EM 605_Assignment1
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School
Stevens Institute Of Technology *
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Course
605
Subject
Industrial Engineering
Date
Dec 6, 2023
Type
Pages
12
Uploaded by PrivateRiverMule27
EM 605 Elements of Operations Research Prof. A. Yassine Homework Assignment #1 1. Consider the linear programming model shown below. Minimize= 2
x
1 +
x
2 a. Graph the feasible region using a 2-dimensional grid. Show an is value contour for the objective function and indicate the direction of decrease. Identify the optimal solution on the graph. Graphically perform a sensitivity analysis for each of the objective function coefficients and each of the right-hand-side constants. subject to (1) –
x
1 + 2
x
2 ≤ 10 (2) x
1 –
2
x
2 ≤ 4 (3) x
1
+
x
2
≥8
(4) x
1
+2
x
2
≥20
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2. Consider the following linear program. Maximize z = 6
x
1 + 3
x
2 subject to x
1 + 2
x
2 2
x
1 +
x
2 x
1 –
x
2 –
x
1 +
x
2 ≥ 10 ≤ 20 ≤ 10 ≤ 3 b. Sketch the feasible region and several is value contours for the objective function in the (
x
1
,
x
2
)- space. Show the optimal solution on the graph.
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3.
Jack is an aspiring freshman at Stevens. He realizes that “all work and no play make jack a dull boy.” As a result, jack wants to apportion his available time of about 10 hours per day (at most) between work and play. He estimates that play is twice as much fun as work. He also wants to study at least as much as he plays. However, Jack realizes that if he is going to get all his homework assignments done, he cannot play more than 4 hours a day. How should Jack allocate his time to maximize his pleasure from both work and play?
4.
Day Trader wants to invest a sum of money that would generate an annual yield of at least $10,000. Two stock groups are available: blue chips and high tech, with average annual yields of 10% and 25%, respectively. Though high-tech stocks provide higher yield, they are riskier, and Trader wants to limit the amount invested in these stocks to no more that 60% of the total investment. What is the minimum amount Trader should invest in each stock group to accomplish the investment goal? Solution: Let's assume Trader wants to invest a total amount of 'T' dollars. The amount invested in blue chip stocks can be calculated as 1 - 0.6 = 0.4 times the total investment amount. So, the amount invested in blue chip stocks is 0.4T dollars. The amount invested in high-tech stocks should not exceed 60% of the total investment. Therefore, the maximum amount that can be invested in high-tech stocks is 0.6T dollars.
To calculate the minimum amount for each stock group, we need to find the appropriate split based on their average annual yields. Let's assume Trader invests an amount 'x' in blue chip stocks. Since the average annual yield for blue chip stocks is 10%, the annual yield from blue chip stocks is 0.1x. The remaining amount, (0.4T - x), is invested in high-tech stocks. Since the average annual yield for high-tech stocks is 25%, the annual yield from high-
tech stocks is 0.25(0.4T - x).
To achieve an annual yield of at least $10,000, the sum of the annual yields from both stock groups needs to be equal to or greater than $10,000. Therefore, we can set up the following equation:
0.1x + 0.25(0.4T - x) ≥ $10,000
Multiplying and simplifying the equation:
0.1x + 0.1T - 0.25x ≥ $10,000
-0.15x + 0.1T ≥ $10,000
To find the minimum value of 'x', we need to maximize (-0.15x) and minimize (0.1T) simultaneously. Since (-0.15x) is a decreasing function and (0.1T) is a constant, the minimum value for 'x' occurs when (-0.15x) and (0.1T) are equal.
Therefore, we can equate (-0.15x) to (0.1T) and solve for 'x': -0.15x = 0.1T
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x = (0.1T)/ (-0.15)
x = -0.6667T Since we cannot invest a negative amount, the minimum value for 'x' is 0. This means the minimum amount Trader should invest in blue chip stocks is $0. Substituting this value into the equation, we can find the minimum amount Trader should invest in high-tech stocks:
0.1(0) + 0.25(0.4T - 0) ≥ $10,000
0 + 0.1T ≥ $10,000
0.1T ≥ $10,000
T ≥ $10,000 / 0.1
T ≥ $100,000
Therefore, the minimum amount Trader should invest in each stock group to achieve an annual yield of at least $10,000 is $0 in blue chip stocks and at least $100,000 in high-
tech stocks.
5.
Dean’s Furniture Company assembles from precut lumber two types of kitchen cabinets: regular and deluxe. The regular cabinets are painted white, and the deluxe are varnished. Both painting and varnishing are carried out in one department. The assembly department can produce a maximum of 200 regular cabinets and 150 deluxe per day. Varnishing a deluxe unit takes twice as much time as painting a regular one. If the painting/varnishing department is dedicated to the
deluxe units only, it can complete 180 units daily. The company estimates that the profits per unit for the regular and deluxe cabinets are $100 and $140, respectively. 1.
Formulate the problem as a linear program and find the optimal production schedule per day. 2.
Suppose that because of completion, the profits per unit of the regular and deluxe units must be reduced to $80 and $110, respectively. Use sensitivity analysis to determine whether or not the optimum solution in (a) remains unchanged.
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