Description:The design and development of the Fuzzy Adaptive Investment Decision (FAID) algorithm comprises of the following procedures:
(1) Information assimilation
1.1 Identification of the key issues in data collection
• Decision on what data to collect: To optimize the collection of relevant data, the objectives of the empirical investigation were defined. Once the objective was defined, the number of variables both in the input and output space could be identified. The search for data could therefore be limited to the factors having significant influence in the IT investment and returns of the organization thereof.
• Develop appropriate instruments for data collection: Several disjointed datasets providing varying information were available in the organization. To enable a judicious selection of dataset, as an initial measure, the data extracted from the different databases were migrated into a common database platform (Windows Access) and thereafter a finer cut was uploaded into Excel for further analysis. Sufficient care was taken to ensure that no quantitative information was lost during this process of data transfer.

1.2 Data Sanitization
• Data Selection: Appropriate adoption and elimination of the data fields from different databases were necessary to arrive at the target data that is relevant to the current investigation.
• Data Cleaning/Preprocessing: This included, among other tasks, noise removal / reduction, missing value imputation, and attribute discretization. The goal of this was to improve the overall quality of any information that may be discovered.
• Data Reduction: As a final step, the dataset thus created contained a certain amount of redundancy that did not aid knowledge discovery and might in fact misled the process. The aim of this step was to find useful features to represent the data and remove non-relevant ones.

(2) Determination of the Input Space

A holistic analysis is conducted by classification of the IT investment (represented by S) of the Manufacturing organization in two parts:-

a) The total IT investment in Manufacturing and Plant Operations. This is represented by S*B-P (where S*B-P is a subset of SB, which is the net IT investment profile of the organization distributed across its various Units e.g. Head Office, Branches, Manufacturing facilities, Logistics etc).

b) The total IT investment in Enterprise Business Management which is the sum of IT investment at Head Office and IT investment at Sales Branch. This is represented by S*B-EA (where S*B-EA is again a subset of SB).

Therefore S = SUM(S*B-P + S*B-EA)
These are treated in the Input Space indicating all possible investments in IT.

(3) Determination of the Output Space

The return from IT investment (represented by R), as reflected in revenue, operational efficiency etc. in the organization, may be classified in the Output Space. Output parameters reflecting the return on IT investment considered are:
(a) Business Process Expenses: This is in-fact an aggregation of the expenses on Communication & service cost, Customer receivables and other direct expense incurred by the organization. This is represented by RBP.
(b) Material and Inventory Carrying Cost: This is in the sum of Material cost (raw material used for finished good production as well as standard materials required for day to day functions of the organization) and various inventory carrying costs (Inventory carrying cost - engineering items, Inventory carrying cost - finished goods, Inventory carrying cost - raw material, WIP inventory cost). This is represented by RIN.
(c) Salary and Wages: This is the sum total of Managerial Salaries, Non-Managerial Salaries, Unskilled Salaries and Wages related to direct and indirect labour. This is represented by RSW.

Therefore R = SUM (RBP + RIN + RSW)

(4) Using Neural Network to Determine the Membership Function (µ)

A membership function (MF) is a curve that defines how each point in the input space is mapped to a membership value (or degree of membership) in between 0 and 1.

(5) Determine the rules for the fuzzy inference engine:
Using Interrogation Techniques - 10-12 Senior Managers of different domain expertise were interrogated.

A set of Rules pertaining to IT Investment and the corresponding returns were thereby formulated based on the educated guess and expertise of the business scenario.

(6) The Model – Fuzzy Adaptive Investment Decision Algorithm

Let us symbolize the hardware, software and communication related expenses as x1. The associated variable cost elements would be skilled manpower (x2) and other support services which we capture by x3.

x11 + x12 + x13 + ……………….. x1k + x1k+1 + ……… x1n = X1
x21 + x22 + x23 + ……………….. x2k + x2k+1 + ……… x2n = X2
x31 + x32 + x33 + ……………….. x3k + x3k+1 + ……… x3n = X3

Where:
The number of Head Office = 1
The number of Branch Offices = k-1
And the number of Plants = (n-k)

Let this input point be represented as a matrix :-
X1
Z1 = [ X2 ]
X3

The collection of all such possible input points may be represented by the following matrix:
X = {Z1, Z2, Z3 …….Zm} where there are m number of points in a 3 dimensional real space.

Formally the Input space is now defined as X belongs to Rm
Where X = {Z1, Z2, Z3, …….Zm}

Similarly we can define the collection of points in the output space. The points in the output space can be derived by considering its constituent elements for example business process expenses (y1), material and inventory carrying cost (y2) and salary and wages (y3). The output profile, i.e. the Output space, of the organization is then given as:

y11 + y12 + y13 + ……………….. y1k + y1k+1 + ……… y1n = Y1
y21 + y22 + y23 + ……………….. y2k + y2k+1 + ……… y2n = Y2
y31 + y32 + y33 + ……………….. y3k + y3k+1 + ……… y3n = Y3

Where:
The number of Head Office = 1
The number of Branch Offices = k-1
And the number of Plants = (n-k)

Let this output point be represented as a matrix :-
Y1
W = [ Y2 ]
Y3

The collection of all such possible output points may be represented by the following matrix:
Y = {w1, w2, w3, …………. ws} where there are s number of points in a 3 dimensional real space.
Therefore collection in the Output space be denoted by
Y = {w1, w2, …. ws}: Y belongs Rs

The problem of optimization can now be stated formally. In IT related investment model:

To find
X , X belongs to X such that
* *
Y = Y (min), Y belongs to Y
i i i

Subject to
X <= B
*
Where B is the total budget of the company on IT related investment.

(All the quantitative elements in the Input space and Output space are in terms of monetary units)

The relation between X and Y is established not by the quantitative techniques, which are usually found in linear and non linear programming models. The points are Fuzzy in nature. The correspondence between points in input space and the points in output space cannot be established by quantitative methods. One shall have to find the Fuzzy correspondence, apply the tools of Fuzzy logic and find the solution to the optimization problem by constructing fuzzy truth table. The results would be mathematically sound because fuzzy logical reasoning is a superset of standard Boolean logic.

(7) Result

The above Fuzzy Adaptive Investment Decision algorithm is used to derive the result as follows:
• Let R be a fuzzy rule
• Suppose the membership function for an input value belonging to fuzzy set Pi is µPi (P in our context denote LOW, MEDIUM or HIGH and i=1, 2 for our 2-input case).
• The input parameters belong to the fuzzy set Pi with degrees of membership µP1 for IT investment in Manufacturing and Plant operations (S*B-P), and µP2 for IT investment for Enterprise Business Management (S*B-EA).
• We implement the “AND” operation by taking the minimum among the two degrees of membership. The result is being called the degree of validity of fuzzy rule R, denoted as DVR. It is a number between zero and one.

DVR(S*B-P, S*B-EA) = min {(µP1(S*B-P), µP2(S*B-EA)}

Test of Robustness of the model
a)With respect to the historical data set the FAID algorithm appears to be result producing (robust).
b)With respect to any systematic distortion of the data set the algorithm produces consistent result.
c)It is checked whether a systematic perturbation in the data set perturbs the result systematically and the result was negative.
d)If one can determine the points in the input space and the points in the output space, whatever be the numerical values, determination of Membership Functions are possible.
Introducing the Fuzzy Rules, which expresses the relation between the input and output linguistically, the algorithm is designed to generate the results in terms of:
– Surface View of input-output fuzzy relationship
– The pattern of behaviour of various variables in the fuzzy domain. Corresponding crisp values can also be derived.

(8) Decision based the model

Let us assume that we have a set of n combination of IT investment alternatives (A) which is quantifiable represented by
A = {A1, A2, · · ·, An}

There will be a possible m qualitative outcome (O) due to these investment alternatives, related by a fuzzy relationship and may be represented as
O = {O1, O2, · · · , Om}

Each alternative Ai in A has a constraint having the following form:
Ci = {[L1(Ai), R1(Ai)], [L2(Ai), R2(Ai)], · · · , [Lm(Ai), Rm(Ai)]}

– Where Lj(Ai) and Rj(Ai) define respectively the minimal degree and the maximal degree of alternative Ai. In this case it clearly implies the maximum and minimum limit of the IT Investment
– Thus the constraint of the alternative Ai is associated with an interval may be defined by an ordered pair as:

= [Lj(Ai), Rj(Ai)] = {A* : Lj(Ai) = A* = Rj(Ai)}

Where A* is the desired level of IT investment, e.g. the preferred alternative.

The objective of the management is then to select the alternative A* (A* belongs to A) such that :-
• A* relates to the desired outcome Oj (Oj belongs to O) and satisfy the constraints
• In case of a deadlock, e.g. a desired qualitative outcome appearing from multiple alternatives, the defuzzification techniques may be adopted.
• The objective is to derive a single crisp numeric value that best represents the inferred fuzzy values of the linguistic output variable.
• This method ascertain the different crisp values of all the possible outcomes based on the different alternative values, e.g. input, and depending on the choice of this crisp output, the related alternative may be chosen.

The invention brings following benefits to the decision makers of IT investment:
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a)From the understanding of the operational design of the organization and the financial data on IT investment, which could be lumpy (one shot) or continuous, by following the Fuzzy Adaptive Investment Decision algorithm, the management can get an answer to the following query – “given a target of operational expenses of the organization, for example raw material cost, salary and wages, communication expense etc., what should be the optimum investment in IT”.

b)The FAID algorithm may be used to implement an Expert System as an interesting tool for valuing IT investments. This may be achieved by compiling an expert database based on the collation of empirical IT investment return analysis data of several organizations operating in different industrial and business domains (e.g. banking, insurance, healthcare, manufacturing, electronics, utilities, mining and metals etc.). This will be possible by implementing the FAID (Fuzzy Adaptive Investment Decision) algorithm proposed, along with different fuzzy rule bases of individual organizations for IT investments and returns.

c)The FAID algorithm may be used to implement an EIS (Executive Information System) which can be used in conjunction with other mathematical tool to automate IT investment decisions in an organization based on various possible market parameters and forecasts.

Claims:(1) An algorithm (Fuzzy Adaptive Investment Decision algorithm), using a Fuzzy Logic framework, that determine the entire IT investment scenario of an organization to maximize the Return on Investment thereof through the following procedures:-
• An initial assessment to establish that the applicability of standard economic models and statistical principles in evaluating IT investments do have certain shortcomings to achieve the goal and therefore an evolutionary algorithmic technique, like Fuzzy Logic, is necessary.
• Identify the IT Investment as ‘inputs’ and Return thereof as ‘outputs’ in the set of fuzzy variables.
• Translate the descriptive variables of IT investments (e.g. High, Medium, Low) in the language of fuzzy mathematics as input variables and the return on investment as output variable.
• Map the IT investment variables in the Input space and the returns of the organization in the Output space.
• Ascertain the Membership Functions of these data points through an evolutionary algorithmic technique.
• Undertake an empirical exercise using Artificial Neural Network and the linguistic rule base to determine the relationship between the variables in the Input and Output spaces.
• Create the Fuzzy inference engine that maps the points in the Input space to the Output space.
• Finally, Fuzzy Adaptive Investment Decision algorithm, is devised to find the optimum point(s) in the output space for a given input.

(2) To determine the Membership Function as in claim 1, a 3-Layered Neural Network model is adopted. A set of historical data from a reputed manufacturing organization is used to train the network. Error Back-Propagation and consecutive iteration is adopted to determine the Membership Function of the input.

(3)To design the Fuzzy Adaptive Investment Decision algorithm as in claim 1, principles of Mamdani Min Implication are adopted. The fundamental optimization problem is stated as:

To find
S* , S* belongs to S*B such that
R* = R (min), R* belongs to R
Subject to

S* <= S (total budget available for IT investment).

S* = Actual Investment in IT to maximize return
S = Total organizational budget available for Investment in IT.
S*B = The net IT investment profile of the organization distributed across its various Units (e.g. Head Office, Branches, Manufacturing facilities, Logistics etc)
R* = Return on Investment in IT
R = Expenses of the organization, based on the possible combinations of outcomes of investment in IT (e.g. Material and Inventory cost, Salary and Wages, Business Process expenses etc.)

(4) The Fuzzy Adaptive Investment Decision (FAID) algorithm, as in Claim 3, is modelled using MATLAB

(5) The FAID algorithm, as in Claim 4, may give rise to multiple optima, then the defuzzification technique is adopted. The objective is to derive a single crisp numeric value that best represents the inferred fuzzy values of the linguistic output variable for a definite decision.