Centralized Decisions for a Two-Stage Supply Chain with Price-Discount Dependent Demand

Xiyang HOU; Yongjiang GUO

doi:10.21078/JSSI-2018-249-11

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Journal of Systems Science and Information ›› 2018, Vol. 6 ›› Issue (3) : 249-259. DOI: 10.21078/JSSI-2018-249-11

Centralized Decisions for a Two-Stage Supply Chain with Price-Discount Dependent Demand

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Abstract

In this paper, we study a centralized supply chain for a two-stage with selling price discount. This supply chain consists of a supplier and a retailer. Based on the feature that the product's selling season is short and the supply chain faces great demand uncertainty. We consider a two-stage scenario where, at the beginning of stage 1, the supplier reserves production capacity based on historic data in advance, stage 2 comes to us after some leadtime, both the supplier and the retailer update the demand information, the retailer then places an order not exceeding the reserved capacity based on the selling-pricing discount dependent demand. We make optimal decisions on the reserved capacity in stage 1, selling price discount and order quantity in stage 2. In this supply chain, the pattern in stage 2 is figured out first, and then stage 1 is cleared as well. Then we present a numerical example to give some insights. Finally we get some conclusions.

Key words

two-stage supply chain / reserved capacity / optimal price discount / optimal order quantity

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Xiyang HOU , Yongjiang GUO. Centralized Decisions for a Two-Stage Supply Chain with Price-Discount Dependent Demand. Journal of Systems Science and Information, 2018, 6(3): 249-259 https://doi.org/10.21078/JSSI-2018-249-11

1 Introduction

In this paper, we consider a two-stage supply chain facing a pricing-discount dependent demand, which is sequent to Petruzzi and Dada^[1] whose model facing a pricing-dependent demand. This scenario occurred in two stages. In stage 1, the supplier reserves a certain capacity accordingly to the historic demand data. After some leadtime stage 2 arrives, the manager updates the supply chain based on the new information, the supplier offers a wholesale price to retailer and the retailer makes an order quantity to the supplier and a selling price discount to market. We suppose that, for retailer, the unmet demand is lost, the unsold items are salvaged, for supplier the unused portion of the reserved capacity is salvaged. We make optimal decisions on the reserved capacity in stage 1, selling price discount and order quantity in stage 2.

Pricing-discount occurred between buying and selling parties is the preferential privilege in commodity procurement. It is often used in group-buying strategy. Group-buying, as a new mechanism for selling through social interactions, has generated increasing attention in recent years due to the simplification of the information and communication technologies. The group-buying model can be looked as a generalization of the classical newsvendor model which is from the market with random situation, such as random information and random demand, see Dvoretzky, et al.^[2] and Scarf^[3]. The group-buying has a history that has been traced back to 1990s (see, e.g., Anand and Aron^[4]), when the internet-based group-buying is being wildly used for both business-to-business (B2B) and business-to-consumer (B2C) transaction. In some industries such as food services and health care, group-buying organizations have a dominate presence, see Hu^[5]. In the scenario of group purchasing, how to develop the optimal discount strategy becomes a key issue which affects the profits of sellers, see Dolan^[6]. There is little work on the group-buying, Luo and Wang^[7] regard a group-buying as a promotion way and consider the group-buying's impact on the sellers. Taking it into account that the seasonal epidemic goods are developing and the life cycle of most products is constantly shortened, the authors introduce traditional newsvender model framework.

Two-stage supply chain model is basic model in management science, and captures researcher's interest in recent years. Barnes-Schuster, et al.^[8] deals with a two-stage correlated demand model, in which the buyer purchases an option besides placing a firm order at the start. After the demand is realized in the first stage, the buyer can partly execute the option to satisfy the demand in the second phase. Mathur and Shah^[9] also consider a two-stage supply chain coordination problem, in which the supplier decides the production quantity before the demand is realized. Mathur and Shah propose to apply the contract with two-way penalties to coordinate the supply chain. Conditions of the contract for achieving supply chain coordination are derived separately for the cases when demand follows an uniform distribution and when production capacity is jointly decided by both parties of the supply chain. Chen, et al.^[10] consider a scenario where the supplier reserves production capacity for the retailer in advance, and permits the retailer to place an order not exceeding the reserved capacity after a demand information update during a lead-time. They formulate a two-stage optimization problem in which the supplier decides the amount of capacity reservation in the first stage, and the retailer determines the order quantity and the retail price after observing the demand information in the second stage. However they do not consider a group-buying strategy.

In this paper, we construct a centralized supply chain together with two characteristics of pricing-discount and two-stage based on the situation of selling a seasonal product. Our work is based on previous work and generalizes it. We consider a two-stage supply chain where, at the beginning of stage 1, the supplier reserves production capacity based on historic data in advance, stage 2 comes to us after some leadtime, both the supplier and the retailer update the demand information, the retailer then places an order not exceeding the reserved capacity based on the selling-pricing discount dependent demand. What should be pointed out is our model's exclusive feature: In our model, the retail price is fixed and the demand function becomes a function of price discount. We make optimal decisions on the reserved capacity in stage 1, selling price discount and order quantity in stage 2. The rest of the paper is organized as follows. In Section 2, we describe the model in details, including the basic assumptions and the decision structures. In Section 3, we make optimal decisions on the reserved capacity in stage 1, selling price discount and order quantity in stage 2. We first analyze stage 2 and then stage 1. We give a numerical example to explain the optimal decisions obtained in Section 3. In Section 4, we make conclusions.

2 Model Description

The supply chain operates in two stages. At stage 1, that is, at the very beginning of the model, the supplier reserves a certain capacity accordingly to the historic demand data. Stage 2 comes to us after some leadtime, based on the new information the manager updates the supply chain, the supplier offers a wholesale price to retailer and the retailer makes an order quantity to the supplier and a selling price discount to market. The demand occurs at the end of stage 2. We suppose that, for retailer, the unmet demand is lost, the unsold items are salvaged, for supplier the unused portion of the reserved capacity is salvaged.

2.1 The Demand Function

In this paper, we assume that the demand function is in the additive form, and is a function of selling price discount, that is,

D (a) = D + α (a),

where

a

is the price discount parameter with its value assumed to be in

(0, 1]

α (a)

is the increasing part of the demand due to the price discount.

D

is the demand when there is no price discount. We claim

α (1) = 0

, because there is no discount when

a = 1

, and then

D (1) = 1

. For the increasing part of the demand

α (a)

, we adopt a linear equation:

α (a) = m + n a,

where

m > 0

is the impact factor of the demand due to the price discount, represents the elastic coefficient of the demand responsiveness to the selling price discount,

n < 0

is a parameter and satisfies

α (1) = m + n = 0

. This implies that the demand function have the following form:

D (a) = D + m (1 - a) .

We give more information for the parameter

m

and

D

. For parameter

m

, in stage

1

m

is thought as a positive random variable, because, in generally,

m

is unknown prior to the selling season when the capacity reservation is made. However, after a leadtime, stage

2

is entered, in which the newsvendor model updates the demand forecast based on some new information, then

m

is realized. Thus, there are two aspects of the demand uncertainty in our model, that is,

m

(in stage

1

) and

D

. We suppose that it is distributed in the range

[L, H]

with a cumulative distribution function (c.d.f.)

G (\cdot)

and its corresponding probability density function (p.d.f.)

g (\cdot)

. The random variable

D

lies in the interval

[A, B]

with c.d.f.

F (\cdot)

, p.d.f.

f (\cdot)

and the mean

μ

2.2 Notations

The needed cost notations are summarized as follows:

M

: The reserved capacity of the supplier;

Q

: The retailer's order quantity to the supplier;

c_{1}

: Per unit cost of the reserved capacity in stage

1

;

v_{1}

: Per unit salvage value of the unused capacity, i.e., the part committed in stage

1

but not actually used for production;

c_{2}

: Per unit product cost in stage

2

;

v_{2}

: Per unit salvage value of the leftover inventory at the end of the selling season;

π

: Per unit penalty cost of stock-out;

p

: The constant selling price to customer.

2.3 Assumptions

To operate the model more reasonable, we need the following assumptions:

v_{1} < c_{1}

, it means that the reserved capacity cost is larger than the salvage value of the unused capacity.

p

is a constant value.

3) The random variable

m

and

D

are assumed to be independent for convenience.

v_{2} < c_{2}

or the retailer will order much more products to extend his profit.

v_{2} < π

a p > c_{1} + c_{2}

, it guarantees

D (a) > 0

7) The hazard rate

k (x) = f (x) / (1 - F (x))

D

satisfies:

2 k^{2} (x) + k^{'} (x) > 0.

3 The Optimal Decision for Supply Chain

In this section, we will study the optimal decision in two steps: Stage

1

and stage

2

. We first go to stage

2

, and then stage

1

. In stage

2

, we will determine the optimal selling price discount and order quantity based on constant

m

. Then in stage

1

, we will make the optimal decision on the capacity

M

based on random variable

m

3.1 The Retailer's Optimal Pricing Discount and Ordering in Stage $2$

In stage

2

, we go to find the optimal decision for the selling price discount and ordering quantity of the retailer. At the beginning of stage

2

m

is thought as a positive constant, denoted by

\tilde{m}

, and the demand function is then updated to

D (a ∣ \tilde{m}) = D + y (a),

where

y (a) = \tilde{m} \cdot (1 - a)

denotes the deterministic part of the demand function. In the following, we first give the profit functions for given capacity

M

. Let

Π_{21} (a, Q)

be the profit function when the reserved capacity is enough to satisfy the order quantity, that is,

Q < M

; and

Π_{22} (a, Q)

be the profit function when the reserved capacity is not enough to satisfy the order quantity, that is,

Q ⩾ M

. For case

Q ⩾ M

, the retailer orders a quantity

Q

, however, the supplier provides

M

only. In this way, it says that the system just determines the optimal selling price discount, which is independent to the order quantity. So,

Π_{22} (a, Q)

is degenerated into a univariate function and denoted by

Π_{22} (a)

. The profit functions are: When

Q < M

Π_{21} (a, Q) = a p [D (a ∣ \tilde{m}) \land Q] - c_{2} Q + v_{2} [Q - D (a ∣ \tilde{m})]^{+} - π [D (a ∣ \tilde{m}) - Q]^{+} + V_{1} (M - Q),

and when

Q \geq M

Π_{22} (a) = a p [D (a ∣ \tilde{m}) \land M] - c_{2} M + v_{2} [M - D (a ∣ \tilde{m})]^{+} - π [D (a ∣ \tilde{m}) - M]^{+},

where

a^{+} = max {a, 0}

and

a \land b = min {a, b}

for any members

a

and

b

Next, we will discuss the above two profit functions and make the optimal decision respectively. First we will discuss the case of

Q < M

, and then

Q \geq M

For case

Q < M

, for the convenience of analysis, we define the following safety stocking factor, see Petruzzi and Dada^[1],

z = Q - y (a) .

We rewrite the profit function

Π_{21} (a, Q)

Π_{21} (a, z) = a p [D (a ∣ \tilde{m}) \land (y (a) + z)] - c_{2} (y (a) + z) + v_{2} [z - D]^{+} - π [D - z]^{+} + V_{1} (M - y (a) - z) .

Taking expectation with respect to

D

we have

\begin{array}{rcl} E [Π_{21} (a, z)] & = & (a p - c_{2} - v_{1}) [y (a) + μ] - (c_{2} + v_{1} - v_{2}) I (z) \\ - (a p + π - c_{2} - v_{1}) L (z) + v_{1} M . \end{array}

(1)

Where

I (z) = \int_{A}^{z} (z - x) d F (x) and L (z) = \int_{z}^{B} (x - z) d F (x) .

Next we give the optimal

a

and

z

. Before doing this, we first present a Lemma in the following, which show us the optimal

a

for any given

z

Lemma 1 Suppose that there is no capacity constraint at the beginning of stage $2$ . $E [Π_{21} (a,$ $z)]$ is concave in $a$ for any given $z$ , the optimal price discount is a unique function of $z$ satisfies the following:

a^{*} \equiv a (z) = a_{0} - \frac{L (z)}{2 \tilde{m}} + \frac{v_{1}}{2 p},

(2)

where

a_{0} = \frac{p \tilde{m} + p μ + c_{2} \tilde{m}}{2 p \tilde{m}}

is the optimal price discount in the absence of uncertainty.

Proof Deriving the first-order and the second-order partial derivatives of

E [Π_{21} (a, z)]

a

, we get

\begin{aligned} \frac{d E [Π_{21} (a, z)]}{d a} & = 2 p \tilde{m} (a_{0} - a) - p L (z) + v_{1} \tilde{m}, \\ \frac{d^{2} E [Π_{21} (a, z)]}{d a^{2}} & = - 2 p \tilde{m} . \end{aligned}

Since

\tilde{m} > 0

in stage

2

, it follows that

\frac{d^{2} E [Π_{21} (a, z)]}{d a^{2}} = - 2 p \tilde{m} < 0.

That is,

E [Π_{21} (a, z)]

is strictly concave in

a

. So,

E [Π_{21} (a, z)]

has a unique optimal solution

a^{*}

for any given

z

. Let

\frac{d E [Π_{21} (a, z)]}{d a} = 0

, we get

a^{*}

. Then Lemma

1

is proved.

Remark 1 The optimal price discount

a_{0}

without uncertainty, in Lemma

1

, appeared in [7, 9], readers can refer to them for details.

Based on Lemma

1

, we can get the optimal

z

in the following Lemma

2

Lemma 2 Suppose that the demand takes the form $D (a ∣ \tilde{m}) = \tilde{m} (1 - a) + D$ in stage $2$ . If there is no capacity constraint, then the optimal stocking and pricing discount policy is to stock $Q^{*} = y (a^{*}) + z^{*}$ units with the unit price discount $a^{*}$ , where $a^{*}$ is specified by Lemma $1$ and $z^{*}$ is the optimal solution that satisfies:

- (c_{2} + v_{1} - v_{2}) + (a (z^{*}) p + π - v_{2}) [1 - F (z^{*})] = 0.

(3)

Proof Deriving the first order derivative of

E [Π_{21} (a^{*}, z)]

z

, we get

\begin{aligned} \frac{d E [Π_{21} (a^{*}, z)]}{d z} & = \frac{\partial E [Π_{21} (a^{*}, z)]}{\partial z} + \frac{\partial E [Π_{21} (a^{*}, z)]}{\partial a^{*}} \frac{d a^{*}}{d z} \\ = - (c_{2} + v_{1} - v_{2}) + (a^{*} p + π - v_{2}) [1 - F (z)], \end{aligned}

where the second equality holds because

\frac{\partial E [Π_{21} (a^{*}, z)]}{\partial a^{*}} = 0

by Lemma

1

. To identify values of

z

that satisfy the above first-order optimality condition, let

\begin{aligned} R (z) & = \frac{d E [Π_{21} (a^{*}, z)]}{d z} \\ = - (c_{2} + v_{1} - v_{2}) + ((a_{0} - \frac{L (z)}{2 \tilde{m}} + \frac{v_{1}}{2 p}) p + π - v_{2}) [1 - F (z)], \end{aligned}

and consider the zeros of

R (z)

\frac{d R (z)}{d z} = - \frac{f (z)}{2 \tilde{m}} [2 \tilde{m} (a_{0} p + π - v_{2} + \frac{v_{1}}{2}) - p L (z) - p \frac{1 - F (z)}{k (z)}],

where

k (z) = \frac{f (z)}{1 - F (z)} .

Also,

\frac{d^{2} R (z)}{d z^{2}} |_{\frac{d R (z)}{d z} = 0} = - \frac{p f (z) [1 - F (z)]}{2 \tilde{m} k^{2} (z)} [2 k^{2} (z) + k^{'} (z)] .

Because

F (x)

is a distribution satisfying the condition

2 k^{2} (z) + k^{'} (z) > 0

, then, it follows that

R (z)

is unimodal or monotonous. So,

R (z)

has at most two roots. Further,

R (A) = \frac{p - c_{2} - v_{1}}{2} + \frac{p A}{2 \tilde{m}} + π > 0,

and

R (B) = - (c_{2} + v_{1} - v_{2}) < 0,

so,

R (z)

has only one root, it indicates that

R (z)

has a change from positive to negative. Therefore,

E [Π_{21} (a^{*}, z)]

has only one unique value

z^{*}

that satisfies

R (z^{*}) = 0

, i.e.,

- (c_{2} + v_{1} - v_{2}) + (a (z^{*}) p + π - v_{2}) [1 - F (z^{*})] = 0.

Then, we get the result.

Remark 2 Be similar with Theorem

2.3

in [7] or Theorem

1

in [9], we can claim that

E [Π_{21} (a, z)]

is unimodal in

z

2 k^{2} (z) + k^{'} (z) > 0

. Thus, there is a unique solution to

(3)

. Note that the solution exists for

z^{*}

. Substituting

z^{*}

into

(2)

, we can get the optimal price discount

a^{*}

Lemma 3 If

\tilde{m} \leq \frac{p (μ - L (z))}{p - c_{2} - v_{1}},

then, $a^{*} \geq 1$ , which means that there is no need price discount, in words, $a^{*} = 1$ ; if

\tilde{m} > \frac{p (μ - L (z))}{p - c_{2} - v_{1}},

then, $a^{*} < 1$ , which means that one can sell product at a optimal price discount $a^{*}$ .

Proof The proof is directly from

(2)

in Lemma

1

, and the proof is omitted.

Next, we consider the case

Q \geq M

. In this case, the reserved capacity is insufficient to warrant the optimal order quantity in stage

2

, then the system just determines the optimal price discount to run out the reserved capacity, i.e.,

Q = M

. So, the expected profit function

Π_{22}

is a univariate function in

a

, its expectation with respect to

D

gives:

\begin{aligned} E [Π_{22} (a)] = & (a p - v_{2}) [y (a) + μ] - (a p + π - v_{2}) \int_{M - y (a)}^{B} (y (a) + x - M) f (x) d x \\ + (v_{2} - c_{2}) M . \end{aligned}

(4)

Lemma 4 For the case $Q \geq M$ , that is, the reserved capacity is less than the optimal order quantity, then the system is optimized by ordering $Q = M$ and selling discount at $a_{2}^{*}$ , where $a_{2}^{*}$ is the unique solution of the following equation

p [y (a_{2}^{*}) + μ] - \tilde{m} (a_{2}^{*} p - v_{2}) - \int_{M - y (a_{2}^{*})}^{B} [p (y (a_{2}^{*}) + x - M) - \tilde{m} (a_{2}^{*} p + π - v_{2})] d F (x) = 0.

(5)

Proof Taking the first and second derivatives of

E [Π_{22} (a)]

, we have

\begin{aligned} \frac{d E [Π_{22} (a)]}{d a} = & p [y (a) + μ] - \tilde{m} (a p - v_{2}) \\ - \int_{M - y (a)}^{B} [p (y (a) + x - M) - \tilde{m} (a p + π - v_{2})] d F (x), \end{aligned}

\frac{d^{2} E [Π_{22} (a)]}{d a^{2}} = - 2 p \tilde{m} \int_{A}^{M - y (a)} - {\tilde{m}}^{2} (a p + π - v_{2}) f (M - y (a)) < 0,

which means

E [Π_{22} (a)]

is concave with respect to the selling price discount of

a

. Letting

\frac{d E [Π_{22} (a)]}{d a} = 0

, then we get the result.

3.2 Capacity Reservation Strategy in Stage $1$

At the very beginning of stage

1

, the manager needs to determine the reserved capacity facing an uncertain market. We do this decision based on the discussion in the above subsection, that is, the cases

Q < M

and

Q \geq M

. However, we noted that, in stage

1

, the parameter

m

D (a) = D + m (1 - a)

is thought to be a positive random variable.

Let

h (\tilde{m}) = \tilde{m} (1 - a^{*}) + z^{*},

which represents the optimal order quantity in stage 2 when the capacity is not constrained. Note that

h (\tilde{m})

is continuous in

\tilde{m}

, so there exist maximum and minimum in the range of

[L, H]

, denoted by

h_{max}

and

h_{min}

respectively. Let

Q < M

, this implies that

m < β (M)

, where

β (M) = \frac{2 p (M - z^{*}) + p (μ - L (z^{*}))}{p - c_{2} - v_{1}} .

Let

Π (m)

be the total profit function, its expected value is

\begin{aligned} E [Π (m)] = - c_{1} M + E [Π_{21}^{*}] I_{{Q \leq M}} + E [Π_{22}^{*}] I_{{Q > M}} . \end{aligned}

Taking expectation with respect to

m

, we get the expected profit function

E [Π (m)] = - c_{1} M + \int_{L}^{β (M)} E [Π_{21}^{*}] \cdot g (m) d m + \int_{β (M)}^{H} E [Π_{22}^{*}] \cdot g (m) d m .

(6)

Where

E [Π_{21}^{*}] = E [Π_{21} (a^{*}, z^{*})]

and

E [Π_{22}^{*}] = E [Π_{22} (a_{2}^{*})]

are the optimal profit in stage

2

fore cases

Q \leq M

and

Q > M

respectively. We assume that the reserved capacity satisfies

h_{min} \leq M \leq h_{max}

. From Lemma

3

, we know, if one can sell product at an optimal price discount

a^{*}

, the condition

m > \frac{p (μ - L (z))}{p - c_{2} - v_{1}}

must be true, that is, the product can sold with the optimal discount

a^{*}

. So, if

m \leq \frac{p (μ - L (z))}{p - c_{2} - v_{1}}

, the cumulative distribution function

G (m) = 0

. To avoid trivial cases, we assume

L \geq \frac{p (μ - L (z))}{p - c_{2} - v_{1}}

Next, we give the optimal reserved capacity.

Theorem 1 The optimal reserved capacity, denoted by $M^{*}$ , is the unique solution to the equation

\int_{L}^{β (M)} v_{1} g (m) d m + \int_{β (M)}^{H} [v_{2} - c_{2} + (a_{2}^{*} p + π - v_{2}) \int_{M - y (a_{2}^{*})}^{B} f (x) d x] g (m) d m = c_{1},

(7)

where

β (M) = \frac{2 p (M - z^{*}) + p (μ - L (z))}{p - c_{2} - v_{1}} .

Proof We note that when

M^{*} > m (1 - a^{*}) + z^{*} = Q^{*},

the optimal

a^{*}

and

z^{*}

can be taken. It is equal to the capacity constraint is unbound. On the other hand, when

M^{*} ⩽ m (1 - a^{*}) + z^{*},

the optimal

a^{*}

can not be taken. It is equal to the capacity constraint is bound. In fact, there are three possible cases.

The first case:

M^{*} > h_{max}

. In this case, when the capacity constraint is unbound, for any given

m

M^{*}

can satisfy the maximum

Q^{*}

. We maximize the objective function:

max_{M > h_{max}} E [Π] = - c_{1} M + \int_{L}^{H} E [Π_{21}^{*}] g (m) d m,

where

E [Π_{21}^{*}]

is the maximum expected profit in stage

2

. Since

\frac{d E [Π]}{d M} = v_{1} - c_{1} < 0,

the optimal capacity is the lower limit of this range, that is,

M^{*} = h_{max}

The second case:

M^{*} < h_{min}

. In this case, when the capacity constraint is bound,

M^{*}

can not satisfy the minimum

Q^{*}

. We maximize the objective function:

max_{M < h_{min}} E [Π] = - c_{1} M + \int_{L}^{H} E [Π_{22}^{*}] g (m) d m,

where

E [Π_{22}^{*}]

is the maximum expected profit in stage

2

. So the optimal order quantity in stage 2 is

M^{*}

The third case:

h_{min} \leq M^{*} < h_{max}

. In this case, we maximize the objective function

max_{h_{min} \leq M \leq h_{max}} E [Π] = - c_{1} M + \int_{L}^{β (M)} E [Π_{21}^{*}] \cdot g (m) d m + \int_{β (M)}^{H} E [Π_{22}^{*}] \cdot g (m) d m .

Taking the derivative of

E [Π]

with respect to

M

, we can get

\begin{aligned} \frac{d E [Π]}{d M} = & - c_{1} + \int_{L}^{β (M)} \frac{\partial E [Π_{21}^{*}]}{\partial M} g (m) d m + \int_{β (M)}^{H} \frac{\partial E [Π_{22}^{*}]}{\partial M} g (m) d m \\ + g (β (M)) \frac{d β (M)}{d M} (E [Π_{21}^{*}] - E [Π_{22}^{*}]) ∣_{m = β (M)} . \end{aligned}

Since

Q^{*} = M^{*}

when

m = β (M)

, we have

E [Π_{21}^{*}] = E [Π_{22}^{*}]

, Then

- c_{2} + v_{2} + (a_{2}^{*} p + π - v_{2}) \int_{M - y (a_{2}^{*})}^{B} f (x) d x = v_{1} .

This leads to

\frac{\partial E [Π_{21}^{*}]}{\partial M} = - c_{2} + v_{2} + (a_{2}^{*} p + π - v_{2}) \int_{M - y (a_{2}^{*})}^{B} f (x) d x = v_{1},

and

\frac{\partial E [Π_{22}^{*}]}{\partial M} = - c_{2} + v_{2} + (a_{2}^{*} p + π - v_{2}) \int_{M - y (a_{2}^{*})}^{B} f (x) d x .

So, we can get

\frac{d E [Π]}{d M} = \int_{L}^{β (M)} v_{1} g (m) d m + \int_{β (M)}^{H} [v_{2} - c_{2} + (a_{2}^{*} p + π - v_{2}) \int_{M - y (a_{2}^{*})}^{B} f (x) d x] g (m) d m - c_{1},

and

\begin{aligned} \frac{d^{2} E [Π]}{d M^{2}} = & v_{1} g (β (M)) \frac{\partial β (M)}{\partial M} - (v_{2} - c_{2} + (a_{2}^{*} p + π - v_{2}) \int_{M - y (a_{2}^{*})}^{B} f (x) d x) g (β (M)) \frac{\partial β (M)}{\partial M} \\ + \int_{β (M)}^{H} [p \frac{\partial a_{2}^{*}}{\partial M} \int_{M - y (a_{2}^{*})}^{B} f (x) d x - (a_{2}^{*} p + π - v_{2}) f (M - y (a_{2}^{*})) (1 + m \frac{\partial a_{2}^{*}}{\partial M})] g (m) d m \\ = & - (a_{2}^{*} p + π - v_{2}) f (M - y (a_{2}^{*})) \int_{β (M)}^{H} g (m) d m \\ + \int_{β (M)}^{H} [p \int_{M - y (a_{2}^{*})}^{B} f (x) d x - m (a_{2}^{*} p + π - v_{2}) f (M - y (a_{2}^{*}))] \frac{\partial a_{2}^{*}}{\partial M} g (m) d m \\ = & - (a_{2}^{*} p + π - v_{2}) f (M - y (a_{2}^{*})) \int_{β (M)}^{H} g (m) d m < 0. \end{aligned}

Therefore,

E [Π]

is concave in

M

when

M

is in the interval

[h_{min}, h_{max}]

. Hence, the optimal capacity to be reserved in stage

1

is the unique solution to

\frac{d E [Π]}{d M} = 0

When the optimal reserved capacity

M^{*}

is determined in stage 1, the expected profit is given as well.

Example 1 We take an example to illustrate the underlying managerial insight. Consider the following problem setting in the numerical study:

\begin{array}{rcl} B = 50, A = 10, μ = 30, c_{1} = 2, c_{2} = 3, v_{1} = v_{2} = 0, \\ π = 3, p = 30, L = 50, H = 100. \end{array}

We assume that the respective distributions of

m

and

D

are taken as uniform. With this date, we would like to see how the impact factor of the demand,

m

, affects the optimal price discount

a^{*}

, the optimal order quantity

Q^{*}

and the safety stocking

z^{*}

As can be seen in Table 1, we can get that

Q^{*}

is increasing in

m

a^{*}

and

z^{*}

are decreasing in

m

. This shows that the retailer will order much more and then set a lower selling price discount, when the impact factor of the demand becomes bigger.

Table 1 Effects of impact factor of the demand

$m$	$a^{*}$	$Q^{*}$	$z^{*}$	$M^{*}$
50	0.8478	53.3896	45.7796	55.5698
60	0.7979	57.6713	45.5453	59.8255
70	0.7624	61.9936	45.3616	64.0951
80	0.7357	66.3577	45.2137	68.4018
100	0.6484	80.1502	44.9902	85.2638

4 Conclusions

In this paper, we investigate a simple centralized supply chain that sells a seasonal product with selling price-discount strategy. At the beginning of the selling season, the supplier needs to reserve capacity, which leads to a meaningful leadtime. There is an opportunity for adjusting the production after demand information updating during the leadtime. The supply chain faces a two-stage strategy problem: firstly the supplier reserves an amount of capacity; secondly the retailer decides the order quantity and the selling price discount. It is especially significant in the business of group-buying. We make optimal decisions on the reserved capacity in stage 1, selling price discount and order quantity in stage 2, we also give a numerical example to present some insights after the optimal decisions is obtained.

References

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Funding

the National Natural Science Foundation of China(11471053)

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Abstract
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Cite this article
1 Introduction
2 Model Description
2.1 The Demand Function
2.2 Notations
2.3 Assumptions
3 The Optimal Decision for Supply Chain
3.1 The Retailer's Optimal Pricing Discount and Ordering in Stage $2$
3.2 Capacity Reservation Strategy in Stage $1$
Table 1 Effects of impact factor of the demand
4 Conclusions
References
Funding

Received	Accepted	Published
2016-12-02	2017-12-07	2018-06-25
Issue Date
2018-06-25

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Abstract

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Cite this article

1 Introduction

2 Model Description

2.1 The Demand Function

2.2 Notations

2.3 Assumptions

3 The Optimal Decision for Supply Chain

3.1 The Retailer's Optimal Pricing Discount and Ordering in Stage $2$

3.2 Capacity Reservation Strategy in Stage $1$

Table 1 Effects of impact factor of the demand

4 Conclusions

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Cite this article

1 Introduction

2 Model Description

2.1 The Demand Function

2.2 Notations

2.3 Assumptions

3 The Optimal Decision for Supply Chain

3.1 The Retailer's Optimal Pricing Discount and Ordering in Stage 2

3.2 Capacity Reservation Strategy in Stage 1

Table 1 Effects of impact factor of the demand

4 Conclusions

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3.1 The Retailer's Optimal Pricing Discount and Ordering in Stage $2$

3.2 Capacity Reservation Strategy in Stage $1$