formulate this Lp model . write in standard form
An investment firm hired you to manage one of their new mutual funds. The fund invests in two main companies/stocks, Pittsburg Steel (X1) and IMB(X2). Pittsburg Steel is currently trading at $40.00 per share and IMB at $30.00 per share. Your goal is to maximize the return on investment (ROI) for the initial investment period. Recent market analysis indicates that both companies will be trading higher at the end of this investment period. The ROI for Pittsburg Steel is expected to be $10 per share and the ROI on the IMB stock is expected to be $15 per share. The firm gives you $60,000 as an initial investment to get your fund going but stipulates that at least 500 shares of each stock must be purchased. Additionally your initial research shows that IMB is a higher risk alternative and you decide that no more than one-half of the total number of shares purchased should be invested in IMB.
a) Formulate this LP model. Write in standard form.
b) Graphically solve the problem to determine the optimal solution on a separate sheet of graph paper. Remember to find and label each constraint line with direction arrows and show the feasible region. Don’t forget to label each axis and show your objective function as a dashed line with some value of Z that you assign to determine the optimal solution. Attach your graph to this paper and turn in with your computer solution.
To formulate this LP model in standard form, we need to identify the decision variables, the objective function, and the constraints.
Decision Variables:
Let X1 be the number of shares of Pittsburg Steel to be purchased.
Let X2 be the number of shares of IMB to be purchased.
Objective Function:
We want to maximize the return on investment (ROI) for the initial investment period. The ROI for Pittsburg Steel is $10 per share, so the objective function is:
Maximize Z = 10X1 + 15X2
Constraints:
1) At least 500 shares of each stock must be purchased:
X1 >= 500
X2 >= 500
2) The total amount invested in IMB cannot exceed half of the total number of shares purchased:
X2 <= (X1 + X2)/2
3) The total cost of the shares purchased cannot exceed $60,000:
40X1 + 30X2 <= 60,000
4) Non-negativity constraints:
X1 >= 0
X2 >= 0
Now, let's write the LP model in standard form by converting the inequality constraints into equality constraints:
Maximize Z = 10X1 + 15X2
Subject to:
X1 - X2/2 = 0 (Constraint 2)
40X1 + 30X2 <= 60,000 (Constraint 3)
X1 >= 500 (Constraint 1)
X2 >= 500 (Constraint 1)
X1, X2 >= 0 (Non-negativity constraints)
This is the LP model formulated in standard form.