Lucius Jesper Chloe

Session 2

Linear Programming

Outline

• Introduction
• Essentials/Properties of Linear Programming
• Process of constructing a linear programming model
• Preparation for war
• War

$100

$40

50 min

19 min

Linear Programming

• Linear programming is a powerful tool used in various fields to solve complex optimization problems.

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• It helps allocate limited resources among competing demands in an efficient and effective manner.

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• The linear programming model is composed of linear objectives and constraints, and this suggests that the variables in the model are directly proportional to each other.

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• Linear programming provides a systematic way to identify the optimal solution for optimizing business processes like production planning, inventory management, and supply chain management.

Essentials/Properties of Linear Programming

• Limited Resources: Linear Programming is used to allocate limited resources among competing demands. The resources may include raw materials, labor, money, space, and time.

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• Objective: Linear Programming requires a specific objective function that needs to be either maximized or minimized. This function can be represented as a linear equation.

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• Linearity: The objective function, as well as the constraints, must be linear equations or inequalities. This means that the relationships between the decision variables and the objective function are linear.

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• Homogeneity: The degree to which each variable contributes to the objective function should be consistent throughout the problem.

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• Divisibility: The decision variables in Linear Programming must be continuous and divisible. This means that the variables can take any value between their upper and lower bounds, including fractional values.

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• Non-negativity constraints: Linear programming assumes that the decision variables cannot take on negative values, as negative solutions are not meaningful in most practical applications.

Process of constructing a linear programming model

8. Final Answer

7. Feasible points and application

6. Drawing graph / Feasible reign

5. Finding points in the graph

4. Non negativity constrains

3. Constrains

2. Objective / Objective Function

1. Define Variables

Define Variables and Objective Function

2. A garment factory manufacture two products: Shirts and Shorts. One shirt takes 15 minutes and a short takes 12 minutes to manufacture.

1. It takes 5 minutes for a toffee manufacturing firm to produce a toffee.

Time

No of toffees - t

Manufacturing Time = 5t

No of shirts - x

No of shorts - y

Manufacturing Time = 15x + 12y

3. ‘Puru Puru Enterprises’ is a well reputed construction company in the industry. They offer two types of products. Below table shows some details regarding these products.

No of Model 1 Houses - x

No of Model 2 Houses - y

Manufacturing Time = 25x + 30y

1. A factory incurs a production cost of 10 dollars per umbrella. This cost includes 3 dollars for the fabric and 7 dollars for the steel required to produce a single umbrella.

Cost

No of umbrellas - u

Manufacturing cost = 10u

2. A factory manufactures two kinds of products. Umbrellas and raincoats. It incurs a production cost of $10 per umbrella and 4$ per raincoat.

No of umbrellas - u

Manufacturing cost = 10u + 4r

No of raincoats - r

1. A bakery incurs a production cost of 2 dollars per cupcake. The bakery sells the cupcakes for 3 dollars per unit. Formulate the Profit function for the bakery.

Profit

No of cupcakes - c

Total Profit = 1c

2. A restaurant offers three main dishes on its menu: chicken fajitas, vegetarian burrito and beef tacos. The restaurant makes a profit of $5 per chicken fajita plate, $3 per vegetarian burrito and $4 per beef taco plate.

No of Chicken Fajita plates - c

Total Profit = 5c + 3v + 4b

No of beef taco plates - b

No of vegetarian burrito plates - v

Constrains

1. ‘Light Picco’ is a new company that manufacture LED stripes. One unit of LED stripe requires 10 pounds of ‘material X’ to manufacture. The total availability of martial X is limited to 1,500 pounds.

No of LED stripes - x

10x

1,500 pounds

2. LV Enterprises manufacture plastic bags. It takes 6 minutes to manufacture 1 plastic bag. The production department only operate 8 hours a day.

No of Plastic Bags - p

6p

480 minutes

3. LV Enterprises manufacture plastic bags and plastic balls. The production department only operate 8 hours a day. Below table shows the production details of the products

No of plastic bags - x

No of plastic balls - y

6x + 10y

480 minutes

4. M B C K L J C A bakery produces two types of cakes - chocolate cake, and vanilla cake. The bakery has a limited amount of ingredient and oven time available to produce the cakes. The following table represents the amount of ingredient required to make each type of cake, the total amount of ingredient available, the baking time required for each cake, and the total oven time available.

No of chocolate cake - c

No of vanilla cake - v

10c + 12v

10,000 grams

Ingredients

30c + 35v

900 minutes

Time

5. A company produces two types of electronic devices - smartphones, and tablets. The following table represents the amount of resources required to produce each type of device, the manufacturing time required for each device, and no of employees required to manufacture each product.

The amount of raw material available daily is limited to 10KG, and the company has 45 employees working 6 hours a day.

No of smartphones - s

No of tablets - t

35s + 50t

10,000 grams

Raw material

30s + 35t

360 minutes

Time

2s + 5t

45 employees

Employees

The amount of raw material available daily is limited to 10KG, and the company has 45 employees working 6 hours a day.

Full Question 1

A company produces two products, X and Y. The production process for each product requires three types of resources - labor, raw materials, and machine time. The table below shows the requirements for each resource and the profit per unit for each product:

The company has a total of 80 hours of labor, 120 kg of raw materials, and 70 hours of machine time available each day. How many units of each product should the company produce to maximize profit?

1. Define Variables

No of units of product X - X

No of units of product Y - Y

2. Objective / Objective Function

Profit Maximization =

20X + 30Y

3. Constrains

1X + 2Y

80 hours

Labor hours

3X + 1Y

120 Kg

Raw Material

1X + 2Y

70 Hours

Machine Hours

4. Non negativity constrains

5. Finding points in the graph

1X + 2Y = 80 hours

Labor hours

1 x 0 + 2Y = 80

2Y = 80

Y = 40

1X + 2Y = 80 hours

Labor hours

1X + 2 x 0 = 80

X = 80

0

40

80

0

3X + 1Y = 120 Kg

Raw material

3 x 0 + 1Y = 120

Y = 120

3X + 1Y = 120 Kg

Raw material

3X + 1 x 0 = 120

X = 40

1X + 2Y = 70 Hours

Machine Hours

1 x 0 + 2Y = 70

Y = 35

1X + 2Y = 70 Hours

Machine Hours

1X + 2 x 0 = 70

X = 70

3X + 1Y = 120 Kg

Raw material

3 x 0 + 1Y = 120

Y = 120

3X + 1Y = 120 Kg

Raw material

3X + 1 x 0 = 120

X = 40

1X + 2Y = 70 Hours

Machine Hours

1 x 0 + 2Y = 70

Y = 35

1X + 2Y = 70 Hours

Machine Hours

1X + 2 x 0 = 70

X = 70

A

B

C

D

A

B

C

D

0

35

0

0

40

0

34

18

7. Feasible points and application

A. Profit = 20X + 30Y

= 20 x 0 + 30 x 35

= Rs 1,050

B. Profit = 20X + 30Y

= 20 x 0 + 30 x 0

= Rs 0

C. Profit = 20X + 30Y

= 20 x 40 + 30 x 0

= Rs 800

D. Profit = 20X + 30Y

= 20 x 34 + 30 x 18

= Rs 1,220

8. Final Answer

In order to maximize the profit to Rs 1,220, the company can manufacture 34 units of Product X and 18 units of Product Y.

“Chill Bill” is a company that makes Lollipops and Toffees. They have 3 main parts to making their candy: mixing and blending the ingredients, molding and shaping the candy, and packaging the candy. The table below shows the time requirements in minutes, and the profit per unit for each product:

Chill Bill has 20 employees in the Mixing and Blending unit, 25 employees in the molding and shaping unit and 10 employees in the Packing unit and if the employees works 8 hours, and they work for six days a week, can you help figure out the best way for Chill Bill to schedule their candy production for the week?

Full Question 2

1. Define Variables

No of Lollipops - L

No of Toffees - T

2. Objective / Objective Function

Profit Maximization =

5L + 1T

3. Constrains

60L + 60T

57,600 minutes

Mixing and Blending

50L + 30T

72,000 minutes (8 x 6 x 60 x 25)

Molding and Shaping

10L + 5T

28,800 minutes (8 x 6 x 60 x 10)

Packing

(8 x 6 x 60 x 20)

4. Non negativity constrains

5. Finding points in the graph

60L + 60T = 57,600 min

Mixing and Blending

60L + 60T = 57,600

60 x 0 + 60T = 57,600

60T = 57,600

T = 960

60L + 60T = 57,600 min

Mixing and Blending

60L + 60T = 57,600

60L + 60 x 0 = 57,600

60L = 57,600

L = 960

50L + 30T =

72,000 min

Molding and Shaping

50L + 30T =

72,000 min

Molding and Shaping

50L + 30 x 0 = 72,000

50 x 0 + 30T = 72,000

30T = 72,000

T = 2,400

50L + 30T = 72,000

50L + 30T = 72,000

50L = 72,000

L = 1,440

10L + 5T =

28,800 min

Packing

10L + 5T =

28,800 min

Packing

10L + 5T = 28,800

10L + 5T = 28,800

10 x 0 + 5T = 28,800

5T = 28,800

T = 5,760

10L + 5 x 0 = 28,800

10L = 28,800

L = 2,880

T

L

A

B

C

T

L

7. Feasible points and application

A. Profit = 5L + 1T

= 5 x 0 + 1 x 960

= Rs 960

B. Profit = 5L + 1T

= 5 x 0 + 1 x 0

= Rs 0

C. Profit = 5L + 1T

= 5 x 960 + 1 x 0

= Rs 4,800

8. Final Answer

In order to maximize the profit to Rs 4,800, the company can manufacture 960 lollipops and 0 Toffees.

Content obtained from:

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Desmos: https://www.desmos.com/calculator

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Byju’s: https://byjus.com/maths/linear-programming/#:~:text=In%20Mathematics%2C%20linear%20programming%20is,in%20the%20form%20of%20inequalities.

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BrainKart: https://www.brainkart.com/article/Linear-programming-problem_37039/

Thank You!