# Formula Sheet: Industrial Engineering Notes | Study Formula Sheets of Mechanical Engineering - Mechanical Engineering

## Mechanical Engineering: Formula Sheet: Industrial Engineering Notes | Study Formula Sheets of Mechanical Engineering - Mechanical Engineering

The document Formula Sheet: Industrial Engineering Notes | Study Formula Sheets of Mechanical Engineering - Mechanical Engineering is a part of the Mechanical Engineering Course Formula Sheets of Mechanical Engineering.
All you need of Mechanical Engineering at this link: Mechanical Engineering
``` Page 1

Forecasting

• Simple Moving Average-
?? ?? =
1
?? (?? ?? + ?? ?? -1
+ ?? ?? -2
+ ?? ?? -3
+ ? )
• Moving Weight Average-
Weight Moving Average =
? W
i
D
i
n
t=1
? W
i
n
i=1

• Single (Simple) Exponential Smoothing-
?? ?? = ?? ?? -1
+ ?? (?? ?? -1
- ?? ?? -1
)
???? ?? ?? = (1 - ?? )?? ?? -1
+ ?? ?? ?? -1

if previous forecasting is not given
?? ?? = ?? ?? ?? + ?? (1 - ?? )?? ?? -1
+ ?? (1 - ?? )
2
?? ?? -2
… ….
Where ?? ?? = Smoothed average forecast for period t
?? ?? -1
=Previous period forecast
?? =Smoothing constant
• Linear Regression-                                ?? = ?? + ????
? ?? = ???? + ?? ? ??
? ???? = ?? ? ?? + ?? ? ?? 2

• Forecasting Error-
?? ?? = (?? ?? - ?? ?? )
• Bias-
???????? =
?? ?? ?(?? ?? - ?? ?? )
?? ?? =??
• Mean Absolute Deviation-
?????? =
?? ?? ? |?? ?? - ?? ?? |
?? ?? =??
• Mean Square Error-
?????? =
?? ?? ?(?? ?? - ?? ?? )
?? ?? ?? =??
• Mean Absolute Percentage Error-
???????? =
?? ?? ?
|?? ?? - ?? ?? |
?? ?? ?? ?? =?? × 100
Inventory
If D= demand/year, ?? ?? =Order cost, ?? ?? = Carrying cost, P= Purchase price/unit
?? *
=Economic Order Quantity, K= Production Rate and ?? ?? =Shortage  Cost/unit/period
• Case-1 Purchase Model With Instantaneous Replenishment and Without Shortage-
1. ?????? ?? *
= v
2?? ?? ?? ?? ??   at EOQ ?????????????????? ???????? = ?????????? ????????
2. ???? . ???? ?????????? =
?? ?? *

3. ???????? ?????????? ?????? ?????????? =
?? *
??
4. ?????????? ???????? = ???????? ???????? + ?????????????????? ???????? + ?????????? ????????
= (?? × ?? ) + (
?? 2
× ?? ?? ) + (
?? ?? × ?? ?? )= (?? × ?? ) + v2?? ?? ?? ?? ??
• Case-2 Manufacturing Model Without Shortage-

Page 2

Forecasting

• Simple Moving Average-
?? ?? =
1
?? (?? ?? + ?? ?? -1
+ ?? ?? -2
+ ?? ?? -3
+ ? )
• Moving Weight Average-
Weight Moving Average =
? W
i
D
i
n
t=1
? W
i
n
i=1

• Single (Simple) Exponential Smoothing-
?? ?? = ?? ?? -1
+ ?? (?? ?? -1
- ?? ?? -1
)
???? ?? ?? = (1 - ?? )?? ?? -1
+ ?? ?? ?? -1

if previous forecasting is not given
?? ?? = ?? ?? ?? + ?? (1 - ?? )?? ?? -1
+ ?? (1 - ?? )
2
?? ?? -2
… ….
Where ?? ?? = Smoothed average forecast for period t
?? ?? -1
=Previous period forecast
?? =Smoothing constant
• Linear Regression-                                ?? = ?? + ????
? ?? = ???? + ?? ? ??
? ???? = ?? ? ?? + ?? ? ?? 2

• Forecasting Error-
?? ?? = (?? ?? - ?? ?? )
• Bias-
???????? =
?? ?? ?(?? ?? - ?? ?? )
?? ?? =??
• Mean Absolute Deviation-
?????? =
?? ?? ? |?? ?? - ?? ?? |
?? ?? =??
• Mean Square Error-
?????? =
?? ?? ?(?? ?? - ?? ?? )
?? ?? ?? =??
• Mean Absolute Percentage Error-
???????? =
?? ?? ?
|?? ?? - ?? ?? |
?? ?? ?? ?? =?? × 100
Inventory
If D= demand/year, ?? ?? =Order cost, ?? ?? = Carrying cost, P= Purchase price/unit
?? *
=Economic Order Quantity, K= Production Rate and ?? ?? =Shortage  Cost/unit/period
• Case-1 Purchase Model With Instantaneous Replenishment and Without Shortage-
1. ?????? ?? *
= v
2?? ?? ?? ?? ??   at EOQ ?????????????????? ???????? = ?????????? ????????
2. ???? . ???? ?????????? =
?? ?? *

3. ???????? ?????????? ?????? ?????????? =
?? *
??
4. ?????????? ???????? = ???????? ???????? + ?????????????????? ???????? + ?????????? ????????
= (?? × ?? ) + (
?? 2
× ?? ?? ) + (
?? ?? × ?? ?? )= (?? × ?? ) + v2?? ?? ?? ?? ??
• Case-2 Manufacturing Model Without Shortage-

1. ?????? ?? *
=
v
2?? ?? ?? ?? ?? (1-
?? ?? )

2. ?? 1
=
?? *
??
3. ?? 2
=
?? *
(1-
?? ?? )
??
4. Total optimum cost = v2?? ?? ?? ?? ?? (1 -
?? ?? )
• Case 3 Purchase Model With Shortage-

1. ?? = ?????? = v
2?? ?? ?? ?? ?? (
?? ?? +?? ?? ?? ?? )
2. ?? 1
= v
2?? ?? ?? ?? ?? (
?? ?? ?? ?? +?? ?? ) ,  ?? 2
= ?? - ?? 1

3. ?? =
?? ?? , ?? 1
=
?? 1
?? and ?? 2
=
?? 2
??
4. Total optimum cost =v2?? ?? ?? ?? ?? (
?? ?? ?? ?? +?? ?? )
• Case 4 Manufacturing Model With Shortfall

1. ?? = ?????? =
v
2?? ?? ?? ?? ?? ×(1-
?? ?? )
(
?? ?? +?? ?? ?? ?? )
2. ?? 1
= v
2?? ?? ?? ?? ?? (
?? ?? ?? ?? +?? ?? ) × (1 -
?? ?? )
Page 3

Forecasting

• Simple Moving Average-
?? ?? =
1
?? (?? ?? + ?? ?? -1
+ ?? ?? -2
+ ?? ?? -3
+ ? )
• Moving Weight Average-
Weight Moving Average =
? W
i
D
i
n
t=1
? W
i
n
i=1

• Single (Simple) Exponential Smoothing-
?? ?? = ?? ?? -1
+ ?? (?? ?? -1
- ?? ?? -1
)
???? ?? ?? = (1 - ?? )?? ?? -1
+ ?? ?? ?? -1

if previous forecasting is not given
?? ?? = ?? ?? ?? + ?? (1 - ?? )?? ?? -1
+ ?? (1 - ?? )
2
?? ?? -2
… ….
Where ?? ?? = Smoothed average forecast for period t
?? ?? -1
=Previous period forecast
?? =Smoothing constant
• Linear Regression-                                ?? = ?? + ????
? ?? = ???? + ?? ? ??
? ???? = ?? ? ?? + ?? ? ?? 2

• Forecasting Error-
?? ?? = (?? ?? - ?? ?? )
• Bias-
???????? =
?? ?? ?(?? ?? - ?? ?? )
?? ?? =??
• Mean Absolute Deviation-
?????? =
?? ?? ? |?? ?? - ?? ?? |
?? ?? =??
• Mean Square Error-
?????? =
?? ?? ?(?? ?? - ?? ?? )
?? ?? ?? =??
• Mean Absolute Percentage Error-
???????? =
?? ?? ?
|?? ?? - ?? ?? |
?? ?? ?? ?? =?? × 100
Inventory
If D= demand/year, ?? ?? =Order cost, ?? ?? = Carrying cost, P= Purchase price/unit
?? *
=Economic Order Quantity, K= Production Rate and ?? ?? =Shortage  Cost/unit/period
• Case-1 Purchase Model With Instantaneous Replenishment and Without Shortage-
1. ?????? ?? *
= v
2?? ?? ?? ?? ??   at EOQ ?????????????????? ???????? = ?????????? ????????
2. ???? . ???? ?????????? =
?? ?? *

3. ???????? ?????????? ?????? ?????????? =
?? *
??
4. ?????????? ???????? = ???????? ???????? + ?????????????????? ???????? + ?????????? ????????
= (?? × ?? ) + (
?? 2
× ?? ?? ) + (
?? ?? × ?? ?? )= (?? × ?? ) + v2?? ?? ?? ?? ??
• Case-2 Manufacturing Model Without Shortage-

1. ?????? ?? *
=
v
2?? ?? ?? ?? ?? (1-
?? ?? )

2. ?? 1
=
?? *
??
3. ?? 2
=
?? *
(1-
?? ?? )
??
4. Total optimum cost = v2?? ?? ?? ?? ?? (1 -
?? ?? )
• Case 3 Purchase Model With Shortage-

1. ?? = ?????? = v
2?? ?? ?? ?? ?? (
?? ?? +?? ?? ?? ?? )
2. ?? 1
= v
2?? ?? ?? ?? ?? (
?? ?? ?? ?? +?? ?? ) ,  ?? 2
= ?? - ?? 1

3. ?? =
?? ?? , ?? 1
=
?? 1
?? and ?? 2
=
?? 2
??
4. Total optimum cost =v2?? ?? ?? ?? ?? (
?? ?? ?? ?? +?? ?? )
• Case 4 Manufacturing Model With Shortfall

1. ?? = ?????? =
v
2?? ?? ?? ?? ?? ×(1-
?? ?? )
(
?? ?? +?? ?? ?? ?? )
2. ?? 1
= v
2?? ?? ?? ?? ?? (
?? ?? ?? ?? +?? ?? ) × (1 -
?? ?? )

3. ?? 1
= (1 -
?? ?? ) ?? - ?? 2

4. ?? =
?? ?? , ?? 1
=
?? 1
?? -?? and ?? 2
=
?? 1
??
1. ?? 3
=
?? 3
?? and ?? 4
=
?? 2
?? -??

• Lead Time Demand + Safety Stock = Reorder Point
PERT and CPM
• EFT = EST + activity time
• LFT = LST + Duration of activity

• Total Float-

• Free Float-  FFo= (Ej-Ei)-Tij

• Independent Float -

Example-

1. Total float = L2 – (E1 + t12) = 57 – (20 + 19) = 18
2. Free float = E2 – E1 – t12 = 0
3. Independent float = E2 – (L1 + t12) = -18
• PERT Expected time- ?? ?? =
?? 0
+4?? ?? +?? ?? 6

1. t0 = Optimistic time i.e., shortest possible time to complete the activity
if all goes well.
2. tp = Pessimistic time i.e., longest time that an activity could take if
everything goes wrong.
3. tm = Most likely time i.e., normal time of an activity would take.
• Standard deviation-

• Variance -

• Crashing-

• Standard Normal Variation (SNV)-

Page 4

Forecasting

• Simple Moving Average-
?? ?? =
1
?? (?? ?? + ?? ?? -1
+ ?? ?? -2
+ ?? ?? -3
+ ? )
• Moving Weight Average-
Weight Moving Average =
? W
i
D
i
n
t=1
? W
i
n
i=1

• Single (Simple) Exponential Smoothing-
?? ?? = ?? ?? -1
+ ?? (?? ?? -1
- ?? ?? -1
)
???? ?? ?? = (1 - ?? )?? ?? -1
+ ?? ?? ?? -1

if previous forecasting is not given
?? ?? = ?? ?? ?? + ?? (1 - ?? )?? ?? -1
+ ?? (1 - ?? )
2
?? ?? -2
… ….
Where ?? ?? = Smoothed average forecast for period t
?? ?? -1
=Previous period forecast
?? =Smoothing constant
• Linear Regression-                                ?? = ?? + ????
? ?? = ???? + ?? ? ??
? ???? = ?? ? ?? + ?? ? ?? 2

• Forecasting Error-
?? ?? = (?? ?? - ?? ?? )
• Bias-
???????? =
?? ?? ?(?? ?? - ?? ?? )
?? ?? =??
• Mean Absolute Deviation-
?????? =
?? ?? ? |?? ?? - ?? ?? |
?? ?? =??
• Mean Square Error-
?????? =
?? ?? ?(?? ?? - ?? ?? )
?? ?? ?? =??
• Mean Absolute Percentage Error-
???????? =
?? ?? ?
|?? ?? - ?? ?? |
?? ?? ?? ?? =?? × 100
Inventory
If D= demand/year, ?? ?? =Order cost, ?? ?? = Carrying cost, P= Purchase price/unit
?? *
=Economic Order Quantity, K= Production Rate and ?? ?? =Shortage  Cost/unit/period
• Case-1 Purchase Model With Instantaneous Replenishment and Without Shortage-
1. ?????? ?? *
= v
2?? ?? ?? ?? ??   at EOQ ?????????????????? ???????? = ?????????? ????????
2. ???? . ???? ?????????? =
?? ?? *

3. ???????? ?????????? ?????? ?????????? =
?? *
??
4. ?????????? ???????? = ???????? ???????? + ?????????????????? ???????? + ?????????? ????????
= (?? × ?? ) + (
?? 2
× ?? ?? ) + (
?? ?? × ?? ?? )= (?? × ?? ) + v2?? ?? ?? ?? ??
• Case-2 Manufacturing Model Without Shortage-

1. ?????? ?? *
=
v
2?? ?? ?? ?? ?? (1-
?? ?? )

2. ?? 1
=
?? *
??
3. ?? 2
=
?? *
(1-
?? ?? )
??
4. Total optimum cost = v2?? ?? ?? ?? ?? (1 -
?? ?? )
• Case 3 Purchase Model With Shortage-

1. ?? = ?????? = v
2?? ?? ?? ?? ?? (
?? ?? +?? ?? ?? ?? )
2. ?? 1
= v
2?? ?? ?? ?? ?? (
?? ?? ?? ?? +?? ?? ) ,  ?? 2
= ?? - ?? 1

3. ?? =
?? ?? , ?? 1
=
?? 1
?? and ?? 2
=
?? 2
??
4. Total optimum cost =v2?? ?? ?? ?? ?? (
?? ?? ?? ?? +?? ?? )
• Case 4 Manufacturing Model With Shortfall

1. ?? = ?????? =
v
2?? ?? ?? ?? ?? ×(1-
?? ?? )
(
?? ?? +?? ?? ?? ?? )
2. ?? 1
= v
2?? ?? ?? ?? ?? (
?? ?? ?? ?? +?? ?? ) × (1 -
?? ?? )

3. ?? 1
= (1 -
?? ?? ) ?? - ?? 2

4. ?? =
?? ?? , ?? 1
=
?? 1
?? -?? and ?? 2
=
?? 1
??
1. ?? 3
=
?? 3
?? and ?? 4
=
?? 2
?? -??

• Lead Time Demand + Safety Stock = Reorder Point
PERT and CPM
• EFT = EST + activity time
• LFT = LST + Duration of activity

• Total Float-

• Free Float-  FFo= (Ej-Ei)-Tij

• Independent Float -

Example-

1. Total float = L2 – (E1 + t12) = 57 – (20 + 19) = 18
2. Free float = E2 – E1 – t12 = 0
3. Independent float = E2 – (L1 + t12) = -18
• PERT Expected time- ?? ?? =
?? 0
+4?? ?? +?? ?? 6

1. t0 = Optimistic time i.e., shortest possible time to complete the activity
if all goes well.
2. tp = Pessimistic time i.e., longest time that an activity could take if
everything goes wrong.
3. tm = Most likely time i.e., normal time of an activity would take.
• Standard deviation-

• Variance -

• Crashing-

• Standard Normal Variation (SNV)-

Linear Programming
Simplex Method Case 1. Maximization Problem
3 1 5 2
/ 3 1 2 2 18 ( )
=+
+ ? -
MaxZ x x
s t x x I

1 4 ( )
2 6 ( )
1, 2 0
?-
?-
?
x II
x III
xx

Standard Form:
Max Z = 3x1 + 5x2 + 0w1 + 0w2 + 0w3
3x1 + 2x2 + w1 + 0w2 + 0w3  =  18
x1 + 0x2 + 0w1 + w2 + 0w3  =  4
0x1 + x2 + 0w1 + 0w2 + w3   =   6
To prepare initial Table:
Table - I

•  Ij = (Zj-cj) = ( ?aij.ci)-cj
Interpretation of  Simplex Table
Table - I

• Key Column ? Min Ij   [ Most Negative ]
• Key Row ? Min positive ratio.
How to get next table ?
• Leaving variable : w3
•  Entering variable : x2
•  Key no. = 1
•  For old key row : New No.= Old No./key No.
•  For other rows:
( .).
( .)
..
.
=-
CorrespondingKeyRowNo
CorrespondingKeyColumnNo
NewNo Old No
KeyNo

• 18 ? 18 - (6*2)/1 = 6
• I(w3) = 0 ? 0 - [1*(-5)]/1 = 5
Page 5

Forecasting

• Simple Moving Average-
?? ?? =
1
?? (?? ?? + ?? ?? -1
+ ?? ?? -2
+ ?? ?? -3
+ ? )
• Moving Weight Average-
Weight Moving Average =
? W
i
D
i
n
t=1
? W
i
n
i=1

• Single (Simple) Exponential Smoothing-
?? ?? = ?? ?? -1
+ ?? (?? ?? -1
- ?? ?? -1
)
???? ?? ?? = (1 - ?? )?? ?? -1
+ ?? ?? ?? -1

if previous forecasting is not given
?? ?? = ?? ?? ?? + ?? (1 - ?? )?? ?? -1
+ ?? (1 - ?? )
2
?? ?? -2
… ….
Where ?? ?? = Smoothed average forecast for period t
?? ?? -1
=Previous period forecast
?? =Smoothing constant
• Linear Regression-                                ?? = ?? + ????
? ?? = ???? + ?? ? ??
? ???? = ?? ? ?? + ?? ? ?? 2

• Forecasting Error-
?? ?? = (?? ?? - ?? ?? )
• Bias-
???????? =
?? ?? ?(?? ?? - ?? ?? )
?? ?? =??
• Mean Absolute Deviation-
?????? =
?? ?? ? |?? ?? - ?? ?? |
?? ?? =??
• Mean Square Error-
?????? =
?? ?? ?(?? ?? - ?? ?? )
?? ?? ?? =??
• Mean Absolute Percentage Error-
???????? =
?? ?? ?
|?? ?? - ?? ?? |
?? ?? ?? ?? =?? × 100
Inventory
If D= demand/year, ?? ?? =Order cost, ?? ?? = Carrying cost, P= Purchase price/unit
?? *
=Economic Order Quantity, K= Production Rate and ?? ?? =Shortage  Cost/unit/period
• Case-1 Purchase Model With Instantaneous Replenishment and Without Shortage-
1. ?????? ?? *
= v
2?? ?? ?? ?? ??   at EOQ ?????????????????? ???????? = ?????????? ????????
2. ???? . ???? ?????????? =
?? ?? *

3. ???????? ?????????? ?????? ?????????? =
?? *
??
4. ?????????? ???????? = ???????? ???????? + ?????????????????? ???????? + ?????????? ????????
= (?? × ?? ) + (
?? 2
× ?? ?? ) + (
?? ?? × ?? ?? )= (?? × ?? ) + v2?? ?? ?? ?? ??
• Case-2 Manufacturing Model Without Shortage-

1. ?????? ?? *
=
v
2?? ?? ?? ?? ?? (1-
?? ?? )

2. ?? 1
=
?? *
??
3. ?? 2
=
?? *
(1-
?? ?? )
??
4. Total optimum cost = v2?? ?? ?? ?? ?? (1 -
?? ?? )
• Case 3 Purchase Model With Shortage-

1. ?? = ?????? = v
2?? ?? ?? ?? ?? (
?? ?? +?? ?? ?? ?? )
2. ?? 1
= v
2?? ?? ?? ?? ?? (
?? ?? ?? ?? +?? ?? ) ,  ?? 2
= ?? - ?? 1

3. ?? =
?? ?? , ?? 1
=
?? 1
?? and ?? 2
=
?? 2
??
4. Total optimum cost =v2?? ?? ?? ?? ?? (
?? ?? ?? ?? +?? ?? )
• Case 4 Manufacturing Model With Shortfall

1. ?? = ?????? =
v
2?? ?? ?? ?? ?? ×(1-
?? ?? )
(
?? ?? +?? ?? ?? ?? )
2. ?? 1
= v
2?? ?? ?? ?? ?? (
?? ?? ?? ?? +?? ?? ) × (1 -
?? ?? )

3. ?? 1
= (1 -
?? ?? ) ?? - ?? 2

4. ?? =
?? ?? , ?? 1
=
?? 1
?? -?? and ?? 2
=
?? 1
??
1. ?? 3
=
?? 3
?? and ?? 4
=
?? 2
?? -??

• Lead Time Demand + Safety Stock = Reorder Point
PERT and CPM
• EFT = EST + activity time
• LFT = LST + Duration of activity

• Total Float-

• Free Float-  FFo= (Ej-Ei)-Tij

• Independent Float -

Example-

1. Total float = L2 – (E1 + t12) = 57 – (20 + 19) = 18
2. Free float = E2 – E1 – t12 = 0
3. Independent float = E2 – (L1 + t12) = -18
• PERT Expected time- ?? ?? =
?? 0
+4?? ?? +?? ?? 6

1. t0 = Optimistic time i.e., shortest possible time to complete the activity
if all goes well.
2. tp = Pessimistic time i.e., longest time that an activity could take if
everything goes wrong.
3. tm = Most likely time i.e., normal time of an activity would take.
• Standard deviation-

• Variance -

• Crashing-

• Standard Normal Variation (SNV)-

Linear Programming
Simplex Method Case 1. Maximization Problem
3 1 5 2
/ 3 1 2 2 18 ( )
=+
+ ? -
MaxZ x x
s t x x I

1 4 ( )
2 6 ( )
1, 2 0
?-
?-
?
x II
x III
xx

Standard Form:
Max Z = 3x1 + 5x2 + 0w1 + 0w2 + 0w3
3x1 + 2x2 + w1 + 0w2 + 0w3  =  18
x1 + 0x2 + 0w1 + w2 + 0w3  =  4
0x1 + x2 + 0w1 + 0w2 + w3   =   6
To prepare initial Table:
Table - I

•  Ij = (Zj-cj) = ( ?aij.ci)-cj
Interpretation of  Simplex Table
Table - I

• Key Column ? Min Ij   [ Most Negative ]
• Key Row ? Min positive ratio.
How to get next table ?
• Leaving variable : w3
•  Entering variable : x2
•  Key no. = 1
•  For old key row : New No.= Old No./key No.
•  For other rows:
( .).
( .)
..
.
=-
CorrespondingKeyRowNo
CorrespondingKeyColumnNo
NewNo Old No
KeyNo

• 18 ? 18 - (6*2)/1 = 6
• I(w3) = 0 ? 0 - [1*(-5)]/1 = 5

Table - II

• Key Column ? Min Ij
• Key Row ? Min positive ratio
Table - III

• This is the final Table
The Optimal Solution is   x1 = 2, x2 = 6
giving Z = 36
Type of Solutions : Basic, Feasible/Infeasible, Optimal/  Non-Optimal,
Unique/Alternative Optimal, Bounded/Unbounded, Degenerate/Non-Degenerate
• Analysis of Solution
1. This is a  Basic solution, as values of basic variables  are Positive
2. This is a  feasible solution, as values of basic variables, not containing
Artificial Variable, are Positive  and all constraints are satisfied
3. This Feasible solution is an Optimal, as all values in Index Row are positive.
4. If there is an Artificial Variable, as Basic variable in final  table,  it is called as
Infeasible solution
5. This solution is unique Optimal, as the number of zeroes are equal to number
of basic variables in Index Row in final Table.
6. If the number of zeroes are more than number of basic variables in Index Row
in final  Table, it is a case of more than one optimal solutions.
7. This is a Bounded Solution, as the values of all Basic variables in final table,
are finite positive.
8. This is a Non-degenerate Solution, as value of none of the basic variables is
Zero ,  in final table.
9. If value of at least one of the basic variables is Zero in Index Row in final
Table, it is a Degenerate Solution.
• Duality With Example
1. Case-1
Max.Z =  x1 - x2  + 3x3
s/t   x1 +  x2  + x3  ?  10
2 x1 -  x2  -  x3 ?  2
2x1   -  2x2  - 3x3 ?  6
x1 , x2, x3  ?  0
Dual of this would be
Min Z = 10y1  +2y2  + 6y3
s/t   y1 + 2 y2  + 2y3   ?   1
y1 -  y2  - 2y3    ?  -1
y1   -  y2  - 3y3   ?   3
y1 , y3 , y3   ?   0 ,
```
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