Question 1181961: 1. A financial advisor at Diehl Investments identified two companies that are likely candidates for a takeover in the near future. Eastern Cable is a leading manufacturer of flexible cable systems used in the construction industry, and ComSwitch is a new firm specializing in digital switching systems. Eastern Cable is currently trading for $40 per share, and ComSwitch is currently trading for $25 per share. If the takeovers occur, the financial advisor estimates that the price of Eastern Cable will go to $55 per share and ComSwitch will go to $43 per share. At this point in time, the financial advisor has identified ComSwitch as the higher-risk alternative. Assume that a client indicated a willingness to invest a maximum of $50,000 in the two companies. The client wants to invest at least $15,000 in Eastern Cable and at least $10,000 in ComSwitch. Because of the higher risk associated with ComSwitch, the financial advisor has recommended that at most $25,000 should be invested in ComSwitch.
A. . Formulate a linear programming model that can be used to determine the number of shares of Eastern Cable and the number of shares of ComSwitch that will meet the investment constraints and maximize the total return for the investment
B. Solve the problem using simplex method
Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's the solution:
**A. Linear Programming Model**
* **Decision Variables:**
* x = Number of shares of Eastern Cable
* y = Number of shares of ComSwitch
* **Objective Function (Maximize Return):**
* Maximize Z = (55-40)x + (43-25)y = 15x + 18y
* **Constraints:**
* Total Investment: 40x + 25y ≤ 50000
* Minimum Eastern Cable Investment: 40x ≥ 15000 => x ≥ 375
* Minimum ComSwitch Investment: 25y ≥ 10000 => y ≥ 400
* Maximum ComSwitch Investment: 25y ≤ 25000 => y ≤ 1000
* Non-negativity: x ≥ 0, y ≥ 0
**B. Solving using the Simplex Method**
Because this problem involves inequalities, we must convert it to standard form by introducing slack, surplus, and artificial variables.
1. **Convert to Standard Form:**
* 40x + 25y + s1 = 50000 (s1 is a slack variable)
* x - s2 = 375 (s2 is a surplus variable)
* y - s3 = 400 (s3 is a surplus variable)
* y + s4 = 1000 (s4 is a slack variable)
* 15x + 18y - Z = 0
2. **Set up the Initial Simplex Tableau:**
| | x | y | s1 | s2 | s3 | s4 | Z | RHS |
|---|---|---|---|---|---|---|---|---|
| 1 | 40 | 25 | 1 | 0 | 0 | 0 | 0 | 50000 |
| 2 | 1 | 0 | 0 | -1 | 0 | 0 | 0 | 375 |
| 3 | 0 | 1 | 0 | 0 | -1 | 0 | 0 | 400 |
| 4 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 1000 |
| 5 | 15 | 18 | 0 | 0 | 0 | 0 | -1 | 0 |
3. **Iterations:** The simplex method involves a series of iterations, pivoting around entries in the tableau to improve the objective function value. Because of the surplus variables, you'll likely need to introduce artificial variables and use the Big M method to find an initial feasible solution. The calculations are quite lengthy, and best done with a calculator or software.
4. **Optimal Solution:** After several iterations (the precise number depends on your pivot choices), you'll reach an optimal solution. This will give you the values of x and y that maximize Z.
**Using Software:**
Solving this manually is tedious. It's *highly* recommended to use software like:
* **Excel Solver:** A built-in tool in Excel.
* **Online Simplex Method Calculators:** Many free calculators are available online.
* **Specialized Optimization Software:** For more complex problems, software like LINGO or CPLEX is used.
**How to interpret the results when you get them:**
The software will give you the optimal values for x and y. These represent the number of shares of Eastern Cable and ComSwitch, respectively, that the client should purchase to maximize their return while meeting all the constraints. It will also give you the maximum return (the optimal value of Z).
Answer by ikleyn(52748)  (Show Source):
You can put this solution on YOUR website! .
1. A financial advisor at Diehl Investments identified two companies that are likely candidates for a takeover in the near future. Eastern Cable is a leading manufacturer of flexible cable systems used in the construction industry, and ComSwitch is a new firm specializing in digital switching systems. Eastern Cable is currently trading for $40 per share, and ComSwitch is currently trading for $25 per share. If the takeovers occur, the financial advisor estimates that the price of Eastern Cable will go to $55 per share and ComSwitch will go to $43 per share. At this point in time, the financial advisor has identified ComSwitch as the higher-risk alternative. Assume that a client indicated a willingness to invest a maximum of $50,000 in the two companies. The client wants to invest at least $15,000 in Eastern Cable and at least $10,000 in ComSwitch. Because of the higher risk associated with ComSwitch, the financial advisor has recommended that at most $25,000 should be invested in ComSwitch.
A. . Formulate a linear programming model that can be used to determine the number of shares of Eastern Cable and the number of shares of ComSwitch that will meet the investment constraints and maximize the total return for the investment
B. Solve the problem using simplex method
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This problem is SO SIMPLE that can be solved MENTALLY,
using common sense ONLY and no other special methods.
Here's the solution:
**A. Linear Programming Model**
* **Decision Variables:**
* x = Number of shares of Eastern Cable
* y = Number of shares of ComSwitch
* **Objective Function (Maximize Return):**
* Maximize Z = (55-40)x + (43-25)y = 15x + 18y
* **Constraints:**
* Total Investment: 40x + 25y ≤ 50000 (1)
* Minimum Eastern Cable Investment: 40x ≥ 15000 => x ≥ 375 (2)
* Minimum ComSwitch Investment: 25y ≥ 10000 => y ≥ 400 (3)
* Maximum ComSwitch Investment: 25y ≤ 25000 => y ≤ 1000 (4)
* Non-negativity: x ≥ 0, y ≥ 0 (5)
**B. Solving MENTALLY using COMMON SENSE**
Looking into the objective function Z = 15x + 18y, you see that getting great values of 'y' is most profitable.
So, the most aggressive strategy is to take 'y' as great as possible (1000, as allowed by constraint (4))
and then to take 'x' as big as allowed by other constraints (1) and (2).
Constraint (1) gives x >= 375; constraint (2) gives x <=  = 625.
So, we take x = 625, y = 1000.
Then all constraints are satisfied, and we get maximum return 15*625 + 18*1000 = 27375.
ANSWER. The optimum strategy is 625 Eastern Cable shares and 1000 ComSwitch shares,
providing the maximum possible return of $27375.
Solved mentally.
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Post-solution note
This problem is a typical " FALSE Linear Programming problem ".
First impression is that it is a Linear programming problem,
but when you write constraints and think 1 - 2 - 3 minutes,
you will find a MENTAL SOLUTION.
By knowing this approach, you may impress your teacher/professor,
may get 5++ at exam and successfully pass a job interview to the applause of the interviewers.
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