**ABSTRACT**

Based on the characteristics of knowledge transfer in innovation networks, an optimization model of the discount expectation of profits is presented, which can determine the optimal time of knowledge transfer. Important factors, such as knowledge absorption capacity, update rate of knowledge in the network, discount rate, the time of knowledge transfer, market share, product life cycle, etc. are taken into account in the model. A large number of simulated experiments are implemented to test the efficiency of the optimization model. Simulation experimental results show that the calculated results are in accordance with the actual economic situation. The optimization model can provide useful decision support in knowledge transfer time for enterprises.

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**How to cite this article**

*Information Technology Journal, 8: 589-594.*

**DOI:**10.3923/itj.2009.589.594

**URL:**https://scialert.net/abstract/?doi=itj.2009.589.594

**INTRODUCTION**

Innovation network is a social context of enterprises and research institutes which are linked to one another to share resources and knowledge to gain critical competencies that contribute to their competitiveness in the marketplace. In this special network, the nodes are enterprises or research institutes, the edges are relations among enterprises or research institutes.

Due to the importance of innovation networks in modern society, various researchers have carried out a lot of researches on the formation and evolution of innovation networks (Debresson and Amesse, 1991; Chang and Harrington, 2007), the factors which influence the knowledge transfer in innovation networks (Cummings and Teng, 2003; Bae and Koo, 2008; Coccia, 2008; Tang *et al*., 2008) and how to enhance the efficiency of knowledge transfer (Reagans and McEvily, 2003; Kotabe *et al*., 2003; Hoffmann, 2008). Research has also showed that knowledge transfer plays a key role in innovation networks (Varga, 2000; Hovorka and Larsen, 2005; Knudsen, 2007; Mihelcic *et al*., 2007). Useful techniques to enhance the efficiency of knowledge transfer have also been developed by Tang *et al*. (2004), Sabherwal and Sabherwal (2005) and Matsuno *et al*. (2007).

However, there are few researches on how to determine the optimal time of knowledge transfer to maximize the profit of an enterprise. Taking into account of several important factors such as knowledge absorption capacity, update rate of knowledge in the network, discount rate, market share and life cycle of a product, we will present an optimization model to determine the optimal time of knowledge transfer. A large number of simulated experiments will be carried out to prove the model is feasible as a basis of decision support of time optimization of knowledge transfer for enterprises.

**BACKGROUND TO TIME OPTIMIZATION IN KNOWLEDGE TRANSFER**

Knowledge transfer can be drawn from the expression: Let G = G (V, E) be an innovation network, where, V = {V_{i}} is the set of nodes (enterprises or research institutes) in the network, E = {e_{i, j}} is the set of edges (knowledge transfer) between nodes, 1≤ i, ≤ j≤ |V| and |V| is the number of nodes in V.

Knowledge in innovation networks can be divided into two types, common knowledge and private knowledge (Khanna *et al*., 1998). It is assumed that common knowledge is the knowledge shared freely by all the nodes in the network. The ability to utilize the common knowledge only depends on the absorptive capacity of a node. On the other hand, private knowledge is the knowledge that can bring innovative profits and expand market share for related nodes. If a node wants to get private knowledge from another node, it should pay knowledge transfer cost. We only consider links of nodes based on private knowledge and we shall limit ourselves to the definition 1 below:

**Definition 1:** Let V_{i} and V_{j} be two nodes in G (V, E), if e_{i, j} = 1 then there exists knowledge transfer between V_{i} and V_{j}, if e_{i, j} = 0 then there is no knowledge transfer between V_{i} and V_{j}.

To make the optimization model simpler and practical, we formulate the following hypothesis.

**Hypothesis 1:** All the nodes in G (V, E) have only one product, whose marginal cost (Mansfield and Gary, 2003) in the starting period is MC. The marginal cost will decline at a rate of α (0<α<1).

**Hypothesis 2:** The total market volume of the product is Q. The market share of each node in the starting period is φ. The total market volume increase at a rate of θ_{1}(0<θ_{1}<1) in the first L_{1} periods and will decrease at a rate of θ (0<θ<1) in other periods.

**Hypothesis 3:** All nodes in G (V, E) share only one private knowledge whose transfer cost in the starting period is k. If the knowledge is transferred to a node, the market share of this node will increase at a rate of ρ (0<θ_{1}<ρ<1) in the first L periods immediately after knowledge transfer.

**Hypothesis 4:** The update rate of the knowledge in G (V, E) is β.

**Hypothesis 5:** The life cycle of the product in G (V, E) is N, which should be reset after knowledge update.

**OPTIMIZATION MODEL**

To maximize the profit of a node V_{i} (enterprise) in G (V, E), a model to determine the optimal time period T to transfer knowledge is presented here.

Given that V_{i} and V_{j} are two nodes in G (V, E) and that V_{i} wants to transfer knowledge from V_{j} in time period T and that the expected total profits of V_{i} is denoted as Ψ(T), we shall have Ψ(T) = – K(T)+ξ(T), where, ξ(T) is the expected profit of V_{i} gotten before knowledge transfer, ξ(T) is the expected profit of V_{i} gotten after knowledge transfer and K(T) is the cost of absorbing knowledge from V_{j}. The calculation of ξ(T), ξ(T) and K(T) is derived as follows:

**Expected profits before knowledge transfer:** Assuming that the price of the product is p, then from hypothesis 2 and 3, the sales revenue in period n can be calculated by:

(1) |

From hypothesis 1, the marginal cost in period n is expressed as MCα". Using hypothesis 2, the total production costs in period n can be calculated by:

(2) |

Therefore, by subtracting the production cost in Eq. 2 from the sales revenue in Eq. 1, the profit in period n is calculated as follows:

(3) |

By discounting the profit in period n to the beginning (with n = 0) at a discount rate r and summing up all the discount profits, the Discount Expectation of Profits (DEP) before knowledge transfer becomes:

(4) |

**Expected profits after knowledge transfer:** If V_{i} transfers knowledge from V_{j} in a period of time T, based on hypothesis 2, the market share of V_{i} in a period of time T can be calculated as follows:

(5) |

From hypothesis 2 and 3, the market share of V_{i} will increase at a rate of ρ in the L periods immediately after the time period T and it will then decay at a rate of θ. Hence, the market share of V_{i} in period n can be denoted as follows:

(6) |

From hypothesis 4, the knowledge adopted by V_{i} in period T has been updated by β^{T}, which will make the marginal cost in period T reduce to MCβ^{T}. If we renumber the periods after knowledge transfer as n starting from 1 to N, where, N is the life circle of the product defined in hypothesis 5, the marginal cost in period n becomes MCβ^{T}α^{n}. Hence, the total production cost in period n after knowledge transfer is Qλ(n, T)MCβ^{T}α^{n}. By subtracting the total production cost Qλ(n, T)MCβ^{T}α^{n} from the sales revenue pQλ(n, T), the profit in period n after knowledge transfer is as follows:

(7) |

Discounting the profits in period n to the starting point by multiplying Eq. 7 by r^{T}r^{n} and summing up all the N discount profits, the Discount Expectation of Profits (DEP) after knowledge transfer becomes:

(8) |

Using Eq. 6 and 8, the expected profits after knowledge transfer can be expressed as follows:

(9) |

**Transfer cost:** The transfer cost of node V_{i} is composed of two parts, the fixed transfer cost k defined in hypothesis 3 and the variable cost. The variable cost is related to the gap between the inside and outside knowledge level. We may compute the variable cost as F(α^{T} – β^{T}), where, F is a constant. After discounting the transfer cost to the starting point, it can be expressed as follows:

(10) |

**The total discount expectation of profits model:** Given that V_{i} and V_{j} are two nodes in G (V, E), V_{i} wants to transfer knowledge from V_{j} in time period T, the optimization problem of knowledge transfer will be to find the optimal time T to maximize – K (T)+ζ (T)+ξ (T). Therefore, the optimization model of knowledge transfer can be expressed as:

(11) |

From Eq. 4, 9 and 10, the Eq. 11 becomes:

(12) |

**OPTIMALITY OF THE MODEL **

**The model itself:** From Eq. 12, it is easy to see that Ψ(T) is a piecewise continuous differential function of T. Therefore, Ψ(T) can reach its maximum in a closed interval 1≤ T≤ N. That is to say, we can find the optimal knowledge transfer time in the life circle N of the product.

Moreover, since generally N is not very large, there is no need to find a higher and efficient algorithm to get the optimal solution, as exhaustive method is enough. Therefore, considering the powerfulness of the numerical calculation and simulation functions of Matlab, we only compiled a very simple program by Matlab.

Fig. 1: | Changes in total DEP |

Table 1: | Profit and cost of knowledge transfer |

We made lots of experiments to test the model by adjusting its parameters.

**Simulation experimental results:** To simulate the practical innovation networks, a great deal of combinations of values of parameters of the model are chosen for testing. One of the sets of parameters considered is: Q = 1000, φ = 8%, θ_{1} = 3%, p = 60, ρ = 6%, θ = 3%, MC = 40, k = 300, F = 1000, α = 95%, β = 88%, N = 10, r = 0.9. We assume the unit of the time period T is year.

Figure 1 and Table 1 show the experimental results of the model and a set of values of parameters used. The results in Table 1 and Fig. 1 imply that the discount expectation of profits is maximal at T = 6 and that the optimal knowledge transfer time is the 6th period.

**3-DIMENSIONAL SIMULATION EXPERIMENTS**

In order to find the influence of the parameters on the optimal knowledge transfer time, one of the parameters was set as a variable with time period set in turn.

Fig. 2: | Changes in EDP with α |

In this way, so many 3-dimensional simulation experiments were carried out at the same time. The 3-dimensional simulation results do not only further verify the correctness of the model but also provide decision support for enterprise management.

**Simulation with α as a variable:** To find the influence of α on the optimal knowledge transfer time, all the parameters except α were set with the same values like the earlier mentioned. Figure 2 shows the Discount Expectation of Profits (DEP) with α varying from 0.88 to 0.99 and T varying from 1 to 10. From Fig. 2, it can be seen that the smaller the α the later the optimal knowledge transfer time becomes.

**Simulation with β as a variable:** To find the influence of β on the optimal knowledge transfer time, all the parameters except β were set with the same values as indicated in the previous section. Figure 3 shows the Discount Expectation of Profits (DEP) with β varying from 0.75 to 0.95 and T varying from 1 to 10. It can be seen that the smaller the β the earlier the optimal knowledge transfer time becomes.

**Simulation with θ as a variable:** To find the influence of θ on the optimal knowledge transfer time, all the parameters except θ were set with the same values as earlier mentioned.

Figure 4 shows the Discount Expectation of Profits (DEP) with θ varying from 0 to 0.15 and T varying from 1 to 10. From Fig. 4, it can be seen that the smaller the θ the later the optimal knowledge transfer time becomes.

**Simulation with ρ as a variable:** To find the influence of ρ on the optimal knowledge transfer time, all the parameters except ρ were set with the same values as earlier mentioned.

Fig. 3: | Changes in EDP with β |

Fig. 4: | Changes in EDP with θ |

Figure 5 shows the Discount Expectation of Profits (DEP) with ρ varying from 0.03 to 0.15 and T varying from 1 to 10. From Figure 5, we can see that the smaller the ρ the later the optimal knowledge transfer time becomes.

**DISCUSSION**

There is no research on time optimization of knowledge transfer in innovation networks. There are only a few qualitative research results which provided various explanations for enhancing the efficiency of the knowledge transfer (Reagans and McEvily, 2003; Kotabe *et al*., 2003; Hoffmann, 2008). The proposed method in this study is quantitative, which can easily calculate the optimal time of knowledge transfer by imputing a set of parameters. The comparison between the proposed method and the current research results is shown in Table 2.

Fig. 5: | Changes in EDP with ρ |

Table 2: | The comparison with current researches |

Table 3: | Comparison of the results of simulations and the actual economic situations |

Meanwhile, from the 3-dimensional simulation experimental results, the influences of a factor on the optimal knowledge transfer time are predicted in the section above. To test the predicted influence trends, we compare them against the actual economic situations. From Table 3, we can see that all the predictions can be verified by the actual economic situations.

**CONCLUSION**

This study analysis the problem of time optimization of knowledge transfer in innovation networks. Based on the analysis of some important factors, a model to maximize the expected discount profit is presented. The validation of the model was done by setting the parameters and deriving the optimal time period of knowledge transfer. Simulation results show that the calculated results are in accordance with the actual economic situation. However, because of the complexity of knowledge transfer process and the diversity of influencing factors, some improvements are needed for the model to be more practical.

**ACKNOWLEDGMENT**

This study is supported by National Natural Science Foundation of China (No. 70572058, 70873038).

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