# Management Science

## Defining "Management Science"

In STREAMS Management Science is a discipline, together with its content, for a community that shares the belief that the methods, techniques and approaches of Science (or Engineering) can and should be applied to the management of enterprises; including their development. Historically Management Science was closely associated with Operational Research and can be traced back to the work of C. West Churchman

## Frameworks, Methodologies, Methods and Techniques of Management Science

### Foundational Mathematical Methods and Techniques

#### Algebra

Algebra is the use and methods of symbols and logic-constrained symbol manipulation in mathematics. As such it is a set of methods and techniques of generalised problem solving where the problems can be represented in mathematical symbology. STREAMS is not a website intended to teach algebra - there are many schools, colleges and universities - locally and globally - and plenty of textbooks and course that teach the methods and techniques of algebra. The Wikipedia page is not a bad place to start - although it does assume its readers have been introduced to algebra in their early schooling.

From a Hard Systems Thinking or Operational Research perspective there are two aspects to using algebra as a problem-solving method: 1) choosing the variables and coefficients that represent the parameters, characteristics and properties of the systems of interest and 2) developing one or a set of algebraic equations, including the boundary conditions, that correctly describe the dynamics of the systems and express the problem situation or question to be answered. Philosophically this is perceiving the real-world situation (which if the STREAMS philosophy is followed may span the Physical, the Mental and the Abstractal - as described in System Conceptualisation and translating that (Mental) perception into a mathematical (Abstractal) representation; this establishes an isomorphism between the real-world system-of-systems and a mathematical (Abstractal) system-of-systems - and the latter serves as a model for the former; and hence this philosophical technique may be called Mathematical Modelling. This is the same (social and personal) process as used to develop precise physical theories in Physics - which are expressed as Mathematical models. So, for example, the famous equation

```   E = mc2
```

where E represents the total energy of the system, m the total mass of the system and c the speed of light (in vacuuo) expresses the physical theory of the equivalence of mass and energy (separate concepts) and quantifies the relationship in terms of a universal constant.

Once an appropriate (correct - or thought to be correct) mathematical model is developed, the methods and techniques of algebra can be used, in some special cases, to obtain a solution (or set of solutions) to solve the problem or answer the question; to produce an algebraic or analytic solution. [This means rearranging and putting together and re-expressing and other principled symbol manipulations in order to isolate the desired variable or variables.] Even, however, when it is simply not possible to solve the equation(s) algebraically it is often possible to solve them - and hence the problem - using numerical techniques on modern computers. Hence there is great value in developing the mathematical model even if it cannot be solved by algebra.

#### Matrix (and Tensor) Algebra

A matrix is a two-dimensional array or structure of numbers. The matrix is a special case of the more general N-dimensional array called a Tensor. It turns out that the logic and rules for manipulating matrices (and tensors) are different from those of ordinary numbers (or variables) - and since they are often represented as any by symbols they follow a different algebra from the algebra of ordinary numbers or variables; this is matrix (or tensor) algebra. The elements of a matrix - ie the numbers in the array - are identified using suffixes; so for example A11 identifies the number in the first row and column of matrix A; these suffixes are also often represented by symbols and so Aij is the whole of matrix A and Tijkl might be a four dimensional tensor or 4-tensor.

Many problems in Hard Systems Thinking or Operational Research are multi-dimensional and can be tackled effectively using mathematical modelling - as described above - that applies multi-dimensional mathematics including matrix algebra. In Physics quantum theory can be expressed in matrix terms and the resulting 'formulation' of the mechanics of subatomic particles is called Matrix Mechanics. This works because 'complementary variables' in quantum theory - paradigmatically position and momentum, energy and temporal position - follow the algebra of matrices rather than that of ordinary numbers. In Relativity, especially General Relativity, the subject of the dynamical equations is the 'spacetime' - a physical entity comprising the dimensions of space and time - which is four-dimensional (in ordinary Relativity), and the theory is expressed in terms of 4D tensor algebraic equations - ie using the Ricci Calculus.

##### Matrices and Transfer Functions

The core of the generic "System Model" is that of a set of inputs to a transformation process that produces a set of output. The transformation process that changes the inputs to the output may be described by a mathematical function (or set of functions). For some systems it is the case that the inputs and outputs may be described by values in a one-dimensional array - or vector. Where this is the case the "transfer function" may be described as a two-dimensional array of mathematical functions - or matrix. To the extent that the elements of such an array are the more complex mathematical functions - of higher order: linear, quadratic, cubic etc. - reflects the complexity of the system described. The more distant the off-diagonal elements of the matrix from zero the greater the coupling between the inputs and the greater the difficulty of changing one input (perhaps for control purposes)without affecting the others - and perhaps leading to wild variation in the outputs. This is the logic behind the difficulty of 'control' in complex systems.

#### Linear Equations, Progressions and Series

Linear equations are a class, type or category of equation whose terms are 'linear' - meaning the variables are simple - not raised to any power - and the coefficients are constant. Many relations - including causal relations - in the real world are linear (where the output of some system is directly proportional to the input) or are describable by linear equations or approximate to linear relations over some range of their behaviours (usually where the inputs and/or outputs are in some sense 'small'). Many real-world 'problem situations' are the subject of 'constraints' which are describable by several linear equations at the same time; these are known as "simultaneous linear equations" or a 'System of Linear Equations' - and they can be solved by a number of methods. The methods of solving simultaneous linear equations are outlined under Linear Programming. The algebra of linear equations is known as 'Linear Algebra'.

A 'Progression' is a series or sequence of mathematical terms in which there is a constant relation between successive terms. Where the relation is one of constant difference - that is subtracting (the value of) one term from the next or previous in the sequence always yields the same number - the progression is an 'Arithmetic Progression'. Where the relation is one of constant ratio - ie dividing one term by the next or previous always yields the same result - then is it a 'Geometric Progression'.

Arithmetic and Geometic progressions are two special cases of the mathematical 'Series'.

### (Enterprise) Change Management, Portfolio Management, Programme Management and Project Management

#### Critical Path Method (CPM)

The essence of Critical Path Method (CPM) is to construct a network model of the activities of the project. This is used to assign optimistic, realistic and pessimistic quantitative estimates of the activity duration. These are then used to calculate the most likely duration of the project as a whole and identify the sequence of dependent activities - the critical path through the network - that drive the project timescale. This is then used to optimise the activities sequence (including activities conducted in parallel) and adjust the allocated resources (people, time and money) so as to minimise the risks in the project to time and budget (ie maximise the probability of delivering on-time and on-budget). The CPM analysis can be carried out with pen-and-paper but in the modern world is often facilitated by software tools.