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# 数学代写|优化和运筹学代写OPERATIONS RESEARCH代写|MAP4231 AllowABLE RANGE OF RHS VALUES (b)

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## 数学代写|优化和运筹学代写OPERATIONS RESEARCH代写|AllowABLE RANGE OF RHS VALUES (b)

The following illustration explains the calculation of change in $\mathrm{b}{3}\left(\Delta \mathrm{b}{3}\right)$ as contribution to profit function with marginal increase in capacity of retailer to cater to demand (constraint 2). Non-basic variable associated with this constraint is $s_{2}$ so in following illustration coefficients of $s_{2}$ have been used.

To understand the range over which $\mathrm{b}_{2}$ can be allowed, the following method should be used.

$$1.12+\Delta \mathrm{b}{2} * 0.16=1.3$$ $$3.46-\Delta \mathrm{b}{2} * 1.24=2.18$$
For feasibility of solution, the RHS values need to be greater than or equal to zero. Thus,

• $1.06-\Delta \mathrm{b}{2} * 0.07 \geq 0 \Delta \mathrm{b}{2} \leq 15.14$
• $1.12+\Delta \mathrm{b}{2} * 0.16 \geq 0 \Delta \mathrm{b}{2} \geq-7$
• $3.46-\Delta \mathrm{b}{2} * 1.24 \geq 0 \Delta \mathrm{b}{2} \leq 2.79$
Calculation of above three equations helps to decide lower and upper bound of ‘ $b_{i}$ ‘ values. It was found that ‘ $b$, can take minimum of $-7$ and maximum of $15.14$. So, the range of ‘ $b_{i}$ ‘ values was found to be:
$$-7 \leq \Delta b_{2} \leq 15.14$$
The initial RHS value of constraint 2 was maximum of 10 units. By adding this 10 into minimum and maximum values of $\Delta b_{3}$ range of feasibility can be found. Thus,
$$\begin{gathered} 10-7 \leq b_{2} \leq 15.14+10 \ 3 \leq b_{2} \leq 25.14 \end{gathered}$$
So, a solution would remain feasible if two retail stores of two geographic areas have the capacity to cater to the demand to the population within the range of minimum value $3^{*} 1,000=3,000$ and maximum value $25.42 * 1,000=25,420$ households.
To validate the range of $b_{2}$, let’s solve the question by increasing $b_{2}$ to 30 , i.e. capacity has been increase to cater to a population of 30,000 households. The following is the final simplex table (Table 4.4) for such scenario.

## 数学代写|优化和运筹学代写OPERATIONS RESEARCH代写|CHANGE IN ObJECTIVE FunCtion Coefficient (NON-BASIC VARIABLE)

In this section, sensitivity analysis technique is applied to understand the impact on optimality of initial solution with change in coefficient of non-basic variable in the objective function. Every objective function consists of certain decision variables denoted by $\mathrm{x}{\mathrm{j}}$ (where $\mathrm{j}=1,2,3 \ldots$.). These are multiplied by some certain constants indicating per unit profit or cost termed as coefficients denoted by $\mathrm{c}{\mathrm{j}}$. We have seen in solutions of LPP by simplex method that either all or some of decision variables are basic variables. The remaining variables are considered in the non-basic variables category. What happens to initial optimal solution when the coefficient of one of these non-basic variables is changed? This section deals with estimating the range of values of such coefficient within which the solution remains optimal.

## 数学代写|优化和运筹学代写OPERATIONS RESEARCH代 写|AllowABLE RANGE OF RHS VALUES (b)

$$\begin{gathered} 1.12+\Delta \mathrm{b} 2 * 0.16=1.3 \ 3.46-\Delta \mathrm{b} 2 * 1.24=2.18 \end{gathered}$$

$1.06-\Delta \mathrm{b} 2 * 0.07 \geq 0 \Delta \mathrm{b} 2 \leq 15.14$

$1.12+\Delta \mathrm{b} 2 * 0.16 \geq 0 \Delta \mathrm{b} 2 \geq-7$

$3.46-\Delta \mathrm{b} 2 * 1.24 \geq 0 \Delta \mathrm{b} 2 \leq 2.79$

$$-7 \leq \Delta b_{2} \leq 15.14$$

$$10-7 \leq b_{2} \leq 15.14+103 \leq b_{2} \leq 25.14$$

## MATLAB代写

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