Nowadays estimation of conversion rates is still one of the most difficult marketing tasks. And it becomes far more difficult in respect to estimation with the use of formal methods, i.e. with the use of complex mathematical, statistical and econometrical models. Brief algorithm of evaluating marketing campaign effectiveness with the use of mathematical methods is demonstrated in this article.
Practically, algorithm consists of two stages:
1. regression analysis is an establishment of functional connection between analytical statement and parameters under study. Regression equation shows how function varies on the average during its parameter’s varying;
2. linear programming problem decision.
Linear programming is a partial branch of optimal programming (subfield of applied mathematics which studies conditional minimization tasks). Flexible and alternative production-economic situations in decision making process are necessary criteria for using of optimal programming in planning and management. Precisely these situations, as a rule, compose everyday production practice (choice of production program, sourcing, distribution logistics, advertising budget planning, material cutting etc.).
The principle of optimality’s core is in convergence to make such plan-economical decision =(x1, x2, x3, …, xn), where xi (i=1..n) – its components, that accounts for internal possibilities and external production conditions of the company in the best way. Words «in the best way» mean the choice of some optimal criterion, i.e. some economical parameter which allows comparing the effectiveness of any given plan-managemental decisions. Traditional optimal criteria: «maximum of profit», «minimum of costs», «maximal rate of return», etc.
But this algorithm does not work without one very important thing – elasticity calculation in order to pass on mathematical programming. Elasticity coefficient is a value that shows how dependant function changes when argument varies per 1%. For illustration purposes consider the following example.
Example. «Ferment» Ltd. produces ice-cream cones. Assumed that price performance is valid, product price changes during the year (i.e. take into account seasonal factor) and delivery costs are a random sequence that does not discolate statistical significance of regression equation. Relevant data is shown in Table 1.
Table 1. Data for profit and costs regression analysis Profit, RUR000’s | Advertising costs, RUR000’s | Price of 1000 ice-cream cones, RUR000’s | Cones’ delivery costs, RUR000’s | 25,7 | 0,500 | 20 | 7 | 35,9 | 1,800 | 21 | 8 | 46,8 | 2,100 | 22 | 8 | 60,9 | 3,800 | 22 | 9,8 | 72,0 | 5,000 | 23 | 11 | 48,7 | 1,800 | 23 | 11,5 | 87,0 | 6,000 | 24 | 12 | 80,0 | 27,900 | 24 | 12,3 | 93,6 | 13,600 | 23 | 9,3 | 102,0 | 8,500 | 22 | 8 | 71,6 | 28,150 | 20 | 6,3 | 88,7 | 28,150 | 20 | 5,4 |
Employ function «Regression» in MS Excel and deduce the following table as a result (Figure 1).
 Figure 1. Results of profit and costs regression analysis.
Deduce the following regression equation: 
Here are advertising costs, price of 1000 ice-cream cones, cones’ delivery costs accordingly. It is important to know that equation is true on the definite interval of parameters’ under study values which are listed in table 7.
It turns out that with advertising costs and product price’s increase profit of the company increases too, but delivery costs’ increase results in lower profit, i.e. maybe «Ferment» Ltd. had delivery work force’s runoff.
According to formula for elasticity calculation
where a – corresponding coefficient of regression equation, the following data can be obtained (Table 2): Check: 
It means that parameters’ coeffect on profit value equals 100%.
Objective function’s coefficients are: 0.2, 12.5, -2.7.
Delimitations:
- company should spend on product delivery more than 8 RUR000’s per month;
- cone’s price should nit be more than 27 RUR (for dimension preserve of objective function let assume that company realizes 1000 cones each month, in this way annual revenues from product realization can not be more than 324 RUR000’s);
- advertising costs should be more than 40 % of labor and production costs;
- company’s budget equals 970 RUR000’s. Now then linear programming problem is the following:
 Turn to MS Excel. Form table of reference (Figure 2).  Figure 2. Data of reference for linear programming problem decision
Invoke the command word «Decision making» and deduce the following data (Table 3).
Table 3. Optimal and actual values of profit and costs
Cost item | Actual value, RUR000’s | Optimal value, RUR000’s | Advertising costs | 127,3 | 168 | Price of 12000 ice-cream cones | 264 | 324 | Cones’ delivery costs | 108,6 | 96 | Profit | 812,9 | 3824 | It is seen from table 3 how company should have allocated its budget in order to get the profit 3,8 million rubles. Results make sit up ant take notice. Company’s profit would have been several times higher, if it had spent on advertising 168 RUR000’s instead of 127,3 RUR000’s per year for described constrained budget. This algorithm can be used for conversion rates’ estimation of real companies and its key point is in elasticity calculation. Because it helps to pass on after regression analysis, which only reproduces measure of dependability of profit from costs, to optimum problem decision, which helps company to answer the main question: «How to set budget properly?» And in many cases the right answer to this question serves as collateral of effective business development.
References
1. Matematicheskie metody i modeli v ekonomike / Gritsyuk S. – M.:«Feniks», 2007. – 348 p.
2. Modelirovanie ekonomicheskikh sistem i prognozirovanie / Chernyshev S.L.. – M.: Izd-vo Moskovskogo gosudarstvennogo tekhnicheskogo universiteta im. N.E.Baumana, 2003. – 232 p. |
