Simulación del modelo con los valores de los parámetros ajustados

This section reports a set of numerical studies that quantify the profit loss of ig-noring network externalities. We also propose and quantitatively evaluate some easy-to-implement heuristics in the presence of network externalities. Our numerical results demonstrate that (1) ignoring network externalities and, thus, employing a myopic pric-ing and inventory policy leads to staggerpric-ing profit losses when the network externalities intensity, the social customer proportion, or the carry-through rate of network size is high;

and (2) the firm can achieve low optimality gaps and effectively exploit network external-ities with heuristic policies that take into account the demand induction opportunexternal-ities in the near future only.

Throughout our numerical studies, we assume that the maximum intrinsic valuation V¯_t is stationary and equals 30 for each period t. The planning horizon length is T = 20.

The network externalities function is γ(N_t) = kN_t(k ≥ 0). The parameter k measures the network externalities intensity. The larger the k, the more intensive network externalities the firm faces. Hence, the demand in each period t is D_t(p_t, N_t) = 30 + kN_t− pt+ ξ_t, where{ξt}^Tt=1follow i.i.d. normal distributions with mean 0 and standard deviation σ = 2.

Note that with the linear network externalities function γ(·), Assumption 2.3.1 does not hold. This slight deviation from our analytical model, however, does not influence the insights obtained in this section. For simplicity, we assume the random perturbation in the market size dynamics ϵt is degenerate, i.e., ϵt = 0 with probability 1. We set the discount factor α = 0.99, the unit procurement cost c = 8, the unit holding cost h = 1, the unit backlogging cost b = 10, and the feasible price range [p, ¯p] = [0, 34]. In the evaluation of the expected profits, we take I_t = 0 as the reference initial inventory level and Nt = 0 as the reference initial network size.

2.6.1 Impact of Network Externalities

This subsection numerically studies the impact of network externalities upon the firm’s profitability under different values of network externalities intensity k, social customer proportion θ, and carry-through rate of network size η. We evaluate the profit of the firm which ignores the tradeoff between generating current profits and inducing future demands in the presence of network externalities. More specifically, we assume that the firm adopts the myopic policy in each period t, i.e., it adopts the pricing and inventory policy that maximizes the expected current-period profit without taking into account future demand-inducing opportunities. Equivalently, the firm employs the optimal final-period policy, (x^∗₁(·, ·), p^∗1(·, ·)), throughout the planning horizon. Let Vm be the expected profit under the myopic policy, and V^∗ be optimal expected profit. Thus, the metric of interest is

λ_m := V^∗− Vm

V^∗ ×100%, which evaluates the profit loss of ignoring network externalities.

We conduct the numerical experiments under the parameters t = 5, 10, 15, 20, k =

Figures 2.1 - 2.3 summarize the results of our numerical study on the impact of ignoring network externalities upon the firm’s profitability. Our results reveal that, when the future demand-inducing opportunity of network externalities is ignored, the firm incurs a significant profit loss, which is at least 4.90% and can be as high as 36.60%, as long as the network externalities intensity k, the proportion of social customers θ, and the network size carry-through rate η are not too low (greater than 0.2 in our numerical case).

If k, θ, and η are higher, the current operations decisions have greater impact upon future network sizes, thus leading to more intensive tradeoff between generating current profits and inducing future demands. Therefore, adopting the myopic policy results in significant losses if k, θ, and η are not too low. Another important implication of Figures 2.1 - 2.3 is that, if k, θ, and η are not too low, the profit loss of ignoring network externalities may be significant even when the planning horizon length is short (i.e., t = 5). This calls for caution that the firm under network externalities should not overlook the tradeoff between generating current profits and inducing future demands even for a short sales horizon.

4 6 8 10 12 14 16 18 20 22 0

5 10 15 20 25 30 35

Planning Horizon Length

Optimality Loss (%)

η=0.2 η=0.5 η=0.8

Figure 2.3. Value of λ_m: k = 0.5, θ = 0.5

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0 5 10 15 20 25 30 35 40

k

Optimality Loss (%)

Myopic Policy 1−Heuristic 3−Heuristic 5−Heuristic

Figure 2.4. Value of λ_m and λⁱ_h: θ = 0.5, η = 0.5

2.6.2 Effective Heuristic Policies under Network Externalities

In this subsection, we propose some easy-to-implement heuristic policies and explore when these heuristics effectively leverage network externalities. As shown in Section 2.6.1, the myopic policy may have a poor performance because it ignores the opportunity of inducing future demands via network externalities. Thus, we consider the heuristic policies that balance generating current profits and inducing demands in the near future (within 5 periods) through network externalities. More specifically, in each period t, the firm dynamically maximizes the expected total discounted profit in the moving time window from period t to period t + i (i = 1, 3, 5). We call the heuristic policy to maximize the profit in the moving time window of length i as the i−heuristic (i = 1, 3, 5). Clearly, obtaining the i−heuristic (i = 1, 3, 5) only involves solving a dynamic program with planing horizon length i + 1, and is, thus, computationally light. Hence, the i−heuristic policy (i = 1, 3, 5) is easy to implement. Let V_hⁱ be the expected total profit under the i−heuristic policy. We have V^∗ ≥ V_h⁵ ≥ V_h³ ≥ V_h¹ ≥ Vm. The metric of interest is

λⁱ_h := V^∗− Vhⁱ

V^∗ × 100% which measures the optimality gap of the i−heuristic policy (i = 1, 3, 5). We conduct the numerical experiments under the parameters t = 20, k = 0.2, 0.5, 0.8, θ = 0.2, 0.5, 0.8, and η = 0.2, 0.5, 0.8.

Figures 2.4 - 2.6 summarize the results of our numerical study on the performance of i−heuritic policies (i = 1, 3, 5). The results show that, compared with the myopic

policy that completely ignores the future demand-inducing opportunities, the i−heuristics (i = 1, 3, 5) significantly improve the profitability of the firm in the presence of network externalities. In particular, the 5−heuristic leads a very low profit loss compared with the optimal policy (no more than 2%, in contrast to the more-than-30% optimality gap of the myopic policy). Therefore, the firm can effectively exploit network externalities by slightly looking into the future and balancing the tradeoff between generating current profits and inducing near future demands. Moreover, as shown in Figures 2.4 - 2.6, if the network externalities intensity k, the social customer proportion θ, or the carry-through rate of network size η is higher, the i−heuristic policies are more valuable relative to the myopic policy. As k, θ, or η increases, the tradeoff between generating current profits and inducing future demands becomes more intensive, and, thus, the forward-looking i−heuristics can deliver higher values to the firm compared with the myopic policy. We have also performed numerical analysis for the i−heuristic policies with i > 5. These more forward-looking heuristic policies cannot generate significantly better performances over the 5−heuristic policy. This further demonstrates that, to exploit network externalities, it suffices for the firm to balance generating current profits and inducing demands in the near future. Finally, we remark that our numerical results are robust and continue to hold in the settings where the planning horizon length T is greater than 20 and/or the market non-stationary (i.e., the maximum intrinsic valuation ¯Vt varies with time t).

For concision, we only present the results for the case where T = 20 and the market is stationary in this chapter.