When Singapore was marked as one of the busiest port for
trading, there was an extensive discussion between the panel consisting of
great minds, how to reduce the congestion by optimising the traffic. The
traders were not satisfied because their freights had to wait for days on the
shore such that they can de board their goods on the port and as we all know,
each hour delay is inversely proportional to the chance you get to do the
business with same entity again.
So with the adverse effect of the traffic in my mind ,
researchers were given task to give the optimised value such that it can be
told to the logistic company earlier such that, there won't be any traffic as
such in future.
While undergoing the study, it was understood that the trade
of crude oil was the important factor for which port traffic was a major issue
of concern. So the primary objective was to optimise the trade of crude oil.
Scientist named Uchiyama.T wrote a paper from which he
formulated an empirical formula which gave the relationship between the cost
and the Size if refinery and tankers
The equation was verified and was ready to solve, to solve a
multi-variable non-linear equation there are many methods to solve, but the
problem arises what to choose, because the each method is problem centric. If
we choose a wrong method, there is a chance for infinite looping or cycling in
lay man's word which may not yield in the answer required.
Thus with intense analysis in this this blog I have showed
Simplex method Algorithm:
1. The iteration was initialised by
guessing three vertices of the triangle i.e. (120000, 181865), (110000,
110000), (130000, 110000).
2. The least required vertices was
found out, that is the coordinate which has the highest cost.
3. The centroid with the two remaining vertices using the formula
Xbar = (X1 + X2)/2
4. Reflection process: Found out the new vertices by the formula
X4 = Xbar + (Xbar - X high)
5. Now the least required vertices
will be replaced by the new vertices.
6. From Step 1, the process was
7. To terminate the cycle:
Xnew = X centroid + alpha (Xcentroid-Xhigh)
Alpha was varied from 0.9 to 0.1 until the cost was decreased.
8. From step 1 the process was
repeated with new alpha
9. The iteration was stopped when any other value of q and t gave
higher C value.
Cauchy's method Algorithm:
1. Starting point of iteration was
guessed which was (110000, 275000)
2. The direction of the search is
determined by (dcdt, dcdq), negative because the objective is to minimise the
cost. (Steepest decent).
3. The formula used to find dcdq:
4. The step size was calculated by solving the quadratic equation
f(x+a) - f(x), where a, is the magnitude of the step size
5. Step size multiplied with
direction gave term to be added to get the new point.
6. Then from step 1 the process was repeated again.
From both the results we can say:
The optimum tanker size = 485272.3 KL
The optimum refinery capacity = 185762.3 bbl /day
The minimum cost = 17.88 $/KL
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