EMIS 8374 The Ratio Test 1 2 3
![EMIS 8374 The Ratio Test EMIS 8374 The Ratio Test](https://slidetodoc.com/presentation_image/dbdeda01720dbb0732aedd9014771fc1/image-1.jpg)
![1. 2. 3. 4. 5. 6. Main Steps of the Simplex Method Put the 1. 2. 3. 4. 5. 6. Main Steps of the Simplex Method Put the](https://slidetodoc.com/presentation_image/dbdeda01720dbb0732aedd9014771fc1/image-2.jpg)
![Example 1: Step 1 Original LP Maximize 4. 5 x 1 + 4 x Example 1: Step 1 Original LP Maximize 4. 5 x 1 + 4 x](https://slidetodoc.com/presentation_image/dbdeda01720dbb0732aedd9014771fc1/image-3.jpg)
![Example 1: Steps 2 and 3 Initial BFS: BV = {z, x 3, x Example 1: Steps 2 and 3 Initial BFS: BV = {z, x 3, x](https://slidetodoc.com/presentation_image/dbdeda01720dbb0732aedd9014771fc1/image-4.jpg)
![Example 1: Steps 4 and 5 x 1 and x 2 are eligible to Example 1: Steps 4 and 5 x 1 and x 2 are eligible to](https://slidetodoc.com/presentation_image/dbdeda01720dbb0732aedd9014771fc1/image-5.jpg)
![Example 1: Step 6 • How much can we increase x 1? • Constraint Example 1: Step 6 • How much can we increase x 1? • Constraint](https://slidetodoc.com/presentation_image/dbdeda01720dbb0732aedd9014771fc1/image-6.jpg)
![Example 1: Step 6 • How much can we increase x 1? • Constraint Example 1: Step 6 • How much can we increase x 1? • Constraint](https://slidetodoc.com/presentation_image/dbdeda01720dbb0732aedd9014771fc1/image-7.jpg)
![Example 1: Step 6 • • How much can we increase x 1? Constraint Example 1: Step 6 • • How much can we increase x 1? Constraint](https://slidetodoc.com/presentation_image/dbdeda01720dbb0732aedd9014771fc1/image-8.jpg)
![Example 1: Step 6 • From constraint 1, we see that we can increase Example 1: Step 6 • From constraint 1, we see that we can increase](https://slidetodoc.com/presentation_image/dbdeda01720dbb0732aedd9014771fc1/image-9.jpg)
![Step 6: The Ratio Test Row 1: 30 x 1 + 12 x 2 Step 6: The Ratio Test Row 1: 30 x 1 + 12 x 2](https://slidetodoc.com/presentation_image/dbdeda01720dbb0732aedd9014771fc1/image-10.jpg)
![Example 1: Ratio Test The minimum ratio occurs in row 1. Thus, x 3 Example 1: Ratio Test The minimum ratio occurs in row 1. Thus, x 3](https://slidetodoc.com/presentation_image/dbdeda01720dbb0732aedd9014771fc1/image-11.jpg)
![Example 1: Step 7 (Pivot) Pivot on the x 1 column of row 1 Example 1: Step 7 (Pivot) Pivot on the x 1 column of row 1](https://slidetodoc.com/presentation_image/dbdeda01720dbb0732aedd9014771fc1/image-12.jpg)
![Example 1: Step 7 (Pivot) First ERO: divide row 1 by 30 12 Example 1: Step 7 (Pivot) First ERO: divide row 1 by 30 12](https://slidetodoc.com/presentation_image/dbdeda01720dbb0732aedd9014771fc1/image-13.jpg)
![Example 1: Step 7 (Pivot) Second ERO: Add – 10 times row 1 to Example 1: Step 7 (Pivot) Second ERO: Add – 10 times row 1 to](https://slidetodoc.com/presentation_image/dbdeda01720dbb0732aedd9014771fc1/image-14.jpg)
![Example 1: Step 7 (Pivot) Third ERO: Add – 4 times row 1 to Example 1: Step 7 (Pivot) Third ERO: Add – 4 times row 1 to](https://slidetodoc.com/presentation_image/dbdeda01720dbb0732aedd9014771fc1/image-15.jpg)
![Example 1: Step 7 (Pivot) Fourth ERO: Add 4. 5 times row 1 to Example 1: Step 7 (Pivot) Fourth ERO: Add 4. 5 times row 1 to](https://slidetodoc.com/presentation_image/dbdeda01720dbb0732aedd9014771fc1/image-16.jpg)
![Example 1: Step 4 BV = {z, x 1, x 4, x 5}, NBV Example 1: Step 4 BV = {z, x 1, x 4, x 5}, NBV](https://slidetodoc.com/presentation_image/dbdeda01720dbb0732aedd9014771fc1/image-17.jpg)
![Example 1: Ratio Test The minimum ratio occurs in row 2. Thus, x 4 Example 1: Ratio Test The minimum ratio occurs in row 2. Thus, x 4](https://slidetodoc.com/presentation_image/dbdeda01720dbb0732aedd9014771fc1/image-18.jpg)
![Example 1: Pivoting x 2 into the basis BV = {z, x 1, x Example 1: Pivoting x 2 into the basis BV = {z, x 1, x](https://slidetodoc.com/presentation_image/dbdeda01720dbb0732aedd9014771fc1/image-19.jpg)
- Slides: 19
![EMIS 8374 The Ratio Test EMIS 8374 The Ratio Test](https://slidetodoc.com/presentation_image/dbdeda01720dbb0732aedd9014771fc1/image-1.jpg)
EMIS 8374 The Ratio Test
![1 2 3 4 5 6 Main Steps of the Simplex Method Put the 1. 2. 3. 4. 5. 6. Main Steps of the Simplex Method Put the](https://slidetodoc.com/presentation_image/dbdeda01720dbb0732aedd9014771fc1/image-2.jpg)
1. 2. 3. 4. 5. 6. Main Steps of the Simplex Method Put the problem in row-0 form. Construct the simplex tableau. Obtain an initial BFS. If the current BFS is optimal then goto step 9. Choose a non-basic variable to enter the basis. Use the ratio test to determine which basic variable must leave the basis. 7. Perform the pivot operation on the appropriate element of the tableau. 8. Goto Step 4. 9. Stop. 1
![Example 1 Step 1 Original LP Maximize 4 5 x 1 4 x Example 1: Step 1 Original LP Maximize 4. 5 x 1 + 4 x](https://slidetodoc.com/presentation_image/dbdeda01720dbb0732aedd9014771fc1/image-3.jpg)
Example 1: Step 1 Original LP Maximize 4. 5 x 1 + 4 x 2 s. t. 30 x 1 + 12 x 2 6000 10 x 1 + 8 x 2 2600 4 x 1 + 8 x 2 2000 x 1, x 2 0 LP in Row-0 Form Maximize z s. t. z - 4. 5 x 1 - 4 x 2 = 0 30 x 1 + 12 x 2 + x 3 = 6000 10 x 1 + 8 x 2 + x 4 = 2600 4 x 1 + 8 x 2 + x 5 = 2000 x 1, x 2, x 3, x 4, x 5 0 2
![Example 1 Steps 2 and 3 Initial BFS BV z x 3 x Example 1: Steps 2 and 3 Initial BFS: BV = {z, x 3, x](https://slidetodoc.com/presentation_image/dbdeda01720dbb0732aedd9014771fc1/image-4.jpg)
Example 1: Steps 2 and 3 Initial BFS: BV = {z, x 3, x 4, x 5}, NBV = {x 1, x 2} z = 0, x 3 = 6, 000, x 4 = 2, 600, x 5 = 2, 000 x 1 = x 2 = 0 3
![Example 1 Steps 4 and 5 x 1 and x 2 are eligible to Example 1: Steps 4 and 5 x 1 and x 2 are eligible to](https://slidetodoc.com/presentation_image/dbdeda01720dbb0732aedd9014771fc1/image-5.jpg)
Example 1: Steps 4 and 5 x 1 and x 2 are eligible to enter the basis. Select x 1 to become a basic variable 4
![Example 1 Step 6 How much can we increase x 1 Constraint Example 1: Step 6 • How much can we increase x 1? • Constraint](https://slidetodoc.com/presentation_image/dbdeda01720dbb0732aedd9014771fc1/image-6.jpg)
Example 1: Step 6 • How much can we increase x 1? • Constraint in Row 1: 30 x 1 + 12 x 2 + x 3 = 6000 => x 3 = 6000 - 30 x 1 - 12 x 2. • x 2 = 0 (it will stay non-basic) • x 3 0 • Thus x 1 200. 5
![Example 1 Step 6 How much can we increase x 1 Constraint Example 1: Step 6 • How much can we increase x 1? • Constraint](https://slidetodoc.com/presentation_image/dbdeda01720dbb0732aedd9014771fc1/image-7.jpg)
Example 1: Step 6 • How much can we increase x 1? • Constraint in Row 2: 10 x 1 + 8 x 2 + x 4 = 2600 => x 4 = 2600 - 10 x 1 - 8 x 2 • x 2 = 0 (it will stay non-basic) • x 4 0 • Thus x 1 260. 6
![Example 1 Step 6 How much can we increase x 1 Constraint Example 1: Step 6 • • How much can we increase x 1? Constraint](https://slidetodoc.com/presentation_image/dbdeda01720dbb0732aedd9014771fc1/image-8.jpg)
Example 1: Step 6 • • How much can we increase x 1? Constraint in Row 3: 4 x 1 + 8 x 2 + x 5= 2000 => x 5 = 2000 - 4 x 1 - 8 x 2 • • • x 2 = 0 (it will stay non-basic) x 5 0 Thus x 1 500. 7
![Example 1 Step 6 From constraint 1 we see that we can increase Example 1: Step 6 • From constraint 1, we see that we can increase](https://slidetodoc.com/presentation_image/dbdeda01720dbb0732aedd9014771fc1/image-9.jpg)
Example 1: Step 6 • From constraint 1, we see that we can increase x 1 up to 200, but we must reduce x 3 to zero to satisfy the constraints. • From constraint 2, we see that we can increase x 1 up to 260, but we must also reduce x 4 to zero to satisfy the constraints. • From constraint 3, we see that we can increase x 1 up to 500, but we must reduce x 5 to zero to satisfy the constraints. • Since x 3 is the limiting variable, we make it nonbasic as x 1 becomes basic. 8
![Step 6 The Ratio Test Row 1 30 x 1 12 x 2 Step 6: The Ratio Test Row 1: 30 x 1 + 12 x 2](https://slidetodoc.com/presentation_image/dbdeda01720dbb0732aedd9014771fc1/image-10.jpg)
Step 6: The Ratio Test Row 1: 30 x 1 + 12 x 2 + x 3 = 6000 => 30 x 1 + x 3 = 6000 => x 1 6000/30 = 200. Row 2: 10 x 1 + 8 x 2 + x 4 = 2600 => 10 x 1 + x 4 = 2600 => x 1 2600/10 = 260. Row 3: 4 x 1 + 8 x 2 + x 5= 2000 => 4 x 1 + x 5 = 2000 => x 1 2000/4 = 500. 9
![Example 1 Ratio Test The minimum ratio occurs in row 1 Thus x 3 Example 1: Ratio Test The minimum ratio occurs in row 1. Thus, x 3](https://slidetodoc.com/presentation_image/dbdeda01720dbb0732aedd9014771fc1/image-11.jpg)
Example 1: Ratio Test The minimum ratio occurs in row 1. Thus, x 3 leaves the basis when x 1 enters. 10
![Example 1 Step 7 Pivot Pivot on the x 1 column of row 1 Example 1: Step 7 (Pivot) Pivot on the x 1 column of row 1](https://slidetodoc.com/presentation_image/dbdeda01720dbb0732aedd9014771fc1/image-12.jpg)
Example 1: Step 7 (Pivot) Pivot on the x 1 column of row 1 to make x 1 basic and x 3 non-basic. 11
![Example 1 Step 7 Pivot First ERO divide row 1 by 30 12 Example 1: Step 7 (Pivot) First ERO: divide row 1 by 30 12](https://slidetodoc.com/presentation_image/dbdeda01720dbb0732aedd9014771fc1/image-13.jpg)
Example 1: Step 7 (Pivot) First ERO: divide row 1 by 30 12
![Example 1 Step 7 Pivot Second ERO Add 10 times row 1 to Example 1: Step 7 (Pivot) Second ERO: Add – 10 times row 1 to](https://slidetodoc.com/presentation_image/dbdeda01720dbb0732aedd9014771fc1/image-14.jpg)
Example 1: Step 7 (Pivot) Second ERO: Add – 10 times row 1 to row 2 13
![Example 1 Step 7 Pivot Third ERO Add 4 times row 1 to Example 1: Step 7 (Pivot) Third ERO: Add – 4 times row 1 to](https://slidetodoc.com/presentation_image/dbdeda01720dbb0732aedd9014771fc1/image-15.jpg)
Example 1: Step 7 (Pivot) Third ERO: Add – 4 times row 1 to row 3 14
![Example 1 Step 7 Pivot Fourth ERO Add 4 5 times row 1 to Example 1: Step 7 (Pivot) Fourth ERO: Add 4. 5 times row 1 to](https://slidetodoc.com/presentation_image/dbdeda01720dbb0732aedd9014771fc1/image-16.jpg)
Example 1: Step 7 (Pivot) Fourth ERO: Add 4. 5 times row 1 to row 0 15
![Example 1 Step 4 BV z x 1 x 4 x 5 NBV Example 1: Step 4 BV = {z, x 1, x 4, x 5}, NBV](https://slidetodoc.com/presentation_image/dbdeda01720dbb0732aedd9014771fc1/image-17.jpg)
Example 1: Step 4 BV = {z, x 1, x 4, x 5}, NBV = {x 2, x 3} z = 900, x 1 = 200, x 4 = 600, x 5 = 1200 Increasing x 2 may lead to an increase in z. 16
![Example 1 Ratio Test The minimum ratio occurs in row 2 Thus x 4 Example 1: Ratio Test The minimum ratio occurs in row 2. Thus, x 4](https://slidetodoc.com/presentation_image/dbdeda01720dbb0732aedd9014771fc1/image-18.jpg)
Example 1: Ratio Test The minimum ratio occurs in row 2. Thus, x 4 leaves the basis when x 2 enters. 17
![Example 1 Pivoting x 2 into the basis BV z x 1 x Example 1: Pivoting x 2 into the basis BV = {z, x 1, x](https://slidetodoc.com/presentation_image/dbdeda01720dbb0732aedd9014771fc1/image-19.jpg)
Example 1: Pivoting x 2 into the basis BV = {z, x 1, x 2, x 3}, NBV = {x 4, x 5} z = 1250, x 1 = 100, x 2 = 200, x 3 = 600 This an optimal BFS. 18
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