Dragos CRISTEA
3. Analysis of the linear programming problem results provided by WinQSB software
The results obtained after the simulation of the linear programming problem emphasize the fact that 5.000 sales agents of the life insurance package, Life Insurance Standard (LIS) contribute each of them with 1.100 EURO to the value of 5.500.000 EURO from the argument of the objective function while the difference of 792.683 EURO up to the total of 6.292.683
EURO is obtained by the contribution of 1.219 sales agents of the life insurance package, Life
Insurance Premium (LIP), each of them with 650 EURO. The ‘0’ value in the ‘Reduced
Costs’ field indicates the fact that there can’t be brought improvements to the objective function without changes of the constraints from the entry data basis.
The ‘Allowable Minimum c[i]’ and ‘Allowable Maximum c[i]’ columns illustrate the value interval in which the decisional variables can be framed. In this practical example, the profit brought by every sales agent of the life insurance package, Life Insurance Standard can’t be lower than 364,63 EURO, but it can be as high as possible and the profit brought by every sales agent of the life insurance package, Life Insurance Premium can’t be higher than 1960,87 EURO.
The inferior part of the results chart emphasizes the constraints, the values that can be assigned and their effects on the objective function. We remark the fact the two constraints ‘ Sales training budget’ and ‘Sales agents number’ present equal values in the ‘Left Hand Side’ and ‘Right Hand Side’ columns and the value associated to the ‘Contribution margin’ constraint differs in the two columns, which determines us to analyze the situations of ‘slack’ variable, ‘surplus’ variable, ‘shadow prices’ and feasibility intervals.
The ‘slack’ variable appears when the optimal values of the decisional variables are assigned to a constraint of „≤” type and the value resulted in the ‘Left Hand Side’ column is smaller than the value in the ‘Right Hand Side’ column. The ‘surplus’ variable appears when the optimal values of the decisional variables are assigned to a constraint of „≥” type and the value resulted in the ‘Left Hand Side’ column overpasses the value in the ‘Right Hand Side’ column. The shadow prices show the measure in which an increase with a unity of a constraint value in the ‘Right Hand Side’ column contributes to the increase of the arguments’ values in the objective function while the feasibility intervals designate the constraints values in the ‘Right Hand Side’ column over which the shadow prices remain constant.
Another method that can be used in order to solve this problem is represented by Simplex Tableau, revealed by the menu Solve and Display steps, which supposes several iterations in the goal to achieve the objective function . (figure no. 3)
Figure no. 3 – Simplex Tableau – alternative method to solve the linear programming problem in the insurance company
We observe from the results chart that only the ‘Contribution margin’ constraint is positioned as ‘surplus’ variable, with a value of 1.560.244 EURO; this denotes that the contribution margin can reach 3.040.244 EURO maximum without affecting the arguments of the objective function.
The value resulted in the ‘Shadow price’ column indicates the fact that every sale agent who is supplementary employed can improve the profit with 735,36 EURO. If the number of the sales
agents weren’t restricted to 5.000, the company of private life insurances could hire 7.173 sales agents, which would generate an additional profit of 1.597.937,28 EURO (2.173 x 735,36). In the same time, the company couldn’t hire a smaller number than 548 sales agents.
4. Conclusions
In order to decide the appropriate assignment of the sales force efforts in order to maximize the profit, our simulation of the linear programming problem took into account the following aspects: the identification of the objective function and its specific variables and constraints, the problem statement planning using proper coefficients to the decisional variables and constraints and the arrangement of the equations system in a form suitable for solving by WinQSB software.
The main strength of this optimization model of the resources assignment in an insurance company, besides the one referring to its projection, is the one that allows the user to simulate the financial impact of the constraints referring to variables such as the budget assigned to the training of the sales team, the dimension of the sale force and the contribution margin generated by it.
Our model reveals the fact that tightening a binding constraint can only worsen the objective function value. Once an optimal solution is found, managers can seek to improve that solution by finding ways to relax binding constraints. This model can be personalized to any type of business activity, revealing the interdependences between its variables and constraints and emphasizing the value of the linear programming approach in the formulation of a business problem.
References
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3. Norton Scott M., Kelly L. – “Resource Allocation – Managing Money and People”, Pearson Education, London, 1997
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Linear Programming”, John Willey & Sons, New York, 2002
6. Tate W. – “Mathematics applied in Business – Creating Opportunities to Learn”, Sage Publications, London, 2005