Question

A firm offers two different prices on its products, depending upon the quantity purchased. Since available resources are limited, the firm would like to prepare an optimal production plan to maximize profits. Product 1 has the following profitability: $75 each for the first 25 units and $60 for each unit over 25. Product 2's profitability is $200 each for the first 50 units and $100 each for each unit over 50. The products each require two raw materials to produce (see table below for usages and available quantities). You may your choice of software tool for this problem

Raw Material	Product 1 usage (gallons per unit)	Product 2 usage (gallons per unit)	Available Quantity (gallons)
A	10	20	1,500
B	5	7	2,000

1. Again, write the model formulation as a separable programming mathematical Linear Optimization model with appropriate mathematical equations and variable definitions as explained in Problem 1 (5 points)

2. Use separable programming to find the optimal production plan. Support your answer with your working documentation. (5 points)

Optimal Profitability: ___________________________

Optimal Production Plan:   Product1 Units __________________    

                                            Product2 Units: ____________________

Answer 1

Answer:

Step A: Formulation of Separable Programming Model

Decision Variables:

Let,

x1 = No. of first 25 units produced for Product 1

x2 = No. of units produced for Product 1 after the first 25 units

x3 = No. of first 50 units produced for Product 2

x4 = No. of units produced for Product 2 after the first 50 units

Objective Function:

As the objective is to maximize the total profit, the objective function=

MaxZ = 75 x1 + 60 x2 + 200 x3 + 100 x4

Subject to Constraints:

10 x1 + 10 x2 + 20 x 3 + 20 x4 ≤ 1500 (Availability of Raw Material A)

5 x1 + 5x2 + 7 x3 + 7 x4 ≤ 2000 (Availability of Raw Material B)

Where,

0 ≤ x1 ≤ 25
0 ≤ x2 ≤ ∞
0 ≤ x3 ≤ 50
0 ≤ x4 ≤ ∞

Step B: Solving separable programming:

As there is no specific information mentioned for which software to use, we will solve this model using MS-Excel Solver as mentioned below:

Step 1: Prepare the following table (Fill the exact details considering the row and column cells)

Step 2: Use the following formulas as mentioned below:

Hence, we get the following screenshot:

Step 3: Open MS-Excel Solver, and type the exact values in solver parameters window as mentioned in the below screenshot:

Then, press Solve

Step 4: Hence, we will get the following solution (As mentoned in below screen shot)

Thus,

Optimal Solution =

Product 1 = 50 Units and Product 2 = 50 Units

Max Z = 25 (75) + 25 (60) + 50 (200) = 13375