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. 2023 Sep 7;9(9):e19844. doi: 10.1016/j.heliyon.2023.e19844

Table 3.

Optimal decision and profit of integrator and provider in two periods.

		Logistics integrator		Logistics provider
		$β$	$π_{i}$	$w$	$π_{p}$
Enabling strategy	The first period	$β_{1}^{e *} = \frac{p - (a - p) k}{2 k v_{1}}$	$π_{i 1}^{e *} = \frac{(p - (a - p) k) (p + 3 (a - p) k)}{8 k}$	$w_{1}^{*} = \frac{p + (a - p) k}{2}$	$π_{p 1}^{e *} = \frac{{(p + (a - p) k)}^{2}}{4 k}$
Enabling strategy	The second period	$β_{1}^{m *} = \frac{p}{k v_{1}}$	$π_{i 2}^{e *} = \frac{p^{2} - 3 k^{2} {(λ a - p)}^{2} + 2 p k (λ a - p)}{8 k}$	$w_{2}^{*} = \frac{k (λ a - p) + p}{2}$	$π_{p 2}^{e *} = \frac{{(k (λ a - p) + p)}^{2}}{4 k}$
Merger strategy	The first period	$β_{2}^{e } = \frac{β_{1}^{m } [p - k (λ a - p)]}{2 p}$	$π_{i 1}^{m *} = (1 - α) (\frac{p^{2} + 2 (a - p) k p}{2 k})$	–	$π_{p 1}^{m *} = α (\frac{p^{2} + 2 (a - p) k p}{2 k})$
Merger strategy	The second period	$β_{2}^{m } = \frac{p}{k v_{2}} = β_{1}^{m }$	$π_{i 2}^{m *} = (1 - α) (\frac{p^{2} + 2 (λ a - p) k p}{2 k})$	–	$π_{p 2}^{m *} = α (\frac{p^{2} + 2 (λ a - p) k p}{2 k})$