Incentive mechanisms for enterprises in networks in case of a lacking financial attractiveness of the order.

Jaehn, Hendrik

Abstract: The process of selecting partners for a networked production process is a very sensible task. In most cases the selection is finally based on the competences and available capacities. But what can be done in case a potential enterprise is qualified from the competence perspective but the cooperation is rejected due to a lacking financial attractiveness of the order? In that case financially oriented incentive mechanisms can convince the enterprise to cooperate and thus allow initiating the production process

Key words: Incentive Mechanisms, Production Network, Network Controlling,

1. MOTIVATION

Incentives are one mechanism for the maximisation of utility of cooperations considering the New Institutional Economics (Furubotn & Richter, 2005). Although highly relevant in real-existing cooperations theoretical disquisitions focusing the theoretical background within network theory rarely can be found. However there is a strong link to the economics of information (Macho-Stadler & Perez-Castrillo, 2001)

The algorithm for the determination of incentives for the participation of one or more enterprises in a value adding process is based on an approach for the profit distribution (Jahn, 2005): every enterprise is given an individual profit share considering a fixed and a variable (value adding dependent) profit share (Jahn, Kaschel & Teich, 2006). This restriction was fixed as an assumption in order to make clear the procedure of the algorithm or respectively the incentive mechanism. Thus, payments to the single enterprises i are fixed during every iteration step within the algorithm in case the total payment sum [z.sub.i] (including profit shares [g.sub.i] and incentives [a.sub.i]) is bigger than the minimum required profit share of an enterprise [g.su.min.sub.i]. In that case [[z.sub.[bar.i]] > [g.su.min.sub.i] must be fulfilled. Those fixed payments are financed from the complete distributable profit G.

The profit is distributed among the enterprises in the network. The enterprises, whose payments were fixed before the calculation of incentives, are not considered any more in the following iteration loop. This gradual procedure however cannot be carried out in case of a profit distribution with a detailed profit distribution model because the composition of the profit (fixed and variable) changes completely when changing the amount of the incentive payments. Thus, this profit composition would need to be further considered in the following procedure of all enterprises. Nevertheless, the calculation of the payments can be carried out using a linear optimisation model.

2. OPTIMISATION APPROACH

In the following, the modelling is illustrated as a linear optimization model. First of all, the objective function will be regarded before the single side conditions are considered in detail. In figure 1 the corresponding equations can be found.

Objective function: The focus of the aims of the single actors in the value adding network is, as mentioned above, the individual maximization of utility in the sense of the new institutional economics. This has to reflect within the target function as well. Thus, the sum of all the individual realised profit shares [g.sub.i] can and has to be maximised.

Side condition 1: The sum of the profit shares of the single enterprises [g.sub.i] results from the generated or distributable network profit G less all the granted incentive payments [a.sub.i]. This side condition implies that the granted incentive payments [a.sub.i] are financed from the profit G.

Side condition 2: The second side condition includes the calculation rule for the profit shares of the enterprises [g.sub.i]. In our case, a two-component-approach is applied including a division parameter [alpha]. Thereby, the profit share of an enterprise is calculated by adding a fixed and a variable profit share.

Side condition 3: The assured minimum profit of an enterprise [g.sup.min.sub.i] consists of an individual realised profit share of this enterprise [g.sub.i] as well as a potential incentive granted to the enterprise [a.sub.i]. In addition, a so-called slack variable [u.sub.i] is introduced. This variable has the function to close a possible gap between the individual profit share [g.sub.i] and the assured minimum profit [g.sup.min.sub.i]. Such a gap might occur if the individual profit share exceeds the assured minimum profit if for example no incentive payment is required. The slack variable takes care that the conditions of the equation are also valid in such cases. For details concerning this slack variable see (Luderer & Wurker, 1995). Every enterprise disposes of a specific slack variable [u.sub.i].

Side condition 4: This side condition is based on the third side condition. Thus, the slack variable is only applied of no incentive payment [a.sub.i] is granted to the corresponding enterprise. This leads to the fact that the slack variable has a value which is unequal to zero if the incentive payment [a.sub.i] is zero or vice versa. Thus, the product from the slack variable [u.sub.i] and incentive payments [a.sub.i] has to be zero in any case.

Further model conditions: For completing the modelling, some further model conditions need to be determined. Those include the corresponding domains and co domains of the variables. Thus, the corresponding assured minimum profit [g.sup.min.sub.i], the incentives [a.sub.i], the individual value adding [w.sub.i] as well as the slack variable [u.sub.i] need to be introduced as non-negativity conditions for the complete profit G. It also has to be determined that the weighting parameter [alpha] is between zero and one because the approach shall be applied solely. Finally, the number of enterprises in the network n must be defined.

Here, it is stated that sum terms were used consciously for variables instead of the new variables that had been introduced in the previous chapters within the scope of the modelling. Thus, for example the sum of all the individual value adding processes of the enterprises [SIGMA][w.sub.i] is used instead of the term W. This procedure is designed to keep the number of different variables in the model as low as possible in order to simplify the solution.

3. LINEAR OPTIMISATION MODEL

In figure 1, the aforementioned linear optimization model will be illustrated. Thereby, the application of a two-component-approach of profit expectations under use of a distribution parameter is assumed in the beginning.

Additionally the following variables must be nonnegative: [a.sub.i], [g.sub.i], [g.sup.min.sub.i], [p.sub.i] and [u.sub.i]. The distribution parameter [alpha] must have a value between 0 and 1 (including). Finally n must be an element of N.

By modelling that approach, it had been considered that only one certain profit distribution model is applied so far with regard to the equation (cf. side condition 2).

The linear optimisation model consists of four equations which are represented by the side conditions including the three unknown variables [a.sub.i], [g.sub.i] and [u.sub.i]. This constellation makes possible a rather easy solution of the task without any occurring problems. However, the complexity of the model grows with a rising number n of the enterprises participating in the value adding process.

4. EXAMPLE

The following six equations disposing of five variables ([a.sub.1], [g.sub.1] [g.sub.2], [g.sub.3], [u.sub.1]) have to be solved in case of a number of just three involved enterprises whereof only one has to be given an incentive (cf. fig. 2.).

Because tasks having an equal or higher number of equations as variables can be solved without any problems by restructuring or replacing, the question arose subsequently by which efficient calculation steps or respectively in which order a solution can be generated. The aforementioned example of the three participating enterprises, out of which one is entitled to receive an incentive, serves as a basis for the following illustration of the solution process. This results in a task composed of six equations and five variables.

In case the calculation of the profit distribution based on the applied profit distribution model (Jahn, 2005) results in the fact that an incentive should be paid to an enterprise, a quick solution can be found. Thereby, the maximisation of the individual profit shares under consideration of the required incentive payment is guaranteed. In that way the utility of the single enterprises and the whole network can be guaranteed.

5. CONCLUSION

The previous sections introduced an approach for the design of incentive mechanisms for the participation of enterprises in a certain value adding process. The model is partly based on several marginal conditions which thus allow several fields of application. Thereby, the most significant restriction proved to be the determination of a certain approach of profit distribution, for example based on two components. However, this deficit might be reduced or abolished by a more flexible design of the side-conditions especially in the case of the approach of linear optimisation.

The allocation of incentives for increasing the attractiveness and profitability of orders has to be considered efficient because it makes possible to accept orders of customers which seem inefficient at first sight.

This on the one hand increases the appearance and thus the competitiveness of the whole network, but also allows every participating enterprise to generate a profit which is easily comprehensible from the perspective of maximising the own utility. Thus, every enterprise will be interested in participating in a value adding process in order to achieve this maximisation of the utility.

6. REFERENCES

Furubotn, E.G. & Richter, R. (2005). Institutions and Economic Theory: The Contribution of the New Institutional Economics, University of Michigan Press, 978-0472086801, Ann Arbor

Jahn, H. (2005). Profit Ascertainment and Distribution in Production Networks, Proceedings of the 16th International DAAAM Symposium 2005, B. Katalinic (Ed.), pp. 169-170, ISBN 3-901509-46-1, Opatija, Croatia, October 2005, DAAAM International, Vienna

Jahn, H.; Kaschel, J. & Teich, T. (2006). Automating of controlling processes in production networks, Chapter 23 in DAAAM International Scientific Book 2006, B. Katalinic (Ed.), pp. 287-304, DAAAM International, ISBN 3-901509-47-X, ISSN 1726-9687, Vienna, Austria

Luderer, B. & Wurker, U. (1995). Einstieg in die Wirtschaftsmathematik, Teubner, 3-519-22098-9, Stuttgart, Leipzig

Macho-Stadler, I. & Perez-Castrillo, J.D. (2001). An Introduction to the Economics of Information--Incentives and Contracts, Oxford University Press, 0-19-924325-5, Oxford, New York

Fig. 1. Model of Linear Optimization

Objective Function: [n.summation over (i=1)][g.sub.i] [right arrow] max

Side Condition 1: [n.summation over (i=1)][g.sub.i] = G -
[n.summation over (i=1)][a.sub.i]

Side Condition 2: MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Side Condition 3: [g.sup.min.sub.i] = [g.sub.i] + [a.sub.i] - [u.sub.i]

Side Condition 4: [a.sub.i] c [u.sub.i] = 0

Fig. 2. Detailed Approach for the determination of incentives

(1) [n.summation over (i=1)[g.sub.i] = G - [a.sub.1]

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(5) [g.sup.min.sub.1] = [g.sub.1] + [a.sub.1] - [u.sub.1]

(6) [a.sub.1] x [u.sub.1] = 0