Optimization Techniques and Their Applications: Assignment

This assignment solution explores various optimization techniques, crucial for business development and decision-making. It begins by formulating optimization problems, defining decision variables, constraints, and objective functions. The solution then delves into stochastic programming, ...Read More

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ASSIGNMENT TITLE: TECHNIQUES OF OPTIMISATION AND
ITS USES

TABLE OF CONTENTS
Formulating optimization problem in terms of decision variables, constraints and objective
function............................................................................................................................................3
Stochastic programming problem of demand uncertainty to minimize expected total cost............5
Expected value of perfect information............................................................................................5
Markowitz’s mean-variance optimization model............................................................................5
Imposing restrictions on the maximum number of assets included in a portfolio for monitoring
and controlling purposes..................................................................................................................6
Dynamic programming model.........................................................................................................6
Linear integer programming model.................................................................................................7
Reference List..................................................................................................................................8
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Formulating optimization problem in terms of decision variables, constraints
and objective function.
Deterministic optimisation is a division of a numerical optimisation which focuses on finding
global solution to an optimisation problem while providing theoretical guarantee for it, with
some predefined tolerances. The terms of deterministic optimisation typically refers to a
complete or rigorous optimisation methods. Rigours methods converge to a global optimum in a
definite time frame state are described by mathematical model. In cases of finding a global
minimum of an optimisation problem it is extremely difficult to find a feasible solution.
The variables in a linear program are a set of quantities that needs to be determined while solving
the problem. The problem is solved by the identification of the best values of the variables; such
variables are called decision variables because the problem is to decide on what value each
variable should take. Typically, the variables represent the amount of a resource to be used.
Defining the variables for the problem is the most crucial step in formulating the problem as a
linear program. Here creative variable definition can be used to reduce the size of the problem or
make a non-linear problem linear. The variables are represented in an abstract manner as X1, X2, .
. ., Xn . (As there are n variables in this list.)
Constrained optimization is the process of optimizing an objective function with respect to some
variables or values in the presence of constraints on those variables or values. In determining the
constraints the constraints can either be hard constraints for which there are set conditions for the
variables which are required to be satisfied, or soft constraints which have some variable values
that are penalized in the objective function based on the extent that, the conditions on the
variables are not satisfied (Graham et al. 1979, p.125 ).
Objective function of a linear programming model is to maximise or minimise a value. The
objective function indicates how much each variable contributes to the value that is to be
optimized in the problem. The objective function takes the following general form:
Where
Ci - the objective function coefficient corresponding to the ith variable, and
Xi - the ith decision variable.
The coefficients of the objective function are to indicate the contribution to the value of the
objective function of one unit of the corresponding variable. As for example, if the objective
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function is to maximize the present value of the project, and Xi is the ith possible activity in the
project, then Ci (the objective function coefficient corresponding to Xi) gives the net present
value generated by one unit of activity i. As another example, if the problem is to minimize the
cost of achieving some goal, Xi might be the amount of resource i used in achieving the goal. In
this case, Ci would be the cost of using one unit of resource i. Of course, some variables may not
contribute to the objective function. In such cases, one can either think of the variable as having
a coefficient of zero, or one can think of the variable as not being an objective function at all.
Therefore as per the given case study the objective functions in production of i-grabber and
walker would be subject to cost constraints.
Therefore the objective function for i-walker would be
Max Z =f(X1,X2)
ST M= C1X1+C2X2+B
Where M= Total cost
X1= Raw material cost
X2= Production cost.
B= Constant fixed cost
Where C1- coefficient of constraint associated with X1
and C2- coefficient of constraint associated with X2
Z- Is the production of i-walker
Similarly, for I-Grabber,
Max Y=f(X1,X2)
ST
N=C1x1+C2x2+G
ST M= C1X1+C2X2+B
Where N= Total cost
x1= Raw material cost
x2= Production cost.
b= Constant fixed cost
Here i=1, 2 as it is a two variable case, the variables are production cost and material cost (Pham
and Karaboga, 2012, p.113).
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Stochastic programming problem of demand uncertainty to minimize
expected total cost
Stochastic programming is similar to other types of optimisations but this method has no definite
formulation. The presence of random quantities in the model under considerations opens the door
to a wealth of different problem settings reflecting different aspects of applied problem at hand.
In order to suppose, the company has decided about order quantity x of either i walker or i
grabber to satisfy demand d. The cost is c>0 per unit. If demand d larger than x, then the
company makes an additional order price b>=0. The cost this is equal to b(d-x) if d>x and is 0
otherwise. On the other hand, if d<x, then a holding cost of h(x-d) is incurred, therefore the total
cost is then equal to F(x, d) = cx+b [d-x] +h[x-d] where b>c is assumed i.e. the penalty cost is
greater than the ordering cost. The objective is to minimise cost where x is the decision variable
and and the demand parameter is d (Samuelson, 1969, p.242).
Expected value of perfect information
In decision theory, the expected value of perfect information (EVPI) is the price or cost that one
would be willing to bear in order to gain access to perfect information. In this context while
looking at a decision of whether to adopt a new treatment technology, there is always some
degree of uncertainty surrounding the decision, because there always remains a chance of the
decision turning out to be wrong (Madan et al. 2014, p.329).
Markowitz’s mean-variance optimization model
Mean-variance portfolio analysis provides the first quantitative treatment of the trade-off
between profit and risk and describing in detail the interplay between objective and constraints in
a number of single-period variants and g semi variance models. Particular emphasis is laid on
avoiding the penalization of over performance. The results are used as building blocks in the
development and theoretical analysis of multi-period models based on scenario trees. A key
property is the possibility of removing surplus in future decisions, yielding approximate risk
minimization (Gökgöz and Atmaca, 2012, p.360).
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Imposing restrictions on the maximum number of assets included in a
portfolio for monitoring and controlling purposes.
Imposition of restrictions by the investor would depend on the case scenario and depending on
the cardinality factor, mean-variance Markowitz model involves two conflicting objectives to be
optimized with respect to the benchmark portfolio. While the expected portfolio return on
investment with respect to the benchmark is maximized, expected portfolio risk that is measured
as the variance of the portfolio return relative to the same benchmark portfolio is minimized and
henceforth the restriction on the imposition on assets will have to be done (Bai et al. 2016.
p.102).
Average: In everyday language, an average is the sum of a list of numbers divided by the
number of numbers in the list. There could be various forms of Averages or means namely
arithmetic mean geometric mean and harmonic mean. The methods of computations are different
(Elliott, 2012, p.106). The formula of average or arithmetic mean is 1/n Σxi where i= 1, 2, 3.n.
Covariance: Covariance is a measure of the joint variability of two random variables and
changes. If the greater values of one variable mainly correspond with the greater values of the
other variables and the same holds for the lesser values, i.e. the variables tend to show similarity
in behaviour, the covariance is positive
Cove (X, Y) = E [(X-E[X]) (Y-E[Y])]
Where E[X] is the expected value of X also recognised as the mean of X (Higle and Sen, 2013,
p.109).
Dynamic programming model
In dynamic programming is a methodology for solving complicated problems by dividing it into
a collection of simpler sub problems and solving each of them one by one and recording the
solutions ideally, using a memory based data structure. in this method if the same sub problem
reoccurs, instead of recomputing its solution, one simply looks up to the previously computed
solution, and thereby saving computational time at the expense of a modest expenditure in
storage space. Each of the sub problem solutions is indexed in some way, typically based on the
values of its input parameters, so as to facilitate its lookup. The technique of storing solutions to
sub problems instead of recomputing them is called "memorization" (Heckman and Raut, 2016,
p.167).
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Linear integer programming model
An integer programming model is a mathematical optimisation or feasibility program in which
some or all of the variables are restricted to be integers. In many settings the term refers to
integer linear programming, in which the objective function and the constraints are linear. There
are two main reasons for using integer variables when modelling problems as linear programs
one is that the integer variables represent quantities that can only be integer. For example, it is
not possible to build 3.7 cars. The other is that the integer variables represent decisions and so
should only take on the value 0 or 1.These considerations occur frequently in practice and so
integer linear programming can be used in many applications areas (Shabani and Sowlati, 2013,
p.357).
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Reference List
Bai, Z., Li, H., McAleer, M. and Wong, W.K., 2016. Spectrally-Corrected Estimation for High-
Dimensional Markowitz Mean-Variance Optimization (No. EI2016-20).
Elliott, P.D., 2012. Probabilistic number theory I: Mean-Value theorems (Vol. 239). Springer
Science & Business Media.
Gökgöz, F. and Atmaca, M.E., 2012. Financial optimization in the Turkish electricity market:
Markowitz's mean-variance approach. Renewable and Sustainable Energy Reviews, 16(1),
pp.357-368.
Graham, R.L., Lawler, E.L., Lenstra, J.K. and Kan, A.R., 1979. Optimization and approximation
in deterministic sequencing and scheduling: a survey. Annals of discrete mathematics, 5, pp.287-
326.
Heckman, J.J. and Raut, L.K., 2016. Intergenerational long-term effects of preschool-structural
estimates from a discrete dynamic programming model. Journal of econometrics, 191(1),
pp.164-175.
Higle, J.L. and Sen, S., 2013. Stochastic decomposition: a statistical method for large scale
stochastic linear programming (Vol. 8). Springer Science & Business Media.
Madan, J., Ades, A.E., Price, M., Maitland, K., Jemutai, J., Revill, P. and Welton, N.J., 2014.
Strategies for efficient computation of the expected value of partial perfect information. Medical
Decision Making, 34(3), pp.327-342.
Pham, D. and Karaboga, D., 2012. Intelligent optimisation techniques: genetic algorithms, tabu
search, simulated annealing and neural networks. Springer Science & Business Media.
Samuelson, P.A., 1969. Lifetime portfolio selection by dynamic stochastic programming. The
review of economics and statistics, pp.239-246.
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