FURTHER EXPLORATION(a) Linear programming is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships. Linear programming is a special case of mathematical programming (mathematical optimization). More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints. Its feasible region is a convex polytope, which is a set defined as the intersection of infinitely many half spaces, each of which is defined by a linear inequality. Its objective function is a real-valued affine function defined on this polyhedron. A linear programming algorithm finds a point in the polyhedron where this function has the smallest (or largest) value if such a point exists. Linear programming can be applied to various fields of study. It is used in business and economics, but can also be utilized for some engineering problems. Industries that use linear programming models include transportation energy, telecommunications, and manufacturing. It hasa proved useful in modelling diverse types of problems in planning, routing, scheduling, assignment and design.

How it started?

LEONID KANTOROVICH

The problem of solving a system of linear inequalities dates back at least as far as Fourier, who in 1827 published a method for solving them and after whom the method of Fourier–Motzkin elimination is named. The first linear programming formulation of a problem that is equivalent to the general linear programming problem was given by Leonid Kantorovich in 1939, who also proposed a method for solving it. He developed it during World War II as a way to plan expenditures and returns so as to reduce costs to the army and increase losses incurred by the enemy. About the same time as Kantorovich, the Dutch-American economist T. C. Koopmans formulated classical economic problems as linear programs. Kantorovich and Koopmans later shared the 1975 Nobel prize in economics. In 1941, Frank Lauren Hitchcock also formulated transportation problems as linear programs and gave a solution very similar to the later Simplex method; Hitchcock had died in 1957 and the Nobel prize is not awarded posthumously. During 1946-1947, George B. Dantzig independently developed general linear programming formulation to use for planning problems in US Air Force. In 1947, Dantzig also invented the simplex method that for the first time efficiently tackled the linear programming problem in most cases. When Dantzig arranged meeting with John von Neumann to discuss his Simplex method, Neumann immediately conjectured the theory of duality by realizing that the problem he had been working in game theory was equivalent. Dantzig provided formal proof in an unpublished report "A Theorem on Linear Inequalities" on January 5, 1948. Postwar, many industries found its use in their daily planning. Dantzig's original example was to find the best assignment of 70 people to 70 jobs. The computing power required to test all the permutations to select the best assignment is vast; the number of possible configurations exceeds the number of particles in the observable universe. However, it takes only a moment to find the optimum solution by posing the problem as a linear program and applying the simplex algorithm. The theory behind linear programming drastically reduces the number of possible solutions that must be checked. The linear-programming problem was first shown to be solvable in polynomial time by Leonid Khachiyan in 1979, but a larger theoretical and practical breakthrough in the field came in 1984 when Narendra Karmarkar introduced a new interior-point method for solving linear-programming problems.

(b)(i)(a) Total Cost ≤ RM1400 100x + 200y ≤ 1400 ÷ 100, x + 2y ≤ 14 Total Space ≤7.2 0.6x + 0.8y ≤ 7.2 x 5, 3x + 4y≤ 36

Ratio: xy ≥

23

3x ≥ 2y 2y ≤ 3x

(b) x + 2y = 14

x 0 14

y 7 0

3x + 4y = 36

2y = 3x

x 0 12

y 9 0

x 0 4

y 0 6

(ii)

Finding maximum storage volume First Method: Find all corner points Corner Points are: 1. (0,0) 3. (8,3) 2. (12,0) 4. (3.5, 5.25) We want to maximize storage volume according to the function:

We will then subtitute the values from each corner points to find the maximum storage value.

V = 0.8x + 1.2γ

Corner Points 0.8x + 1.2γ Answer(s)

(0,0) 0.8(0) + 1.2(0) 0(12,0) 0.8(12) + 1.2(0) 9.6

(8,3) 0.8(8) + 1.2(3) 10

(3.5,5.25) 0.8(3.5) + 1.2(5.25) 34.3

Therefore, the maximum storage volume is 10. To get this value we have to use 8 cabinets of x and 3 cabinets of y to achieve maximum storage value.

Second Method: Using set squareIn this method, we use the assistance of a set square to find the maximum storage volume. To do this, we have to use and objective function.

So, the value is 0.8x + 1.2γ = k. we put in the value of k as 10 to get the new equation 8x +12γ = 10. We then have to draw a line. The new graph is as follows;

From the new line, with the aid of a set square, we move it from the new line until it touches the last point in the feasibility region. The last point is (8,3).Therefore, the maximum storage is; k = 0.8(8) + 1.2(3) = 10Therefore, the maximum storage volume is 10. To obtain 10, we have to use 8 cabinets of x and 3 cabinets of γ .

ax + bγ = k

Cabinet x

Total price of cabinet

Balance money

Cabinet γ can be

purchase

Total area Total volume Final combination

4 RM 400 RM 1000100200

= 5 (4 × 0.6) + (5×

0.8)= 6.4

(4 × 0.8) + (5×1.2)= 9.2

4x + 5yArea = 6.4 m²Volume = 9.2 m³

5 RM 500 RM 900900200

=4 (5 × 0.6) + (4×

0.8)= 6.2

(5 × 0.8) + (4×1.2)= 8.8

5x + 4yArea = 6.2 m²Volume = 8.8 m³

6 RM 600 RM 800800200

= 4 (6 × 0.6) + (4×

0.8)= 6.8

(6 × 0.8) + (4×1.2)= 9.6

6x + 4y Area = 6.8 m²Volume = 9.6 m³

7 RM 700 RM 700700200

=3 (7 × 0.6) + (3×

0.8)= 6.6

(7 × 0.8) + (3×1.2)= 9.2

7x + 3yArea = 6.6 m²Volume = 9.2 m³

8 RM 800 RM 600600200

= 3 (8 × 0.6) + (3×

0.8)= 7.2

(8 × 0.8) + (3×1.2)= 10

8x + 3yArea = 7.2 m²Volume = 10 m³

9 RM 900 RM 500500200

=2 (9 × 0.6) + (2×

0.8)= 7

(9 × 0.8) + (2×1.2)= 9.6

9x + 2yArea = 7 m²Volume = 9.6 m³

iv) Justification

If I was Aaron, I would choose the fifth combination, which is 8 cabinets of x and 3 cabinets of γ . My reasons are as follows;

- It uses the maximum space that can be held in the office room, which is 7.2 metres squared. - It can store up to 10 cubic metres of file, which is the maximum storage volume. - It comes at a reasonable price at RM 1 400, which is not too

(iii)

expensive.

- It is the most suitable combination for the future of Aaron’s company.