Question

Suppose the firm in this example considers a second product that has a unit profit of $5 and requires 2 hours for each unit produced. Assume total production capacity remains 40. Use y as the number of product 2 produced.
a.) Show the mathematical model when both products are considered simultaneously.
b.) Identify the controllable and uncontrollable inputs for this model.
c.) Draw the flowchart of the input-output process for this model
d.) What are the optimal solution values of x and y?
e.) Is the model developed a deterministic or a stochastic model? Explain.

Answer 1

Answer:

Answer is explained below in the explanation section.

Explanation:      

a) Mathematical Model:

Data Given:

Unit Profit = $5 for product 2

Unit Profit = $10 for product 2

Time = 5 hours for product 1 each unit

Time = 2 hours for product 2 each unit

Production Capacity = 40 units

y = number of product 2 produced.

x = number of product 1 produced.

Profitability from product 1 = $10/5hours

Profitability from product 1  = $2

Profitability from product 2 = $5/2 hours

Profitability from product 2 = $2.5

Profitability from product 2 > Profitability from product 1

Max 10x + 5y subject to constraints.

5x + 2y $\leq$ 40

where,

x $\geq$ 0 and y $\geq$ 0

Hence,    

If all the hours are used to produce the product 2 then, the profit will be maximum.

b)

Simply, we can see that the controllable  inputs are product 1 and product 2 or x and y.

and uncontrollable inputs are labor hours here.

c) Flowchart is attached in the attachment below. Please refer to the attachment for the flowchart of the input output process for this model.

d) Optimal solution values for x and y:

If we put the values of the x and y into the equation 10x + 5y, we will get 100 as the objective function. where, x $\geq$ 0 and y $\geq$ 0

Hence, objective function = $100

Answer 2

a)  We are using the Mathematical Model:

                    Data is Given:

• Then Unit Profit = $5 for product 2
• Then Unit Profit = $10 for product 2
• Now, Time = 5 hours for product 1 each unit
• After that Time = 2 hours for product 2 each unit
• Now Production Capacity is = 40 units
• After that y = several products 2 produced.
• Then x = number of products 1 produced.
• Then the Profitability from product 1 is = $10/5hours
• After that Profitability from product 1 = $2
• Now when Profitability from product 2 = $5/2 hours
• Then the Profitability from product 2 = $2.5
• Then Profitability from product 2 > Profitability from product 1
• Now Max 10x + 5y subject to constraints.
• Now, 5x + 2y 40
• x 0 and y 0
• After that If all the hours are used to produce product 2 then, the profit will be maximum.

(b) Simply, when we can see that the controllable inputs are product 1 and also that product 2 or x and then y.

• After that uncontrollable inputs are labor hours here.

(c) When Flowchart is attached in the attachment below. then please refer to the attachment for the flowchart of the input-output process for this model.

(d) when Optimal solution values for x and y:

• Then If we put the values of the x and also that y into the equation 10x + 5y, we will get 100 as the objective function. where x 0 and y 0

• Hence proof that, objective function = $100

(e) When A deterministic model is a model where:

1. When the material properties are well known, that is. deterministic. none of them is random
2. After that The applied load are also deterministic

• When Stochastic model has on the other hand:

1 -  When we random properties, e.g. Young's modulus is a random variable with uniform distribution [E1, E2]; when normal distribution (of a given mean or standard deviation)

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