**Assignment**

**Question 1:**

The local bookstore must determine how many of each of the four new books on photonics it must order to satisfy the new interest generated in the discipline. Book 1 costs $40, will provide a profit of $8, and requires 1 inch of shelf space. Book 2 costs $55, will provide a profit of $15, and requires 2 inches of shelf space. Book 3 costs $75, will provide a profit of $9, and requires 5 inches of shelf space. Book 4 costs $55, will provide a profit of $30, and requires 1 inch of shelf space. In addition, the bookstore now offering a promotion that purchasing one Book 4 will get one Book 1 free.

Find the number of each type that must be ordered to maximize the profit. Total shelf space is 200 inches. Total amount available for ordering is $5000. It has been decided to order at least 10 Book 2.

(1) Establish the model of the optimisation problem. State clearly in words what the decision variables are, what the objective function and the equation/inequality constraints represent.

(2) Suppose the number of Book 2 is twice the number of Book 3, and the number of Book 4 is fixed as 5, establish the model of the new optimisation problem.

(3) For the model established in step (2), obtain the graphical solution using MATLAB.

(4) Change one of the parameters to cause the optimal solution (optimal design variables) changes and solve the problem again.

**Question 2:**

Determine the objective function for building a minimum-cost cylindrical tank of volume 160??! and the height should be at least 2?? longer than the radius, it should be noted that the cylindrical tank does not have a lid. If the circular ends cost $18 per ??", the cylindrical wall costs $5 per ??" , and it is necessary to spray a protective material over the whole surface and inside of the tank at a cost of $30 per ??". Moreover, due to safety restrictions, the total height of the tank must not exceed 10 m.

(1) Establish the mathematical model of the optimisation problem. State clearly in words what the decision variables are, what the objective function and the equation/inequality constraints represent.

(2) Draw the graphs and find the optimal solution (graphical solution).

(3) Change one of the parameters to cause the optimal solution (optimal design variables) changes and solve the problem again.

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