In this video, I'll talk about how to formulate a type of LP problem called the production
process problem.
Let's see an example.
A company uses raw material (RM) and labor to produce 4 types of products.
The company can purchase at most 2000 units of RM per week, and they have 6000 labor hours
available per week.
These 4 types of products are Product 1 (P1), Product 2 (P2), Luxury Product 1 (L1), and
Luxury Product 2 (L2).
We assume the demand for these products are unlimited.
Processing one unit of RM will need 1 hour of labor and yield 3 units of P1 and at the
same time 4 units of P2.
If we further spend $5 and 3 hours to process one unit of P1, we will get one unit of L1.
If we further spend $4 and 2 hours to process one unit of P2, we will get 1 unit of L2.
The purchasing price of RM is $3/unit.
The selling prices of P1 and P2 are $7 and $6 per unit, respectively.
The selling prices of L1 and L2 are $17 and $16 per unit, respectively.
We want to maximize the company’s weekly profit.
The key step for this type of problem is to determine how the outputs from a later stage
of the process are related to the outputs from an earlier stage.
In order to calculate the profit, we need to know how much raw material should be purchased
and how many of each type of product should be sold.
Therefore, we define xrm as the units of RM purchased weekly.
xp1 as the units of p1 sold weekly, and so on.
With these definitions, the profit = the revenues – the costs.
The total revenue is equal to the sum of the unit selling price times the quantity of each
type of product.
The total cost is equal to the raw material cost plus the further processing cost.
The raw material cost is equal to the unit purchasing price times the quantity of the
raw material.
The further processing cost is equal to the unit processing cost times the quantity of
each time of luxury product.
As for the constraints, we know no more than 2000 units of RM can be purchased weekly.
No more than 6000 labor hours can be used each week.
xrm represents the total labor hours needed to process the raw material to get P1 and
P2. 3 times xl1 is the total labor hours needed to further process P1 to get L1. 2 times xl2
is the total labor hours needed to further process P2 to get L2.
All the decision variables are greater than or equal to 0.
This is the initial formulation.
Is this formulation complete?
No, we still have some important relations that are not considered yet.
Each unit of RM yields 3 units of P1 AND 4 units of P2.
If later we decide to further process one unit of P1, we can get one unit of L1.
But the sum of P1 and L1 is still 3 times RM.
Similarly, if later we decide to further process two units of P2, we can get two unit of L2.
But the sum of P2 and L2 is still 4 times RM.
Therefore we need these two constraints.
This is the complete problem formulation.
This is the optimal solution.
That means, we need to purchase 2000 units of RM, sell 6000 units of P1, 6000 units of
P2, 0 units of L1, 2000 units of L2, and the max profit is $96,000 per week.
What if we change the problem a little bit.
Each unit of RM yields 3 units of P1 OR 4 units of P2?
In this case, we can assume xrm1 is the amount of the RM that is used to make P1, and xrm2
is the amount of the RM that is used to make P2.
Now this will also be changed to xrm1 and this will also be changed to xrm2.
We have an additional relation: the portion of the RM used to make P1 plus the portion
of the RM used to make P2 is equal to the total amount of the RM.
Combining these three equations, we will have this new constraint.
This is the new problem formulation.
This constraint replaced the previous two constraints.
This is the optimal solution.
That means, we need to purchase 2000 units of RM, sell 6000 units of P2, 2000 units of
L2, and the max profit is $54,000 per week.
Okay, that is how to formulate the production process problem.
Thanks for watching.