OPS910 Linear Programming Assignment Sample

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OPS910 Linear Programming Assignment Sample

Please include your name and student number on the cover page of your submission. Write concisely, check for spellings, right justify the text, use 1.5 line spacing at the minimum and use 12pt font Calibri or Times New Roman, and include page numbers.

1. Ronald Mcdonald oversees loading cargo ships for Brisbane Cargo Company (BCC) at the port in Brisbane. Ronald is preparing a loading plan for a BCC freighter destined for the US. An agriculture cargos dealer wants to transport the following cargo products aboard this ship. Ronald learned that this ship has space to take 480,000 cubic feet of cargo, and also has a cargo weight limit of 13,000 tons. Ronald can elect to load any of the available cargos shown in the table below.

Ronlad plans to solve the following optimization problem: Maximize 75 X1 +55 X2 +70 X3 +60 X4

Subject to:

X1+X2+X3+X4 ≤ 13,000
40 X1+35 X2 +60 X3 +55 X4 ≤ 480,000 X1 ≤4800
X2 ≤ 5000
X3≤ 1200
X4 ≤ 1700
X1, X2, X3, X4 20
Where X1, X2, X3, and X4 represent the amount (tons) to load to the ship for Cargos 1, 2, 3 and 4, respectively.

(a) Solve this Linear Programming for university assignment help problem in Excel using Solver and report the results.

(b) Present a table showing how sensitive Roland's loading strategy is to the profit per ton for each type of cargo. Which cargo's profit per ton is the loading strategy most sensitive to? Why?

(c) How much of each type of cargo should Ronald accept, if the profit per ton for Cargo 1 decreases to $52? Why?

(d) Now, the dealer makes further requirement about loading the ship: if a type of cargo is to be shipped, at least 500 tons of that type need to be shipped; otherwise, that type of cargo should not be shipped at all.

How much of each type of cargo to accept, to maximize the profit?

Please describe how you will model this problem differently from (a) and report the result. Be specific in your answer.

(e) List all assumptions you got directly from the question description to complete (a) -(c). List and justify all additional assumptions you needed to make.

(f) Think broadly about this decision process. Write-up a discussion of questions like the following. What hasn't been included above? Do you think these missing things should be included? What information is missing? Which of the assumptions in (e) don't you like and why? How realistic is this problem? What extensions would you suggest?

2. Let x be a decision variable. Set up a spreadsheet to
maximize 25-76x +61x2 - 17x3 +1.94x4-x3/13
subject to 0 ≤x≤ 10.

With the initial value of x=2 run standard Solver GRG algorithm, what value do you get?

Now run Solver's standard evolutionary algorithm. What answer do you get?

If this answer is different from the first, which answer is correct and why?

Explain what is going on.

Solution

Solution 1

The given constrains are

a)

By using Excel Solver,

Substitute the values as per this

Thus, by using solver, we get 763,727.27 as the result.

b)

The tabular form to show the profit per ton for each type of cargo is given below:

Therefore, the most sensitive strategy is of for Cargo 4 as its profit changes from 1700 to 745 while other variables profit remains the same.

c)

If the profit per ton for Cargo 1 decreases to $52, then

The objective function will changes accordingly as given below:
New objective function will be
Max: 52x_1+55x_2+70x_3+60x_4

Now, by using solver, the solution will be

Hence, the maximum profit will be 653,327.27

d)

If the dealer makes the changes and atleast 500 cargo need to be shipped, then

Cargo 1, 2, 3 and 4 will become 4800, 5000, 1200, and 745 respectively to maximum the profit.

The maximum profit will be
=52*4800+55*5000+70*1200+60*745=653,300

e)

The assumptions directly derived from the part (a) – (c) are

Objective Function: Function that needs to be maximized or minimized.

Constraints: Constraints are typically provided in the question, specifying limitations or restrictions on the decision variables.

Decision Variables: The question should mention decision variables that need to be determined to optimize the objective function.

Linearity: Linear programming assumes that the objective function and all constraints are linear functions of the decision variables.

Additional assumptions are:

Non-negativity: In linear programming, decision variables are typically assumed to be non-negative, unless explicitly stated otherwise.
Continuity: Linear programming assumes that decision variables can take on any real value within a certain range, rather than discrete values.
Independence: Linear programming assumes that decision variables are independent of each other.

f)

The assumptions which are missing can be considered as:

Boundedness: Decision variables typically have upper and/or lower bounds unless otherwise specified.

Convexity: Linear programming problems are convex, meaning that the feasible region formed by the constraints is a convex set.

Certainty: The coefficients of the objective function and constraints are assumed to be known with certainty and do not involve uncertainty or randomness.

Rationality: Linear programming assumes rational decision-making, where decisions are made to optimize a certain objective.

These assumptions are standard in the context of linear programming unless stated otherwise in the problem statement. Depending on the specific problem, additional assumptions may be necessary.

Solution 2

The given constraints are

Max

                  25-76x+61x^2-17x^3+1.94x^4-x^5/13

Where

                    0≤x≤10

If the initial value of x = 2, then from Excel

Now, by using the Excel Solver,



Hence, by using Solver, the maximum profit will changes from 9.58 to 28.14

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