A review of linear programming Example CTB sells bagels and - - PowerPoint PPT Presentation

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Jul 05, 2023 217 likes •465 views

A review of linear programming Example CTB sells bagels and cupcakes, earning a profit of $6 for each dozen of bagels and $8 for each dozen of cupcakes. We have the following information: 1 dz Bagels 1 dz Cupcakes Amount available Eggs 3 6

SLIDE 1

A review of linear programming

Example

CTB sells bagels and cupcakes, earning a profit of $6 for each dozen of bagels and $8 for each dozen of cupcakes. We have the following information: 1 dz Bagels 1 dz Cupcakes Amount available Eggs 3 6 50 Flour 7 5 100 Butter 3 4 75 Additionally, CTB must produce at least 3 dozens bagels everyday for its regulars. How many dozens of bagels and cupcakes should CTB produce each day to maximize total profit?

SLIDE 2

A review of linear programming

Example

CTB sells bagels and cupcakes, earning a profit of $6 for each dozen of bagels and $8 for each dozen of cupcakes. We have the following information: 1 dz Bagels 1 dz Cupcakes Amount available Eggs 3 6 50 Flour 7 5 100 Butter 3 4 75 Additionally, CTB must produce at least 3 dozens bagels everyday for its regulars. How many dozens of bagels and cupcakes should CTB produce each day to maximize total profit?

SLIDE 3

A review of linear programming

Formulating a problem as an LP:

SLIDE 4

A review of linear programming

Formulating a problem as an LP:

1. Decision variables:

SLIDE 5

A review of linear programming

Formulating a problem as an LP:

1. Decision variables:

x1 = number of dozens of bagels to produce x2 = number of dozens of cupcakes to produce

SLIDE 6

A review of linear programming

Formulating a problem as an LP:

1. Decision variables:

x1 = number of dozens of bagels to produce x2 = number of dozens of cupcakes to produce

2. Objective function:

SLIDE 7

A review of linear programming

Formulating a problem as an LP:

1. Decision variables:

x1 = number of dozens of bagels to produce x2 = number of dozens of cupcakes to produce

2. Objective function:

Total profit = 6x1 + 8x2

SLIDE 8

A review of linear programming

Formulating a problem as an LP:

1. Decision variables:

x1 = number of dozens of bagels to produce x2 = number of dozens of cupcakes to produce

2. Objective function:

Total profit = 6x1 + 8x2

3. Constraints:

SLIDE 9

A review of linear programming

Formulating a problem as an LP:

1. Decision variables:

x1 = number of dozens of bagels to produce x2 = number of dozens of cupcakes to produce

2. Objective function:

Total profit = 6x1 + 8x2

3. Constraints:

3x1 + 6x2 ≤ 50

SLIDE 10

A review of linear programming

Formulating a problem as an LP:

1. Decision variables:

x1 = number of dozens of bagels to produce x2 = number of dozens of cupcakes to produce

2. Objective function:

Total profit = 6x1 + 8x2

3. Constraints:

3x1 + 6x2 ≤ 50 7x1 + 5x2 ≤ 100

SLIDE 11

A review of linear programming

Formulating a problem as an LP:

1. Decision variables:

x1 = number of dozens of bagels to produce x2 = number of dozens of cupcakes to produce

2. Objective function:

Total profit = 6x1 + 8x2

3. Constraints:

3x1 + 6x2 ≤ 50 7x1 + 5x2 ≤ 100 3x1 + 4x2 ≤ 75

SLIDE 12

A review of linear programming

Formulating a problem as an LP:

1. Decision variables:

x1 = number of dozens of bagels to produce x2 = number of dozens of cupcakes to produce

2. Objective function:

Total profit = 6x1 + 8x2

3. Constraints:

3x1 + 6x2 ≤ 50 7x1 + 5x2 ≤ 100 3x1 + 4x2 ≤ 75 x1 ≥ 3

SLIDE 13

A review of linear programming

Formulating a problem as an LP:

1. Decision variables:

x1 = number of dozens of bagels to produce x2 = number of dozens of cupcakes to produce

2. Objective function:

Total profit = 6x1 + 8x2

3. Constraints:

3x1 + 6x2 ≤ 50 7x1 + 5x2 ≤ 100 3x1 + 4x2 ≤ 75 x1 ≥ 3 x1, x2 ≥

SLIDE 14

A review of linear programming

So, the LP is max 6x1 +8x2 s.t. 3x1 +6x2 ≤ 50 7x1 +5x2 ≤ 100 3x1 +4x2 ≤ 75 x1 ≥ 3 x1, x2, ≥

SLIDE 15

A review of linear programming

So, the LP is max 6x1 +8x2 s.t. 3x1 +6x2 ≤ 50 7x1 +5x2 ≤ 100 3x1 +4x2 ≤ 75 x1 ≥ 3 x1, x2, ≥

SLIDE 16

A review of linear programming

So, the LP is max 6x1 +8x2 s.t. 3x1 +6x2 = 50 7x1 +5x2 = 100 3x1 +4x2 = 75 x1 = 3 x1, x2, ≥

SLIDE 17

A review of linear programming

So, the LP is max 6x1 +8x2 s.t. 3x1 +6x2 + x3 = 50 7x1 +5x2 = 100 3x1 +4x2 = 75 x1 = 3 x1, x2, x3, ≥

SLIDE 18

A review of linear programming

So, the LP is max 6x1 +8x2 s.t. 3x1 +6x2 + x3 = 50 7x1 +5x2 + x4 = 100 3x1 +4x2 = 75 x1 = 3 x1, x2, x3, x4, ≥

SLIDE 19

A review of linear programming

So, the LP is max 6x1 +8x2 s.t. 3x1 +6x2 + x3 = 50 7x1 +5x2 + x4 = 100 3x1 +4x2 + x5 = 75 x1 = 3 x1, x2, x3, x4, x5, ≥

SLIDE 20

A review of linear programming

So, the LP is max 6x1 +8x2 s.t. 3x1 +6x2 + x3 = 50 7x1 +5x2 + x4 = 100 3x1 +4x2 + x5 = 75 x1 − x6 = 3 x1, x2, x3, x4, x5, x6 ≥

SLIDE 21

A review of linear programming

So, the LP is (after adding slack variables x1, x2, . . . , x6) max 6x1 +8x2 s.t. 3x1 +6x2 + x3 = 50 7x1 +5x2 + x4 = 100 3x1 +4x2 + x5 = 75 x1 − x6 = 3 x1, x2, x3, x4, x5, x6 ≥

SLIDE 22

A review of linear programming

LP in standard form: max cTx s.t. Ax = b x ≥ 0, where c, x are n-vectors, b is an m-vector, and A is an m × n matrix.

SLIDE 23

A review of linear programming

In our example, n = 6, m = 4, x =      x1 x2 . . . x6      and c =         6 8         , A =     3 6 1 7 5 1 3 4 1 1 −1     , b =     50 100 75 3     .