{r} 2 X_1+3 X_2 \geq 5 \ X_1+7 X_2 \geq 9 \ X_1, X_2 \geq 0 \end{array} $$ Normally, we would have added two artificial variables $a_1$ and $a_2$ to get an initial basic feasible solution. We do not add these now but write the constraints as equations with slack variables only. The equations are written with a negative RHS (something that the simplex method does not approve). We also convert it as a maximization problem by multiplying the objective function with $-1$. The problem becomes Maximize $Z=-4 X_1-7 X_2-0 X_3-0 X_4$ Subject to $$ \begin{array}{r} 2 X_1+3 X_2-X_3=5 \ X_1+7 X_2-X_4=9 \ X_1, X_2 \geq 0 \end{array} $$ We set up the simplex table as shown in Table $3.4$ with slack variables $X_3$ and $X_4$ as basic variables and with a negative RHS.
数学代写|运筹学代写Operations Research代考|KMA255 SIMPLEX METHOD SOLVES BOTH THE PRIMAL AND THE DUAL
现代博弈论始于约翰-冯-诺伊曼（John von Neumann）提出的两人零和博弈中的混合策略均衡的观点及其证明。冯-诺依曼的原始证明使用了关于连续映射到紧凑凸集的布劳威尔定点定理，这成为博弈论和数学经济学的标准方法。在他的论文之后，1944年，他与奥斯卡-莫根斯特恩（Oskar Morgenstern）共同撰写了《游戏和经济行为理论》一书，该书考虑了几个参与者的合作游戏。这本书的第二版提供了预期效用的公理理论，使数理统计学家和经济学家能够处理不确定性下的决策。
