Mathematical model is an idealized representation expressed in mathematical symbols and expressions. A mathematical model of a business problem might be in the form of a set of equations and related mathematical expressions that describe the essence of the problem. An economic order quantity model is given by: EOQ = √2AO/C where A – annual requirement, O – ordering cost and C – carrying cost. A linear programming model is given by objective function: Say Maximize Z = 10a + 12b, subject to 2a + b < = 60, 3a + 4b < = 120, a, b > = 0, where a and are units of products A and B, respectively to be produced to maximize total contribution given individual contribution of Rs.10 per unit of A and Rs.12 per unit of B, the resource constraints being that resource 1 and 2 are available respectively to the extent of 60 and 120 units only. An internal rate (IRR) model is given by: -I + Cft/(1 + k)t = 0, where ‘k’ is the IRR to be found, while ‘I’ is the initial investment, CFt refers to periodic cash flows. To construct a mathematical model objective of the firm, variables, constants and constraints must be known, besides the relationship amongst the variables. The relationship may be linear or non-linear. The EOQ and linear programming models given above are linear relationship based, while the IRR model is non-linear based as its exponential form.
Variables in Mathematical Models
Variables are something whose magnitude can change. These can take different values. Price, profit, revenue, cost, quantity produced, quantity sold, imports, exports, income tax, excise duty, national income, savings, consumption, investment, inflation rate, cost of living, etc., are all variables and can take different values entity to entity, time to time, place to place and so on.
While variables can take any value, we may give them particular values and ‘freeze’ them in a given context.
Variables may be 1) dependent or independent, 2) stochastic, probabilistic or deterministic, 3) endogenous or exogenous, 4) continuous or discrete, 5) slack or surplus, 6) choice or random, 7) real or artificial, 8) non-negative or unrestricted, 9) integer or non-integer variables.
Let E = (P-V)Q –F-D-I, where, E- earnings, P- price per unit, V- variables cost per unit, Q- quantity produced and sold, F- fixed cost for the period, D- depreciation for the period and I- interest on borrowed capital. Here, ‘E’ is the dependent variable and all those on the right side of the equation are independent variables. This is a profit model. The set of all permissible values that the independent variables can take in a given context is known as domain of the profit function and the values of E, i.e., profit for the domain values are called the range of the function.
A stochastic variable takes any value and its value cannot be predicted at all. The rate of change in sales per time is a stochastic variable. A probabilistic variable takes values according to a given probability distribution. The number of hit that a web-site gets per unit time is perhaps Poisson distribution based. A deterministic variable gets a value that is known beforehand.
Endogenous variables originates from within, say the organization. The ‘Q’, ‘D’ F and V in the profit function are mostly endogenous, i.e., emanating from inside the organization. But, P (assumed to be competition triggered) and I (assumed to be money triggered) are exogenous, emanating from outside environment.
Continuous variables take any value in a continuum, fractional, integer in the continuum. In normal probability distribution the variable is assumed to be continuous, while in binomial distribution the variable is discrete, i.e., the variables ‘jumps’, not ‘moves’.
Slack variable is introduced to an inequality of the “<=” type on the left hand side to convert the same into an equality. A stack thus represents a shortfall being met. Against this, the magnitude of random variables is the excess of actual over minimum expected. In respect of a cheaper input, a surplus usage and in respect of a dearer output a surplus achievement are generally worked out, if possible.
Choice variables, alternatively referred to decision or policy variables are the variables whose magnitude the organization can pick and choose. Against this, the magnitude of random variables is not in the hands of the organization. Choice variables can be also called controllable as to magnitude, while random variables are uncontrollable.
Real variables are variables that enter the solution set and exist there. Artificial variables are mathematical artifice and are involved to find a solution, but cannot be part of the solution.
Non-negative variables can take only positive values or zero. They can’t take negative magnitude. These are referred to a, b>=0. There cannot be situation of -5 units of product A or -3 units of product B being produced. Unrestricted variables can take positive and/or negative values.
Integer variables can take only whole number values while non-integer variables can take any value. Number of chairs/ships produced cannot take fractional value, but amount of time, consumed can take fractional days/hours.
In constructing a mathematical model all the variables and their nature must be known.
Constants in Mathematical Models
A constant can take only a specified magnitude. Hence a constant is the antithesis of a variable. A constant when added to a variable, it is called coefficient of that variable. However, a coefficient may be symbolic instead of numeric. Suppose let the symbol ‘C’ stand for a given constant and the use of the expression C in lieu of 6C in a model is perfectly all right and this expression permits greater level of generality. The symbol is rather a peculiar case. It is suppose to represent a given constant. Yet since it is not assigned a specific numeric value, it can virtually take any value. In other words, it is a ‘constant’ that is ‘variable’!! Such constants are known as parametric constants or parameters.
Take the famous equation of Einstein: E=MC2. Here, E- energy produced, M- mass of the object and C- velocity of light. The velocity of light, C, is a constant. In the Poisson distribution we use. ‘e’ is a constant, whose value is corrected to four decimals at 2.7183. The above two constants are classical constants, never change. These are non-changing constants. But, the parametric constants are constant at the given time and space. If any of these is changed, the parameters also changes. Thus, a parametric constant changes with the time and space.
Total cost of production for a given level of output, can be expressed as sum of fixed cost + Total variable cost. Let the output level be, Q, fixed cost, F, Rs.10, 00, 000 and variable cost, V, Rs.5000 per unit.
Then Total Cost = TC = F + VC OR TC (Q) = 10, 00, 000 + 5000Q
Here, total cost varies as ‘Q’ changes. The fixed cost in total is a constant. But, these constants are not universal constants. For the time being these are constants. Fixed cost may change as rent, administrative and depreciation change and unit variable cost will change if cost of direct material and labour change. Thus these are two constants. Universal constants and temporal constants.