give answer in 2 step with explanation at the end of each step and final answer at the end: A,B AND C.

Maximize
Z
=
-
5
x
1

5
x
2

1
3
x
3
subject to:
-
1
x
1

1
x
2

Question

give answer in 2 step with explanation at the end of each step and final answer at the end: A,B AND C.


Maximize 
Z
=
-
5
x
1
 
5
x
2
 
1
3
x
3
subject to:
-
1
x
1
 
1
x
2
 
3
x
3
≤
2
0
1
2
x
1
 
4
x
2
 
1
0
x
3
≤
9
0
x
1
≥
0
,
x
2
≥
0
,
x
3
≥
0
a
.
 ﻿Construct the augmented form of this LP by introducing slack variables. 
(
5
 ﻿pts
)
b
.
 ﻿The final simplex tableau for the above LP is given below.
\
table
[
[
Basic Variable,Eq
,
Coefficient of:
,
Right Side
]
,
[
Z
,
x
1
,
x
2
,
x
3
,
x
4
,
x
5Maximize Z=-5xi+5%:+ 13%
subject to:
Axitixr 3x S20
12x + 4x4 10x $90
x120,%20,%20.
a. Construct the augmented form of this LP by inroducing slack variables. (5 pts)
b.. The final simplex tableau for the above LP is given below.
Basic Coefficient of: Right
Variable - — RE Sido
Ea |Z [x |u| x | ou | x
z © 1 0 0 2s 0 | 100
wo | mo [oo | a 1 3 1 o | =»
2
x |[@ | ow o 2 4 1 |
What is the optima solution and optimal value? (10 points)
Now suppose that the right-hand side of the second constraint is changed from b = 9010 b;
= 70. Using the sensitivity analysis procedure covered in

VIPSolutions · Accepted Answer

Let's solve the question step by step as requested.

---

### **Step A: Construct the Augmented Form with Slack Variables**

**Original Problem:**
Maximize $ Z = -5x_1 + 5x_2 + 13x_3 $

Subject to:
1. $ -x_1 + x_2 + 3x_3 \leq 20 $
2. $ 12x_1 + 4x_2 + 10x_3 \leq 90 $
3. $ x_1, x_2, x_3 \geq 0 $

**Introducing Slack Variables:**
- Let $ x_4 $ be the slack variable for the first constraint.
- Let $ x_5 $ be the slack variable for the second constraint.

**Augmented System:**
$$
\begin{align*}
\text{Maximize:} \quad & Z = -5x_1 + 5x_2 + 13x_3 \
\text{Subject to:} \quad & -x_1 + x_2 + 3x_3 + x_4 = 20 \
& 12x_1 + 4x_2 + 10x_3 + x_5 = 90 \
& x_1, x_2, x_3, x_4, x_5 \geq 0
\end{align*}
$$

**Explanation:**  
We add slack variables to turn inequalities into equalities, enabling us to use the simplex method.

---

### **Step B: Find the Optimal Solution and Value from the Final Tableau**

Given Final Simplex Tableau:

| Basic Variable | Eq | Z | $x_1$ | $x_2$ | $x_3$ | $x_4$ | $x_5$ | Right Side |
|----------------|----|---|---------|---------|---------|---------|---------|------------|
| Z              | 0  | 1 | 0       | 2       | 5       | 0       | 100     |
| $x_2$        | 1  | 0 | -1      | 1       | 3       | 1       | 0       | 20         |
| $x_5$        | 2  | 0 | 16      | 0       | -2      | -4      | 1       | 10         |

**Step 1: Read Solution from Tableau**

- The basic variables are $x_2$ and $x_5$.
- Non-basic variables (set to 0): $x_1 = 0$, $x_3 = 0$, $x_4 = 0$
- $x_2 = 20$ (from row 2, right side)
- $x_5 = 10$ (from row 3, right side)

**Step 2: Compute the Optimal Value**

Plug values back into the original objective function:
$$
Z = -5x_1 + 5x_2 + 13x_3 = -5(0) + 5(20) + 13(0) = 100
$$

**Explanation:**  
The optimal solution is where non-basic variables are zero and the basic variables take the right-side values from the tableau.

---

### **Final Answers**

**a. Augmented LP:**
$$
\text{Maximize } Z = -5x_1 + 5x_2 + 13x_3
$$
$$
\begin{align*}
-x_1 + x_2 + 3x_3 + x_4 &= 20 \
12x_1 + 4x_2 + 10x_3 + x_5 &= 90 \
x_1, x_2, x_3, x_4, x_5 &\geq 0
\end{align*}
$$

**b. Optimal Solution:**
$$
x_1 = 0, \quad x_2 = 20, \quad x_3 = 0, \quad x_4 = 0, \quad x_5 = 10
$$
**Optimal Value:** $\boxed{Z = 100}$

---

Let me know if you'd like to proceed with part (c)!

VIPSolutions · Answer

**Part A: Construct the augmented LP with slack variables**

Original LP:
$$
\text{Maximize } Z = -5x_1 + 5x_2 + 13x_3
$$
Subject to:
$$
\begin{cases}
 -x_1 + x_2 + 3x_3 \leq 20 \
 12x_1 + 4x_2 + 10x_3 \leq 90 \
 x_1, x_2, x_3 \geq 0
\end{cases}
$$

**Introduce slack variables $x_4, x_5 $:**
$$
\begin{cases}
 -x_1 + x_2 + 3x_3 + x_4 = 20 \
 12x_1 + 4x_2 + 10x_3 + x_5 = 90 \
 x_1, x_2, x_3, x_4, x_5 \geq 0
\end{cases}
$$

**Explanation:**  
Adding slack variables converts inequalities into equalities, forming the augmented system suitable for the simplex method.

---

**Part B: Find the optimal solution and optimal value**

Given the final simplex tableau:

| Basic Variable | Eq | $Z$ | $x_1$ | $x_2$ | $x_3$ | $x_4$ | $x_5$ | Right Side |
|------------------|-----|-------|---------|---------|---------|---------|---------|--------------|
| $Z$           |     | 1     | 0       | 2       | 5       | 0       | 0       | 100          |
| $x_2$       | 1   | 0     | -1      | 1       | 3       | 1       | 0       | 20           |
| $x_5$       | 2   | 0     | 16      | 0       | -2      | 0       | 1       | 10           |

**Step 1: Determine basic variables and solution:**

- Basic variables are $x_2$ and $x_5$.  
- Non-basic variables $x_1, x_3, x_4 = 0$.  
- From the tableau:  
  $$
  x_2 = 20, \quad x_5 = 10
  $$

- Values of non-basic variables:
  $$
  x_1 = 0, \quad x_3 = 0, \quad x_4 = 0
  $$

**Step 2: Compute the optimal value:**

$$
Z = -5x_1 + 5x_2 + 13x_3 = -5(0) + 5(20) + 13(0) = 100
$$

**Final Answer:**

$$
\boxed{
\text{Optimal solution:} \quad x_1=0, \; x_2=20, \; x_3=0, \; x_4=0, \; x_5=10
}
$$

$$
\boxed{
\text{Maximum value of } Z = \mathbf{100}
}
$$

---

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