CAPITULO II MARCO TEÓRICO
2.4. Influencias en la obra de Chambi
of active supply channels from which the firm orders.
To conclude this subsection, we characterize some preliminary concavity and differ-entiability properties of the value and objective functions in the following lemma.
Lemma 17 For t = T, T − 1, · · · , 1 and any given (It, θt), the following statements hold:
(a) Ψt(·|θt) is concave and continuously differentiable in z.
(b) Jt(·, ·, It|θt) is strictly jointly concave and continuously differentiable in (dt, qt).
(a) Vt(·|θt) is concave and continuously differentiable in It.
It follows immediately from Lemma 17 that the optimal joint pricing and ordering policy (d∗t(It, θt), q∗t(It, θt)) is well-defined and unique in the feasible setF.
6.4.2 Comparative Statics Analysis with Our New Method
First we observe that, by Equation (6.13), the objective function in each period Jt(·, ·, It|θt) is of the similar form to our illustrative optimization problem (i.e.,
Equa-tion (6.1)) in SecEqua-tion 6.3. Therefore, following the same argument as the discussion in Section 6.3.1, the standard IFT and MCS approaches are generally not applicable to com-parative statics analysis in our general joint pricing and inventory management model with demand segmentation, supply diversification and fluctuating market environment.
Therefore, we employ our new comparative statics method to study this model. Moreover, Lemma 15 applies to the proofs of several comparative statics results in this subsection, including Theorem 6.4.1 and Theorems 6.4.4-6.4.6. To begin with, we apply our new comparative statics method to characterize the optimal policy structure in the following theorem.
Theorem 6.4.1 (Optimal policy structure.) For t = T, T − 1, · · · , 1 and any given θt, the following statements hold:
(a) For each i ∈ N , di∗t (It, θt) is continuously increasing in It. Moreover, there exists
Theorem 6.4.1 shows that, in each period, the optimal policy is a state-dependent threshold policy. More specifically, for each demand segment i ∈ N [supply channel j ∈ M], the firm should sell to this segment [order from this channel] if and only if the starting inventory level It is above [below] the corresponding threshold Itd,i(θt) [Itq,j(θt)].
This optimal policy structure is characterized by employing our new method to establish the monotonicity of the optimal sales price/order quantity with respect to the starting inventory level. More specifically, both the optimal sales price for each demand segment, pi(dit∗(It, θt)), and the optimal order quantity from each supply channel, qtj∗(It, θt), are
decreasing in the starting inventory level, whereas the optimal total order-up-to level x∗t(It, θt) is increasing in the starting inventory level. Consequently, the optimal set of active demand segments, Nt∗(It, θt) [active supply channels, M∗t(It, θt)], is increasing [decreasing], in the set inclusion order, in the starting inventory level. Theorem 6.4.1 generalizes the base-stock list-price policy in the joint pricing and inventory management literature to the general setting with demand segmentation, supply diversification, and market environment fluctuation. Due to the diseconomy of scale and supply diversi-fication, the order-up-to level and sales prices are inventory-dependent in this general setting. Finally, we remark that if the multiplicative random perturbation in market size is demand-segment-dependent (i.e., Dti(pit, Λit) = Λitdi(pit)ςti + ϵit for i ∈ N and ςti’s are independent for different i’s), parts (b) and (c) of Theorem 6.4.1 still hold but part (a) doesn’t. It is well established in the inventory management literature that when there exist multiple multiplicative random perturbations in the system, the optimal order quan-tities and/or sales prices are, in general, not monotone in the starting inventory level (see [72]).
A key question in this inventory system is, for a given starting inventory level and market state, how to determine the optimal set of active demand segments, Nt∗(It, θt), and the optimal set of active supply channels, M∗t(It, θt). The following theorem par-tially addresses this issue by comparing the optimal purchasing probabilities for different demand segments, and the optimal order quantities from different supply channels.
Theorem 6.4.2 For t = T, T− 1, · · · , 1 and any given θt, the following statements hold:
In Theorem 6.4.2, we show that the firm sells more to a demand segment with higher marginal revenue with respect to demand, and it orders more from a supply channel with lower marginal procurement cost. Moreover, when the marginal revenues with respect to demand [marginal procurement costs] for different demand segments [supply channels]
have the same order for all purchasing probabilities [order quantities], the optimal set of active demand segments [supply channels], Nt∗(It, θt) [M∗t(It, θt)], is consecutive in the marginal revenue with respect to demand [marginal procurement cost].
Next, we employ our new comparative statics method to study the impact of market fluctuation upon the firm’s optimal pricing and ordering policy. In this application, we integrate our new method with the standard backward induction argument to perform comparative statics analysis in a dynamic program. More specifically, by employing Lemma 16, we iteratively link the comparison between optimizers and that between partial derivatives of the value functions and objective functions by backward induction.
This treatment is necessary because the current market state also impacts future market states and, thus, the value functions in the future. For the rest of this subsection, we make the additional assumption that ςt = 1 with probability 1 for all t, i.e., the demand process follows an additive form. The additive demand assumption is commonly imposed in the joint pricing and inventory management literature for tractability (see, e.g., [112, 136, 189]). In our model, this assumption enables us to iteratively link the monotone relationship between the optimizers and that between the partial derivatives. For the rest of this subsection, since ςt = 1 with probability 1 for all t, we rewrite the objective function in period t as stock in period t with starting inventory level It and market state θt. The following theorem characterizes the impact of current market size on the optimal sales prices and order quantities.
Theorem 6.4.3 (Impact of market size.) Assume that, for each t = T, T −1, · · · , 1, ςt = 1 with probability 1. For any given t, let θt = (Λt, ct) and ˆθt= ( ˆΛt, ct) with ˆΛt> Λt. For any It, the following statements hold:
(a) ∂ItVt(It|ˆθt)≥ ∂ItVt(It|θt).
(b) For each i ∈ N , dit∗(It, ˆθt) ≤ dit∗(It, θt), Itd,i(ˆθt) ≥ Itd,i(θt), and, thus, Nt∗(It, ˆθt) ⊂ Nt∗(It, θt).
(c) For each j ∈ M, qjt∗(It, ˆθt) ≥ qjt∗(It, θt), Itq,j(ˆθt)≥ Itq,j(θt), and, thus, M∗t(It, θt)⊂ M∗t(It, ˆθt).
(d) x∗t(It, ˆθt)≥ x∗t(It, θt).
Theorem 6.4.3 proves that an increase in the current market size of any demand segment has the following impacts: (a) it prompts the firm to increase the sales price for each demand segment; (b) it drives the firm to order more from each supply channel;
and (c) it motivates the firm to set a higher total order-up-to level. As the market size of one demand segment increases, the firm should increase its order quantities from all the supply channels to match supply with demand, so the optimal set of active supply channels is enlarged. At the same time, the firm should increase its sales prices in all demand segments, and the optimal set of active demand segments is smaller. Moreover, since the potential market size is more likely to become larger with a larger current market size, it is optimal for the firm to keep a higher total order-up-to level.
The risks and opportunities of procurement cost fluctuation have been extensively studied in [173]. In a model with one demand segment and two supply channels, the paper shows that inventory becomes more valuable under a higher current procurement cost, and the optimal sales price is increasing in the current procurement cost so that the firm should pass part of the cost fluctuation risk to its customers. In Theorem 6.4.4 below, we generalize these results to our joint pricing and inventory management model with demand segmentation, supply diversification, and market environment fluctuation.
More specifically, we show that, with a higher reference procurement cost of any supply channel, the marginal value of inventory is higher, and the firm charges a higher sales price in each demand segment. As a result, the demand in each segment and the optimal set of active segments are decreasing in the reference procurement cost of any supply channel.
On the other hand, [173] show that the impact of cost on the firm’s replenishment policy is more involved, because the current procurement cost also summarizes the infor-mation on future costs. When facing a higher current procurement cost, the firm faces the tradeoff between ordering less to save current cost and ordering more to speculate
on higher future costs. Numerical studies in [173] demonstrate that the optimal order quantities may not be monotone in the current procurement cost when the firm orders its inventory either from a spot market or through a forward-buying contract. In our model, the optimal order quantity from a supply channel continues to be non-monotone in its own reference procurement cost. However, we are able to show, in the following theorem, that as the reference procurement costs of one or more supply channels increase, the optimal order quantities and ordering thresholds of the supply channels with unchanged reference procurement costs increase as well.
Theorem 6.4.4 (Impact of current reference procurement cost.) Assume that, for each t = T, T−1, · · · , 1, ςt= 1 with probability 1. For any given t, let θt= (Λt, ct) and ˆθt = (Λt, ˆct) with ˆct> ct. For any It, the following statements hold:
(a) ∂ItVt(It|ˆθt)≥ ∂ItVt(It|θt).
(b) For each i ∈ N , dit∗(It, ˆθt) ≤ dit∗(It, θt), Itd,i(ˆθt) ≥ Itd,i(θt), and, thus, Nt∗(It, ˆθt) ⊂ Nt∗(It, θt).
(c) If ˆcjt = cjt, qjt∗(It, ˆθt)≥ qtj∗(It, θt) and Itq,j(ˆθt)≥ Itq,j(θt).
In addition to the current market condition, the firm should also take into account the future market trend to achieve the long-run optimality. Our new comparative statics method enables us to offer insights on the optimal responses of the firm to potential changes in the future market condition. We first study the impact of future market size trend on the firm’s optimal decisions.
Theorem 6.4.5 (Impact of market size trend.) Assume that, for each t = T, T − 1,· · · , 1, ςt= 1 with probability 1. Let the two systems be equivalent except that ˆξtΛ,i(Λit)≥s.d.
ξtΛ,i(Λit) for any t, i∈ N , and Λt. For any t and (It, θt), the following statements hold:
(a) ∂ItVˆt(It|θt)≥ ∂ItVt(It|θt).
(b) For each i ∈ N , ˆdit∗(It, θt) ≤ dit∗(It, θt), ˆItd,i(θt) ≥ Itd,i(θt), and, thus, ˆNt∗(It, θt) ⊂ Nt∗(It, θt).
(c) For each j ∈ M, ˆqjt∗(It, θt) ≥ qjt∗(It, θt), ˆItq,j(θt)≥ Itq,j(θt), and, thus, M∗t(It, θt)⊂ Mˆ∗t(It, θt).
(d) ˆx∗t(It, θt)≥ x∗t(It, θt) and ˆ∆∗t(It, θt)≥ ∆∗t(It, θt).
Theorem 6.4.5 shows that, under a higher market size trend for any demand segment, it is optimal to charge higher sales prices to all demand segments and, thus, sell to a smaller set of segments. On the other hand, a higher market size trend implies higher future demand, so the firm should order more from all supply channels, expand the set of active supply channels, and set a higher safety stock to hold more inventory for future consumption.
As shown by [173], a higher procurement cost trend increases the marginal value of inventory and prompts the firm to increase its order quantities both from the spot market and through the forward-buying contract so as to save the future cost. A higher safety stock should also be kept. In addition, the firm should raise its sales price to consume its inventory in the most profitable way. In our general model, we show that, when the reference procurement cost trend in one system is higher than that in the other, all of the comparative statics results in [173] continue to hold for each demand segment and supply channel. In addition, with a higher cost trend, the optimal set of active demand segments [supply channels] is smaller [larger].
Theorem 6.4.6 (Impact of cost trend.) Assume that, for each t = T, T − 1, · · · , 1, ςt = 1 with probability 1. Let the two systems be equivalent except that ˆξtc,j(cjt)≥s.d. ξtc,j(cjt) for any t, j ∈ M and ct. For any t and (It, θt), the following statements hold:
(a) ∂ItVˆt(It|θt)≥ ∂ItVt(It|θt).
(b) For each i ∈ N , ˆdit∗(It, θt) ≤ dit∗(It, θt), ˆItd,i(θt) ≥ Itd,i(θt), and, thus, ˆNt∗(It, θt) ⊂ Nt∗(It, θt).
(c) For each j ∈ M, ˆqjt∗(It, θt) ≥ qjt∗(It, θt), ˆItq,j(θt)≥ Itq,j(θt), and, thus, M∗t(It, θt)⊂ Mˆ∗t(It, θt).
(d) ˆx∗t(It, θt)≥ x∗t(It, θt) and ˆ∆∗t(It, θt)≥ ∆∗t(It, θt).
In addition, our new method enables us to perform comparative statics analysis for the optimal decisions in different models with non-parameterizable changes. More specifically, we employ our method to characterize the impact of sales and procurement flexibilities (i.e., additional demand segments and supply channels) upon the firm’s optimal pricing
and replenishment policy. When the firm is blessed with the opportunity to sell to additional demand segments, the marginal value of inventory increases, and the firm should charge higher prices in the original segments. Moreover, the firm should increase its replenishment quantities from all supply channels and expand the set of active supply channels, so as to match supply with the higher demand from a larger pool of segments.
These intuitions are formalized in the following theorem.
Theorem 6.4.7 (Impact of additional demand segments.) Assume that, for each t = T, T− 1, · · · , 1, ςt= 1 with probability 1. Let the two systems be equivalent except for N ⊂ ˆN . For t = T, T − 1, · · · , 1, and any (It, θt), the following statements hold:
(a) ∂ItVˆt(It|θt)≥ ∂ItVt(It|θt).
(b) For each i ∈ N , ˆdit∗(It, θt) ≤ dit∗(It, θt), ˆItd,i(θt) ≥ Itd,i(θt), and, thus, ( ˆNt∗(It, θt)∩ N ) ⊂ Nt∗(It, θt).
(c) For each j ∈ M, ˆqjt∗(It, θt) ≥ qjt∗(It, θt), ˆItq,j(θt)≥ Itq,j(θt), and, thus, M∗t(It, θt)⊂ Mˆ∗t(It, θt).
(d) ˆx∗t(It, θt)≥ x∗t(It, θt).
On the other hand, the supply diversification strategy enables the firm to hedge against the procurement cost fluctuation risk and the diseconomy of scale of the supply channels.
By sourcing from a larger supply pool, the firm enjoys more procurement flexibility, and orders less from each of the original supply channels. Moreover, the marginal value of inventory is smaller with a larger supply pool, and, to match supply with demand, the firm should set lower sales prices in all demand segments and sell to more segments.
Theorem 6.4.8 (Impact of additional supply channels.) Assume that, for each t = T, T− 1, · · · , 1, ςt= 1 with probability 1. Let the two systems be equivalent except for M ⊂ ˆM. For t = T, T − 1, · · · , 1, and any (It, θt), the following statements hold:
(a) ∂ItVˆt(It|θt)≤ ∂ItVt(It|θt).
(b) For each i ∈ N , ˆdit∗(It, θt) ≥ dit∗(It, θt), ˆItd,i(θt) ≤ Itd,i(θt), and, thus, Nt∗(It, θt) ⊂ Nˆt∗(It, θt).
(c) For each j∈ M, ˆqtj∗(It, θt)≤ qjt∗(It, θt), ˆItq,j(θt)≤ Itq,j(θt), and, thus, ( ˆM∗t(It, θt)∩ M) ⊂ M∗t(It, θt).
To sum up, comparative statics analysis is essential in our general joint pricing and inventory management model with demand segmentation, supply diversification, and market environment fluctuation. Although the standard IFT and MCS approaches do not apply to this complex model, our new comparative statics method enables us to characterize its optimal policy as a state-dependent threshold policy, and to analyze the impact of market fluctuation and operational flexibilities upon the optimal policy.
6.5 Application of the New Comparative Statics Method in a Competition