Multi-product expedited ordering with demand forecast updates

Accepted Manuscript Multi-Product Expedited Ordering with Demand Forecast Updates

Bin Zhang, Yurui Ma PII:

S0925-5273(18)30404-3

DOI:

10.1016/j.ijpe.2018.09.034

Reference:

PROECO 7188

To appear in:

International Journal of Production Economics

Received Date:

12 April 2018

Accepted Date:

28 September 2018

Please cite this article as: Bin Zhang, Yurui Ma, Multi-Product Expedited Ordering with Demand Forecast Updates, International Journal of Production Economics (2018), doi: 10.1016/j.ijpe. 2018.09.034

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT Multi-Product Expedited Ordering with Demand Forecast Updates

Bin ZHANG1 Lingnan College, Sun Yat-sen University, Guangzhou 510275, China

Tel: 86-20-84110649, Fax: 86-20-84114823. E-mail: [email protected]

Dongxia Duan Lingnan College, Sun Yat-sen University, Guangzhou 510275, China

Tel: 86-20-84110649, Fax: 86-20-84114823.

Yurui MA Lingnan College, Sun Yat-sen University, Guangzhou 510275, China

Tel: 86-20-84110649, Fax: 86-20-84114823.

1

Corresponding author

ACCEPTED MANUSCRIPT Multi-Product Expedited Ordering with Demand Forecast Updates

Abstract This paper studies a purchasing problem with multiple products, resource constraints, demand forecast updates and expedited ordering. The retailer procures the initial quantities of multiple products before the selling season, which are restricted by limited resource, and the retailer can improve the demand forecasts through observing the initial sales. Based on the updated demand forecasts and inventories on hand, the retailer can place an expedited order at higher costs for products during the selling season, which is restricted by the minimum quantity of reordering. We analyze the retailer’s optimal procurement policies by trading off between forecast reliability, procurement costs and resource availability. The bi-level programming is used to model the purchasing problem and the binary search solution method is developed for solving the optimal solution. We extend the proposed algorithm to the case with two resource constraints, and also replace the expedited ordering with option contract in the extended study. Numerical experiments are designed to show the value of expedited ordering with demand forecast updates and to obtain managerial insights in comparison with the model with the option contract.

Keywords: Purchasing; Expedited ordering; Forecast updates; Resource constraints; Quick response; Option contract

1

ACCEPTED MANUSCRIPT 1. Introduction In today’s highly competitive market environment, retailers often face demand uncertainty when managing perishable or fashion products. The retailers often place orders before the selling season with unreliable demand forecasts due to long production and delivery lead time. In many industries, typical production lead-time is usually longer than the length of selling season (Fisher and Raman, 1996; Miltenburg and Pong, 2007). It’s very difficult to obtain reliable demand forecasts for the perishable or fashion products before the selling season, and it’s very common that retailers are suffering simultaneously from lost sales of popular products and markdown loss of unpopular products (Caro and Gallien, 2007). For example, in fast fashion industry, China-based sports fashion company Li Ning faced stock out of popular products and also carried 3.66 billion RMB inventory (1 US$≈6.05 RMB) in June 2012, which approaches to its total income 3.88 billion RMB1. The inventory level far exceeded its initial expectation. In 2013, the company declared to raise about 1.48 billion RMB through convertible securities for cutting down its inventory and reengineering its business process. It is challenging to manage demand uncertainty for the industries with perishable or fashion products. There are two common ways to manage uncertain demand based on updated demand forecasts: dynamic pricing and flexible purchasing. After the replenishment is finished, the retailers adopt dynamic pricing (especially price markdown) to manage uncertain demand based on updated demand forecasts. The enterprises use real-time sales data to update future demand and dynamically set the price, and many researchers studied these problems, such as Lin (2006), Şen and Zhang (2009), and Araman and Caldentey (2009). More relevant research can refer to Chen and Chen (2015) and den Boer (2015) who reviewed the developments in dynamic pricing research and demand learning. Flexible purchasing becomes popular in practice since quick response can be easily implemented with modern information systems. Based on the updated demand forecasts, some flexible purchasing policies including flexible contract, dual sourcing and expedited ordering are implemented to derive the value from demand forecast updates. For example, in computer manufacturing industry, some major manufacturers (e.g., Hewlett Packard and Sun Microsystems) are adopting flexible The report: The Li Ning Interim Financial Report, 2012, p31-33. http://www.irasia.com/listco/hk/lining/interim/ir96999-e02331.pdf. 1

2

ACCEPTED MANUSCRIPT contract, which allows the buyer to postpone the ordering time after receiving the improved demand forecasts. In most of study on flexible purchasing with demand learning, the demand forecasts are updated on new market information. Although demand forecasts can be improved significantly using the initial sales during the selling season, as showed in Fisher and Raman (1996), demand forecasts updating with the initial sales has obtained little attention in literature. The possible reason is that demand forecasts updating with the initial sales makes the flexible purchasing problems more difficult to solve. In this paper, we study a purchasing problem with an opportunity to place expedited in-season orders for multiple products after updating uncertain demand forecasts. The retailer places an initial order for all products before the selling season, and the initial order quantities are restricted by limited resources. The retailer improves the demand forecasts by observing the initial sales. Based on the updated demand forecasts and inventories on hand, the retailer places an expedited order at higher costs to fulfill future demand, and the expedited order quantities are restricted by the minimum reordering quantities. The objective of the retailer is to maximize the total expected profit by determining the order quantities for each product at the two stages. We model the purchasing problem with one and two resource constraints as bilevel programming problem, and efficient solution methods are developed for solving the optimal purchasing decisions. To show the performance of expedited ordering, we also study the problem with option contract and budget constraint. Numerical experiments are given to show the value of expedited ordering with demand forecast updates and to obtain managerial insights in comparison with the model with the option contract. This paper contributes to the literature on flexible purchasing problem and demand learning. Firstly, we use the initial sales data instead of new market information to update demand forecasts, which make the studied problem more difficult to solve, since the initial sales simultaneously affect the updated demand forecast and the inventory level at the instant of placing the expedited order. In most of research in literature, the new market information is used to update demand forecast, which only affects the updated demand forecast, and the on-hand inventory level is fixed before placing the second order. Secondly, we consider multi-product problem with two practical restrictions: 1) the resources are limited in the initial sales season, and 2) there is a minimum ordering quantity for the expedited delivery for 3

ACCEPTED MANUSCRIPT each product. Most studies on flexible purchasing problem with demand forecast updates focused on single product setting. The multi-product problems with restrictions are not easily solved. In addition, we also study the multi-product purchasing problem with option contract, and compare the efficiency of expedited ordering and option contract in the studied problem. The option contract model in our study is different from the option contract model in literature, since the works in literature considered demand forecast updates based on new market information in single-product setting. The numerical comparison of the expedited ordering model with the option contract model shows that which model is better depends on the cost parameters. The expedited ordering model outperforms the option contract model if the expedited ordering cost is small, and the option contract model is better if the unit reservation cost of the option contract is small. Our numerical study also shows effectiveness of the proposed solution method, and indicates that the value from the expedited ordering is very large when the retailer faces very limited resources. The rest of this paper is organized as follows. Section 2 presents a literature review. In Section 3, we describe the problem with one resource constraint. In Section 4, the structural properties of the optimal ordering policy are established, and the solution method is developed. In Section 5, we extend the problem to the twoconstraint problem and model the purchasing problem with option contract instead of expedited ordering. Section 6 is dedicated to numerical study. This paper is concluded in Section 7. All proofs are given in Appendix. 2. Literature review Several streams of research are relevant to this paper: dual sourcing strategy, flexible contract and expedited ordering with demand forecast updates, and multiproduct purchasing problems. Dual sourcing is common in supply chain practice. It becomes more popular to pair one responsive but costly supplier with a less responsive but less costly supplier to meet uncertain demand (Zhou and Chao, 2014; Ju et al., 2015; Biçer, 2015). Some researchers studied dual sourcing problems with demand forecast updates. For examples, Yan et al. (2003) studied a dual sourcing problem, in which one supplier is fast but expensive and the other is cheap but slow, and that demand forecast accuracy increases with time. Martínez-de-Albéniz and Simchi-Levi (2005) studied a 4

ACCEPTED MANUSCRIPT purchasing problem in which the buyer chooses a procurement portfolio from multiple contract suppliers and a spot market, and the demand information is updated during multiple periods. Cheaitou et al. (2014) proposed a two-period inventory control model combines demand forecast updating with the flexibility of two supply sources, the information quality has an impact on optimal policy. Li et al. (2017) considered a dual sourcing problem where the firm satisfies future demand with uncertainty risk via forward sourcing and spot trading. Some research has been done to investigate the purchasing problems with flexible contracts and demand forecast updates. For instances, Sethi et al. (2004) studied quantity flexible contracts with demand forecast updates in multi-period setting. Huang et al. (2005) investigated a two-stage ordering contract, in which the buyer has an opportunity to adjust the initial order with updated demand forecast. Chen et al. (2006) considered a risk sharing contract in which the manufacturer decides the initial production quantity in the first stage and the retailer determines the specific ordering quantity in the second stage when demand information is updated. Cai et al. (2015) considered minimum order quantity commitments in a two-period setting where retailers have the opportunity to update their orders before the selling season. Option contracts have also been considered in procurement problems with demand forecast updates. Wang and Tsao (2006) developed a supply contract with bidirectional options and formulated the buyer’s optimal policies. Nosoohi and Nookabadi (2016) investigated option contracts to provide the manufacturer with the flexibility to adjust his initial orders. Cheaitou and Cheaytou (2018) modeled a twoperiod purchasing problem with option contract and risky supplier, and demand forecast is updated with exogenous information. Zhang and Hua (2010) studied multiproduct purchasing problem with option contract and budget constraint, but they did not consider demand forecasts updates. Flexible supply can also be implemented by expedited ordering. Researchers considered different situations about expedited ordering with demand forecast updates. For examples, Gurnani and Tang (1999) considered a situation where demand forecast can be improved using market signals between the first and second ordering instants. Choi et al. (2003) studied a two-stage ordering problem in which the retailer can observe market information after placing an initial order, and updates the demand forecast before making the second order. Sethi et al. (2003) investigated a 5

ACCEPTED MANUSCRIPT multi-period inventory problem, in which the products can be delivered in either slow or fast mode, and demand forecasts are regularly updated. Sethi et al. (2005) analyzed a two-stage problem where the buyer places an additional order after forecast update, and the procurement cost at the second stage is uncertain at the first stage. Some researchers studied supply chain coordination in two-stage ordering problem with expedited ordering and demand forecast updates, such as Donohue (2000), Chen and Xu (2001) and Özer et al. (2007), and other researchers examined the use of order expediting in dealing with lead-time uncertainties (Kouvelis and Tang, 2012; Kim et al., 2015). In recent years, multi-product setting with demand forecast updates started to attract some attention. Miltenburg and Pong (2007) studied a constrained multi-item ordering problem with two order opportunities and Bayesian demand updates. Serel (2012) considered a multi-item quick response problem with budget constraint, in which the retailer has two order opportunities before the selling season and demand forecasts can be updated. Song et al. (2014) investigated a multi-item dual sourcing problem with budget and space constraints and Bayesian demand updates. ValenciaCárdenas et al. (2016) presented a comparison of four different demand forecast models for multiple products. Biçer and Seifert (2017) studied a dynamic order scheduling problem where the schedules dynamically update as the demand forecast evolves over time, and extended the model to a multi-product case. Our work differs from most of the previous literature in three aspects. Firstly, the demand forecasts in our study are updated based on the initial sales instead of new market information. Secondly, we study the two-stage purchasing problem in multiproduct setting with one and two limited resources and the minimum expedited ordering quantity constraints. These two aspects make the studied problem more challenging to solve. Finally, we compare the efficiency of expedited ordering and option contract in the studied problem to obtain some managerial insights. 3. Problem description Before modeling the problem, we first give the notation in Table 1. Insert Table 1 here Insert Fig. 1 here

6

ACCEPTED MANUSCRIPT The sequence of the events is shown in Fig. 1. The selling season is divided into two periods. The retailer places the initial order Q  (Q1 , , Qn ) for all products before the selling season based on the initial demand forecasts X  ( X 1 , , X n ) and

Y  (Y1 , , Yn ) , and receives the products at the start of the selling season. The retailer satisfies the demands in the first period with on-hand inventory Q  (Q1 , , Qn ) , and the demands exceeded on-hand inventories are lost. At the end of the first period, the retailer uses the sales data x  ( x1 , , xn ) in the first period to update the demand forecasts Y  (Y1 , , Yn ) , and places an expedited order q  (q1 , , qn ) at higher costs

e  (e1 , , en ) for all products which are delivered immediately. Then the retailer fulfills the demands of the second period with the on-hand inventories, which includes the leftover inventories in the first period and the expedited order quantities. In the retailer’s problem, the initial order quantities of multiple products in the first order are restricted by limited resource B1 , which can be budget, capacity or other resource in practice. The expedited order quantities of multiple products in the second order are restricted by supplier’s requirements and the minimum reordering quantity

l  (l1 , , ln ) . Two-period correlated demand is often modeled by joint distribution, especially by bivariate normal distribution (Fisher and Raman, 1996; Gurnani and Tang, 1999). In this study, we also model two-period demand ( X i , Yi ) using bivariate normal distribution, i.e., ( X i , Yi ) ～ N (  X i , Yi ,  X2 i ,  Y2i , i ) , i  0 , i  1, , n . Notice that

i  0 implies the two period demands are positively correlated. As shown in Fisher and Raman (1996), the two period demands are highly positively correlated in fashion retailing, and the correlation coefficients are from 0.8 to 0.9 for most products in their study. We have an investigation of a retail company and the data shows that the demand forecast precision of the whole selling season will significantly increase based on the sales data of first 10 days in selling season, there is a huge gap between initial demand forecast and actual sales of most product (see Fig. 2(a)), but the demand forecast based on the sales data of first 10 days in the selling season have high degree of coincidence with actual sales (see Fig. 2(b)). Insert Fig. 2 here 7

ACCEPTED MANUSCRIPT

Based on this observation, the retailer can improve demand forecast

Y  (Y1 , , Yn ) after observing the sales x  ( x1 , , xn ) . It’s well known (Bickel and Doksum, 1977) that the updated demand forecast Yi xi based on the initial sales xi follows normal distribution N ( Yi  i

Y ( x   X ),  Y2 (1  i2 )) , i  1, , n . X i i

i

i

i

For given (q, Q, x) . the retailer’s total expected profit is

 n  pi  min(Qi , xi )  min(qi  (Qi  xi )  , yi )       .  (q, Q, x)  EY x      i 1   si  qi  (Qi  xi )  yi    ci Qi  ei qi   

(1)

In Eq. (1), ()   max(, 0) , the first term is the total selling revenue in two periods, the second term is the total salvage value at the end of the selling season, and the last term the total procurement cost. It is not uncommon to assume pi  ei  ci  si (refer to Dong and Rudi, 2004; Prasad et al. 2011; Zhou and Chao, 2014), which means the expedited ordering is expensive than the initial ordering, and the unit cost of the expedited product does not exceed the selling price, and the retailer has no motivation to procure any product for obtaining its salvage value. Using min( x, y )  x  ( x  y )  , we can rewrite Eq. (1) as n   (q, Q, x)   ( pi  ci )Qi  ( pi  ei )qi  ( pi  si ) EY x  qi  (Qi  xi )   yi   . (2) i 1





 

Now we model the retailer’s purchasing problem as the following bi-level programming problem (denoted as problem P1). max  (Q)  EX  (Q, x)  , n

s.t.  bi ,1Qi  B1 , i 1

(3)

Q  0.

 (Q, x)  max  (q, Q, x), s.t. q  0, qi  li , if qi  0, i  1, , n.

(4)

 (Q, x) is the maximal total expected profit for given (Q, x) by optimizing the expedited order quantity q .  (Q) is the retailer’s total expected profit for given Q . The retailer’s problem before the selling season is to maximize  (Q) by choosing Q subject to the resource constraint. Notice that the resource constraints only occur in 8

ACCEPTED MANUSCRIPT the first period but not in the second period, because the resources must be shared among multiple products in the first period and the resources are dynamically released during the selling season since the products sold in the first period will generate some resources which can be used in the second period. 4. Structural properties and solution method In this section, we first establish the structural properties of problem P1, and then we develop a solution method based on the structural properties for solving the optimal solution to problem P1. 4.1 Structural properties By analyzing the objective of the studied problem, we have the following results. Proposition 1. (a)  (q, Q, x) is jointly concave in (q, Q) for any x . (b)  (Q) is jointly concave in Q .

We denote by q* the optimal solution to the problem defined in Eq. (4). According to Proposition 1(a), we know  (q, Q, x) is concave in q for given Q and

x . Solving the first order optimal conditions of  (q, Q, x) given in Eq. (A1) of

 p e Appendix A, we have qi  FYi 1xi  i i  pi  si

 1    (Qi  xi ) , i  1, , n , where FYi xi () is the 

inverse distribution function of the updated demand forecast Yi xi . Combining it with the non-negative constraint q  0 , we can rewrite the expedited order quantity for product i  1, , n , as follows:   0,    qi   FYi 1xi      F 1   Yi xi 

if xi  Si1 pi  ei pi  si

 2   xi  Qi , if xi  Si . 

pi  ei pi  si

 , 

(5)

if xi  Si3

9

ACCEPTED MANUSCRIPT

where

  p e Si1   xi 0  xi  Qi  FYi 1xi  i i  pi  si 

   ,  

  p e Si2   xi Qi  FYi 1xi  i i  pi  si 

    xi  Qi   

and Si3   xi xi  Qi  . Let’s combine Eq. (5) with the constraint qi  li , if qi  0 . If qi  0 , then we have

qi*  0

qi*  li

. If

qi  li

, then

qi*  li

. If

0  qi  li

, we must decide whether

qi*  0

or

x  Si1 xi  Si2 by comparing  (0, Q, x) and  (l , Q, x) . By analyzing i , and

xi  Si3 , respectively, we can calculate the optimal expedited order quantity qi* for product i  1, , n . According the property of normal distribution, we have

Y  p e   p e  FYi 1xi  i i   FYi 10  i i   i i xi  Xi .  pi  si   pi  si 

(6)

* Using Eq. (6), qi can be written as:

0,  li ,   p e   FY1x  i i   xi  Qi , qi*   i i  pi  si  0,    1  p  e   max  FYi xi  i i  , li  ,  pi  si    

if xi  Ai1 , if xi  Ai2 , if xi  Ai3 ,

(7)

if xi  A , 4 i

if xi  Ai5 ,

where  0  xi    Y  p  e    Qi  FYi 10  i i   1  i i     Xi Ai1   xi   pi  si    max    Y  e s  Qi  li i i   X i i i  Yi       pi  si  Xi   

      ,  ,     Yi   1  i    X i   

10

ACCEPTED MANUSCRIPT     Y   p  e     Qi  FYi 10  i i   1  i i  ,    X i       pi  si     max            e  s Y Y  i i i i Ai2   xi (Qi  li p  s   X i i   Yi ) 1  i     , i i Xi Xi          Yi      1  pi  ei    xi  min  Qi  li  FYi 0  p  s   1  i   , Qi  Xi   i i          p e Ai3   xi min  Qi  li  FYi 10  i i  pi  si   

  

 Y 1  i i  Xi 

    , Qi   xi  Qi  ,   

  c  s  Y Ai4   xi Qi  xi  max  li i i   X i i i  Yi     Xi    pi  si   c  s  Y A   xi xi  max  li i i   X i i i  Yi     Xi    pi  si 5 i

  Yi  i   Xi

  Yi  i   Xi

  , Q  i  ,  

   , Qi  .  

2 3 Notice that Ai and Ai may become empty for some cases, which does not affect

our analysis and computation. * Taking the derivative of qi by Qi , we have

if xi  Ai1  Ai2  Ai4  Ai5 qi* 0  . Qi 1 if xi  Ai3

(8)

According to Eq. (7), we know that qi*  Qi is constant over the domain xi  Ai3 , otherwise, Qi has no impact on qi* . Now we analyze the first stage problem. By substituting qi* into Eq. (3), the objective function of problem P1 can be written as

 (Q)  EX  (Q, x) n    EX ( pi  ci )Qi  ( pi  ei )qi*  ( pi  si ) EY x  qi*  (Qi  xi )   yi   .     i 1

(9)

Using Eq. (8), we can calculate the first derivative of  (Q) ,

11

ACCEPTED MANUSCRIPT  pi  ci  ( pi  ei ) qi* Qi   (Q)    EXi   ( pi  si ) EY x  qi*  (Qi  xi )   yi   Qi  Qi i i       pi  ci  E X A3 ( pi  ei )  ( pi  si ) E X A2 FYi xi [(li  (Qi  xi )  )  ] i

i

i

i

(10)

 ( pi  si ) E X A1 FYi xi (Qi  xi )  . i

i

where FYi xi () is the probability distribution function of the updated demand forecast

Yi xi . From Proposition 1(b), we know  (Q) is concave in Q . Since the feasible domain of problem P1 is convex, the optimality condition for problem P1 can be characterized via KKT conditions. Let 1  0 be the Lagrange multiplier for the n

constraint  bi ,1Qi  B1 , and wi  0 , be the Lagrange multipliers for Q  0 , i  1, , n . i 1

We denote by w  ( w1 , , wn ) , then the Lagrange function for problem P1 is n

n

i 1

i 1

L(Q, 1 , w )   (Q)  1 ( B1   bi ,1Qi )   wi Qi .

(11)

KKT conditions for problem P1 are as follows

 pi  ci  E X A3 ( pi  ei )  ( pi  si ) E X A2 FYi xi [(li  (Qi  xi )  )  ] i i i i    0,i  1, , n, (12)   ( pi  si ) E X A1 FY x (Qi  xi )  1bi ,1  wi  i i i i   n

 wi Qi  0,

(13)

i 1

n

1 ( B1   bi ,1Qi )  0.

(14)

i 1

n

Notice that 1 is the Lagrange multiplier of  bi ,1Qi  B1 . Eq. (14) is met if either i 1

n

1  0 or  bi ,1Qi  B1 . i 1

We denote by Q* the optimal solution to problem P1 and 1* the corresponding Lagrange multiplier. Let Q(1 ) be the optimal solution of Eqs. (12) and (13) with given 1  0 . We denote by

i (Qi , 1 )  E X A1 FYi xi (Qi  xi )  E X A2 FYi xi (li  Qi  xi ) i

i

i

i

 p  e   pi  ci  1bi ,1   E X A3  i i   , i  1, , n. i i pi  si  pi  si  

(15)

12

ACCEPTED MANUSCRIPT then Q(1 ) has the following properties. Proposition 2. (a) For any given 1  0 , Q(1 ) satisfies Eqs.(12) and (13) if and only if Qi (1 )  arg Qi i (Qi , 1 )  0, i  1, , n.

(16)

n

(b) If (Q(1 ), 1 ) satisfies 1  0 or  bi ,1Qi (1 )  B1 , then we have Q*  Q(1 ) . i 1

(c) Qi (1 ) is nonincreasing in 1 , i  1, , n. In the case of 1  0 , we have Qi (0)  arg Qi i (Qi , 0)  0, i  1, , n.

(17)

Notice that Qi (0) is the optimal solution to the problem without the resource constraint. If

n

 bi ,1Qi (0)  B1 , then the resource constraint is inactive. If

i 1

n

 bi ,1Qi (0)  B1 , then the resource constraint is active, the optimal ordering quantities

i 1

n

must satisfy  bi ,1Qi (1 )  B1 . i 1

4.2 Solution Method According to Proposition 2(c), Qi (1 ) is nonincreasing in 1 . In the case of n

 bi ,1Qi (0)  B1 ,

i 1

we

need

to

find

1  0

such

that

n

 bi ,1Qi (1 )  B1 .

i 1

If

1  1  max i 1,,n ( pi  ci ) / bi ,1 , according to Proposition 2(a), then we have pi  ci  bi ,11  0 and Qi (1 )  0 , i  1, , n , which violates the slackness condition n

1 ( B1   bi ,1Qi )  0 . Thus, we can find 1* using a binary search over [0, 1 ] such that i 1

n

n

i 1

i 1

*  bi ,1Qi (1 )  B1 for the case of  bi ,1Qi (0)  B1 .

We denote problem P1 as P1 (c) with c  (c 1 , , cn ) . Main steps of the solution method are given in Algorithm 1. Algorithm 1. SolveP1 (c) 13

ACCEPTED MANUSCRIPT Step 1: Let 1L  0 , 1U  max i 1,,n ( pi  ci ) / bi ,1 ; Step 2: Let 1  (1L  1U ) / 2 ; Step 3: If 1  0 , then let 1*  0 and let Q*  Q(0) , stop; Step 4: Calculate Q(1 ) from Eq. (16); n

Step 5: If  bi ,1Qi (1 )  B1 , then let 1L  1 , go to Step 2; i 1 n

If  bi ,1Qi (1 )  B1 , then let 1U  1 , go to Step 2; i 1

Step 6: Let Q*  Q(1 ) and 1*  1 , stop. Algorithm 1 applies a binary search procedure to determine 1* over [0, 1 ] . The n

binary search terminates if either 1  0 or  bi ,1Qi (1 )  B1 . If the constraint i 1

n

L U  bi ,1Qi (1 )  B1 is inactive, then the iterating process will lead to 1  1  0 , and

i 1

the solution procedure will stop at Step 3 with 1*  0 and Q*  Q(0) . If the constraint n

n

i 1

i 1

 bi ,1Qi (1 )  B1 is active, then the iterating process will lead to  bi ,1Qi (1 )  B1 with

1  0 , and the solution procedure will stop at Step 6 with Q*  Q(1 ) and 1*  1 . Step 4 calculates Q(1 ) by using Eq. (16). Since Qi (1 ) is decreasing in 1 , Step 5 n

chooses half-interval for 1 in the binary procedure by comparing  bi ,1Qi (1 ) with B1 i 1

n

using an implicit stopping condition  bi ,1Qi (1 )  B1 . i 1

The loop of binary search method has constant complexity O(log 2 (1/  )) , where

 is the error target for the binary search. The calculation procedures of Steps 3 and 4 have complexity O(n) . Thus, the computational complexity of Algorithm 1 is

O((log 2 (1/  ))n) , which is polynomial in n . 5. Extensions In this section, we extend to investigate the two-constraint problem, and we also replace the expedited ordering with the option contract in the studied problem.

14

ACCEPTED MANUSCRIPT 5.1 The two-constraint problem In retailing industry, retailers often face two resource constraints, such as budget and space. In this subsection, we extend the purchasing problem to investigate the two-constraint case, i.e., the problem in Eq. (3) is extended by adding one additional n

resource constraint  bi ,2Qi  B2 . We denoted the two-constraint purchasing problem i 1

as problem P2. n

Let λ  (1 , 2 ) be the Lagrange multiplier vector for constraint  bi ,1Qi  B1 and i 1

n

 bi ,2Qi  B2 . Then the Lagrange function for problem P2 is

i 1

n

n

n

i 1

i 1

i 1

L(Q, λ , w )   (Q)  1 ( B1   bi ,1Qi )  2 ( B2   bi ,2Qi )   wi Qi .

(18)

KKT conditions for problem P2 are as follows

 pi  ci  E X A3 ( pi  ei )  ( pi  si ) E X A2 FYi xi [(li  (Qi  xi )  )  ] i i i i    0, i  1, , n, (19)   ( pi  si ) E X A1 FY x (Qi  xi )  1bi ,1  2bi ,2  wi  i i i i   n

 wi Qi  0,

(20)

i 1

n

1 ( B1   bi ,1Qi )  0,

(21)

i 1 n

2 ( B2   bi ,2Qi )  0.

(22)

i 1

Note that  j is the Lagrange multiplier of the resource j  1, 2 . Eqs. (21) and n

(22) are met if either  j  0 or  bi , j Qi  B j , j  1, 2 . i 1

We also denote by Q* the optimal solution to problem P2 and λ * the corresponding Lagrange multiplier vector. We denote by  E X i Ai1 FYi xi (Qi  xi )  E X i Ai2 FYi xi (li  Qi  xi )     i (Qi , 1 , 2 )   p  c   b   b  p  e   i i 1 i ,1 2 i ,2   ,i  1, , n.   E X A3  i i    i i pi  si  pi  si   

and let



n

(23)



ˆ , ˆ )  arg (Q,  )  ( B   b Q )  0 and  (Q ,  ,  )  0, i  1, , n . (24) (Q 1 1 1 1 i ,1 i i i 1 2 i 1

ˆ  (Qˆ , , Qˆ ) and ˆ are functions of   0 . By analyzing Eq. (24), we Notice that Q 1 n 1 2

15

ACCEPTED MANUSCRIPT have the following results. Proposition 3.

ˆ , ˆ ) is the optimal solution of Eqs. (19)-(21). (a) (Q 1 n

ˆ , ˆ ,  ) satisfies   0 or  b Qˆ  B , then we have Q*  Q ˆ. (b) If (Q 1 2 2 i ,2 i 2 i 1

ˆ is the optimal solution to problem P (cˆ ( )) , with cˆ ( )  c   b , (c) Q 1 2 i 2 i 2 i ,2 i  1, , n .

n

Proposition 4.  bi ,2Qˆ i is nonincreasing in 2 , i  1, , n . i 1

ˆ 0  (Qˆ 0 , , Qˆ 0 ) and ˆ 0 to denote Q ˆ and ˆ , respectively, for the case We use Q n n 1 1

ˆ 0 is the optimal solution to problem P2 of 2  0 . The condition 2  0 implies that Q n

ˆ 0 can be solved using without the second resource constraint  bi ,2Qi  B2 , thus Q i 1

Algorithm 1. n

n

i 1

i 1

If  bi ,2Qˆ i0  B2 , then the second resource constraint is inactive. If  bi ,2Qˆ i0  B2 , then the second resource constraint is active, we need to find 2  0 such that n

n

i 1

i 1

 bi ,2Qˆ i  B2 . According to Proposition 4,  bi ,2Qˆ i is nonincreasing in 2 . Denote by

2  max i 1,,n ( pi  ci ) / bi ,2  ,

2  2  0 ,

if

we

have

pi  ci  bi ,1ˆ1  bi ,2 2  pi  ci  bi ,2 2  0 and Qˆ i  0 , i  1, , n , from Eq. (24), which n

violates the slackness condition 2 ( B2   bi ,2Qi )  0 . Therefore, we can find 2* by i 1

n

n

i 1

i 1

using a binary search over [0, 2 ] such that  bi ,2Qi*  B2 for the case of  bi ,2Qˆ i0  B2 . We present main steps of the solution method for solving problem P2 in Algorithm 2. Algorithm 2. SolveP2

ˆ 0 using SolveP (c) ; Step 1: Solve Q 1 16

ACCEPTED MANUSCRIPT Step 2: Let 2L  0 , 2U  max i 1,,n ( pi  ci ) / bi ,2  ; Step 3: Let 2  (2L  2U ) / 2 ;

ˆ 0 , stop; Step 4: If 2  0 , then let 2*  0 and Q*  Q ˆ and ˆ using SolveP (cˆ ( )) ; Step 5: Solve Q 1 1 2 n

Step 6: If  bi ,2Qˆ i  B2 , then let 2L  2 , go to Step 3; i 1 n

If  bi ,2Qˆ i  B2 , then let 2U  2 , go to Step 3; i 1

ˆ ,  *  ˆ and  *   , stop. Step 7: Let Q*  Q 1 1 2 2 The solution process of Algorithm 2 is similar to that of Algorithm 1 except that

ˆ and ˆ by calling Algorithm 1 to solve Steps 1 and 5 of Algorithm 2 obtains Q 1 problems P1 (c) and P1 (cˆ (2 )) , respectively. Since the loop of 1-tier binary search method has constant complexity O(log 2 (1/  )) , 2-tier binary search procedure has computational complexity O((log 2 (1/  )) 2 ) . So the computational complexity of Algorithm 2 is O((log 2 (1/  )) 2 n) , which is polynomial in n . Thus, the proposed algorithms are efficient for solving large-scale problems. 5.2 The problem with option contract In this section, we use option contract instead of the expedited ordering to flexibly adjust the order quantities after updating demand forecast. The sequence of the events is as follows. The retailer places the initial order Qi and determines the reservation quantity K i of the option contract for product i,

i  1, , n by giving unit reservation cost vi . At the start of the second period, the retailer orders qi  K i by giving unit execution cost ri based on on-hand inventories and updated demand forecasts. The retailer faces budget constraint for signing all n

wholesale price contracts and option contracts, i.e.,  ci Qi vi K i  B1 . i 1

Without loss of generality, we assume that pi  vi  ri  ci  si . The total cost of the option contract is larger than the cost of wholesale price contract, vi  ri  ci since the retailer has flexibility with the option contract. The unit reservation cost is 17

ACCEPTED MANUSCRIPT assumed to be smaller than the unit pure procurement cost, vi  ci  si , otherwise, the retailer will always choose the wholesale price contract for the lower cost whether the product can be sold or not. Let Q  (Q1 , , Qn ) , K  ( K1 , , K n ) and q  (q1 , , qn ) . For any given

(q, Q, K , x) , the retailer’s total expected profit is

 n  pi  min(Qi , xi )  min(qi  (Qi  xi )  , yi )        O (q, Q, K , x)  EY x      i 1   si  qi  (Qi  xi )  yi   ci Qi  ri qi  vi K i   n    ( pi  ci )Qi  ( pi  ri )qi  vi K i  ( pi  si ) EY x  qi  (Qi  xi )   yi   .     i 1 

(25)

Now we model the retailer’s purchasing problem with option contract as the following bi-level programming problem (denoted as problem PO ). max  O (Q, K )  EX  O (Q, K , x)  , n

s.t.  ci Qi vi K i  B1 ,

(26)

i 1

Q  0, K  0.

 O (Q, K , x)  max  O (q, Q, K , x), s.t. q  K , q  0.

(27)

Using the similar process in the proof of Proposition 1, we can prove the following results. Proposition 5. (a)  O (q, Q, K , x) is jointly concave in (q, Q, K ) for any x . (b)  O (Q, K ) is jointly concave in (Q, K ) . By calculating the first derivative of  O (q, Q, K , x) with qi , we have





 O (q, Q, K , x)    pi  ri    pi  si  FYi | xi qi   Qi  xi  . qi

(28)

 p r   Setting Eq. (28) to zero, we obtain qi  FYi |1xi  i i    Qi  xi  . Combining it with  pi  si  the non-negative constraint q  0 , we have 18

ACCEPTED MANUSCRIPT  0,     qi   FYi |1xi      FY|1x   i i 

 p r  if 0  xi  Qi  FYi |1xi  i i  ,  pi  si  pi  ri   1  pi  ri     Qi  xi  , if Qi  FYi | xi    xi  Qi , pi  si   pi  si  pi  ri  , pi  si 

(29)

if xi  Qi .

Combining Eq. (29) with the constraint q  K , we have the optimal quantity of

qi for the given (Q, K , x) as follows: 0,   F 1   Yi | xi    * qi   K i ,   F 1   Yi | xi    K i ,

if xi  Ri1 , pi  ri pi  si pi  ri pi  si

 2   Qi  xi , if xi  Ri ,  if xi  Ri3 ,  , 

(30)

if xi  Ri4 , if xi  Ri5 ,

where    p r Ri1   xi | 0  xi   Qi  FYi |01  i i  pi  si  

    ,  

 Y 1  i i  Xi 

  

  Y   p  r    xi |  Qi  FYi |01  i i   1  i i   xi  X i     pi  si    2 Ri    Y    1  pi  ri   min Q  K  F  1  i i    i i Yi |0     Xi  pi  si         p r Ri3   xi | min   Qi  K i  FYi |01  i i   pi  si  

  

 Y 1  i i  Xi 

   p  r  Ri4   xi | Qi  xi  max  K i  FYi |01  i i    pi  si        p r Ri5   xi | xi  max  K i  FYi |01  i i  pi  si  

  

   ,    , Qi           , Qi   xi  Qi  ,    

  Yi  i   Xi

  Yi  i   Xi

   , Qi  ,  

   , Qi  .  

Taking the derivative of qi* by Qi and K i , we have

qi* 0, if xi  Ri1  Ri3  Ri4  Ri5 ,  Qi 1, if xi  Ri2 .

(31) 19

ACCEPTED MANUSCRIPT qi* 0, if x  Ri1  Ri2  Ri4 ,  K i 1, if xi  Ri3  Ri5 .

(32)

Now we analyze the first stage problem. By substituting qi* in Eq. (30) into Eq. (26), we have

 n ( pi  ci )Qi  ( pi  ri )qi*  vi K i     .  O (Q, K )  EX    i 1  ( pi  si ) E  qi*  (Qi  xi )   yi     Yx       

(33)

By calculating the first derivative of  O (Q, K ) , we have   O (Q, K )  pi  ci   pi  si  E X i Ri1 FYi | xi  Qi  xi  ,  Qi   E X R2  pi  ri    p  s  E X R3 FYi | xi  Qi  K i  xi   i i i i  

(34)

 O (Q, K )  E X i Ri3  Ri5  pi  ri    pi  si  E X i Ri5 FYi | xi  K i   .  K i    pi  si  E X R3 FYi | xi  Qi  K i  xi   vi  i i  

(35)

From Proposition 5(b), we know  O (Q, K ) is jointly concave in (Q, K ) . Let

O  0 ,  i  0 and i  0 , be the dual variables corresponding to the constraints in Eq. (26), respectively. Since the feasible domain of this problem is convex, the following KKT conditions are necessary and sufficient for optimality for problem PO .  pi  ci   pi  si  E X R1 FYi | xi  Qi  xi   E X R2  pi  ri   i i i i    0 , i  1, , n    p  s  E X R3 FYi | xi  Qi  K i  xi   O ci   i  i i    E X R3  R5  pi  ri    pi  si  E X R5 FYi | xi  K i   i i  i i i   0,    pi  si  E X R3 FYi | xi  Qi  K i  xi   vi  O vi  i  i i   n

  i Qi  i K i  0,

O  B1    ci Qi  vi K i    0. n

i 1

(37)

(38)

i 1



i  1, , n

(36)

(39)



Due to the concavity of  O (Q, K ) , for any given O  0 , we can solve Qi (O ) n

and K i (O ) from Eqs. (36)-(38) and we know that   ci Qi (O )  vi K i (O )  is noni 1

increasing in O . Thus, we can develop the following algorithm for solving problem

PO .

20

ACCEPTED MANUSCRIPT Algorithm 3. SolvePO Step 1: Let O  0 , solve Q(O ) and K (O ) from Eqs. (36)-(38); n

Step 2: If   ci Qi (O )  vi K i (O )   B1 , let Q*  Q(O ) and K *  K (O ) , stop; i 1

Step 3: Let OL  0 , OU  max i 1,,n ( pi  ci ) / ci  ; Step 4: Let O  (OL  OU ) / 2 ; Step 5: Solve Q(O ) and K (O ) from Eqs. (36)-(38); n

Step 6: If   ci Qi (O )  vi K i (O )   B1 , then let OL  O , go to Step 4; i 1 n

If   ci Qi (O )  vi K i (O )   B1 , then let OU  O , go to Step 4; i 1

Step 7: Let Q*  Q(O ) and K *  K (O ) , stop. Notice that 2-tier binary search procedure can be used in Steps 1 and 5 to solve

Q(O ) and K (O ) from Eqs. (36)-(38) due to the concavity of  O (Q, K ) . Since the loop of 1-tier binary search method has constant complexity O(log 2 (1/  )) , the computational complexity of Algorithm 3 is O((log 2 (1/  ))3 n) . 6. Numerical study In numerical study, we show the value of expedited ordering and the efficiency of our solution methods, and compare the efficiency of expedited ordering and option contract in the studied problem We use the problem without the expedited ordering (i.e., q*  0 ) as the benchmark problem, the retailer places the order before the selling season and the expedited ordering is not allowed. The benchmark problem can be rewritten as the constrained multi-product newsvendor problem, which is well solved in literature (Abdel-Malek and Montanari, 2005; Niederhoff, 2007; Zhang, 2012). The cost parameters and demand data in the numerical example are obtained from our investigation of a retail industry, and the range for parameters in the randomly generated instances are also set based on our investigation. * We denote by Q*BM the optimal solution to the benchmark problem, let  BM and

 * the optimal profits of the benchmark problem and our problem respectively. We 21

ACCEPTED MANUSCRIPT * * use   ( *   BM )  BM 100% to measure the value of expedited ordering. We set

bi ,1  ci , i  1, , n in our numerical study since B1 is the budget constraint. All computation experiments are conducted on a desktop computer (dual processor 3.60GHz, memory 8G) with Matlab 2016. 6.1 Illustrative example In the illustrative example, there are five products and two resource constraints

B1  20000 and B2  22000 , other parameters are given in Table 2, where  X i , Yi ,

 X ,  Y , and i are the parameters of mean, standard deviation and the correlation i

i

coefficient. The optimal solutions to one-constraint problem, two-constraint problem and the benchmark problems are reported in Table 3. Insert Tables 2 and 3 here From Table 3, we observe that the expedited ordering can significantly improve the retailer’s expected profit. The initial order quantity Q*i may also be larger or smaller than the order quantity of benchmark Q*i , BM , but the retailer’s expected profit of the expedited ordering is higher than the model without it. We illustrate the efficiency of our solution methods for solving large-scale problems. We use the notation x ～ U ( ,  ) to denote that x is uniformly generated over [ ,  ] . The parameters of problems are randomly generated as follows: B1 ～

n  U (5000, 6000) ,

B2 ～ n  U (8000,9000) ,

pi ～ U (15, 20) , ci ～ U (8,13) , si ～

U (2,5) , li ～ U (200, 250) , bi ,2 ～ U (8, 25) ,  X i ～ U (150, 200) ,  X i ～ U (40,50) , Yi ～ U (450, 700) ,  Yi ～ U (90,100) , i ～ U (0.5, 0.9) and ei  1.1 ci , for i  1, , n ,

j  1, 2 . We set n  100, 500, and 1000, respectively. For each problem size, 30 test instances are randomly generated. The statistical results of number of iterations and computation times are reported in Table 4 and Table 5. In these tables, 95% C.I. stands for 95% confidence interval. Insert Tables 4 and 5 here 22

ACCEPTED MANUSCRIPT

From Tables 4 and 5, we can conclude that our method can solve large-scale multi-product expedited ordering problem efficiently in limited iterations. The standard deviations of number of iterations and computation times are low in Tables 4 and 5, indicating that our method is effective and robust. Robustness of our method should be attributed to the effectiveness of binary search procedure. 6.2 Sensitivity analysis To show the impact of resource availability on the value of the expedited ordering, we investigate more cases by varying B1 . The value of expedited ordering becomes larger when the limited resource becomes smaller in both one-constraint and two-constraint problems, with the increase of B1 in both problem P1 and problem P2, the value of expedited ordering  decreases. This downward trend continues until the first resource constraint becomes inactive, and the expedited ordering still has significant value even if there is no resource constraint. The results are plotted in Figs. 3 and 4. Insert Figs. 3-4 here In one-constraint problem (Fig. 3), the condition B1  37555 in this illustrative example implies that the first resource constraint is inactive. The expedited ordering still has significant value (   3.41 % ) even if there is no resource constraint. In twon

constraint problem (Fig. 4), the second constraint  bi ,2Qi  B2 is always active, the i 1

first resource constraint becomes inactive when the condition B1  22691 , and the minimal value of the expedited ordering is   50.89 % . To show the impact of additional expedited costs on the value of the expedited ordering, we let ei  a  ci , i  1, , n , where a is an indicator of the additional expedited costs. a  1 means the expedited ordering has no additional cost, and a  1 implies the expedited ordering has additional cost. The result is plotted in Fig. 5. Insert Fig. 5 here

23

ACCEPTED MANUSCRIPT From Fig. 5, we observe that the value of expedited ordering becomes smaller when the additional expedited costs become larger. When the expedited costs are equal to the selling prices ( a  1.5 in this example), there is no value of the expedited ordering. Fig. 5 illustrates that the expedited ordering is still valuable with large additional expedited costs as long as the expedited costs do not exceed the selling prices. 6.3 Comparison with the option contract model In this subsection, we compare the expedited ordering model (problem P1) with the option contract model (problem PO ). Since the performance of these two models depends on the model parameters, especially, on the expedited ordering cost ei ,

i  1, , n and the option contract cost (vi , ri ) , i  1, , n , it’s not easy to obtain clear comparison in the multi-product setting, we compare these two models in singleproduct setting with budget constraint. In addition, we set ei  vi  ri , since the expedited ordering has more flexibility than the option contract for the retailer, and the retailer should pay higher cost when using expedited ordering. In the base case, we set p1  15 , c1  10 , e1  13 , r1  3.11 , v1  6.99 , s1  3 ,

l1  100 ,  X1  150 ,  X1  40 , Y1  450 ,  Y1  90 , 1  0.9 , and B1  1600 . We let

 * and  O* be the optimal expected profits of the expedited ordering model and the option contract model, respectively, and we calculate 1  ( *   O* )  O* 100% to compare the two models. We change the reservation cost v1 from 0.01 to 6.99 and hold v1  r1  10.1 and other parameters unchanged, the comparison between the expedited ordering model and the option contract model is given in Fig. 6. To show the impact of the expedited ordering cost on the model comparison, we change e1 from 13 to 14.9, and hold other parameters unchanged, and the comparison between the two models with different e1 is illustrated in Fig. 7. Insert Figs. 6 and 7 here According to Fig. 6, for smaller reservation cost v, we have 1  0 , which implies the option contract model outperforms the expedited ordering model since small v 24

ACCEPTED MANUSCRIPT saves more budget. For larger v, we have 1  0 , which means the expedited ordering model is better than the option contract model if unit reservation cost is high. There exists a medium value for v such that the expected profits of the two models are same. From Fig. 7, we know that the expedited ordering model is better than the option contract model for small expedited ordering cost and it is worse than the option contract model for large expedited ordering cost. 7. Conclusions This paper studies the value of the expedited delivery option when a retailer can use the initial sales information to adjust his ordering quantities in a multi-product, two-period setting. The retailer is subject to the resource constraints and requirements on minimum ordering quantity for the expedited delivery. We formulate the twoperiod problem as a bi-level programming, investigate the structural properties of the solution, and we develop an algorithm based on the binary search procedure to solve for the problem. We extend our study to investigate two-constraint case and we also replace the expedited ordering with option contract in the extended study. By investigating the structural properties of these two extended models, we develop efficient solution methods. Numerical results show that the proposed solution methods can solve large-scale multi-product expedited ordering problem efficiently in limited iterations, and the methods are effective and robust. Comparing the solution to the problem when the expedited delivery option is not available, we show that expedited ordering can improve the retailer’s expected profit. Moreover, the decreasing impact of the resource constraints is also illustrated numerically. The comparison of the expedited ordering model with the option contract model shows that which model is better depends on the cost parameters, and the expedited ordering model does well if the expedited ordering cost is small, and the option contract model is better if the unit reservation cost of the option contract is small. This study demonstrates the significant value of expedited ordering in retailing. Since the initial sales data is easily obtained from the retailing system rather than other market information, managers should pay more attention to improving their demand forecasts with sales data in season, and attempt to implement the policy of expedited ordering when managing the demand uncertainty. The research of this

25

ACCEPTED MANUSCRIPT paper can be extended in several different directions. Since the retailer often faces understock and overstock situation, expedited ordering is used to deal with the understock case, other policy with demand forecast updates such as markdown can be applied to address the overstock situation. One extended work is to investigate the markdown problem with demand forecast updates in multi-product setting. Another extension is to simultaneously analyze the expedited ordering and markdown options with demand forecast updates for managing the demands of multiple products. Our initial assessment shows that this problem needs developing more complex solution methods. Acknowledgements We are very grateful to the reviewers for their valuable suggestions and helpful comments which improved the manuscript significantly. This work was supported by the National Natural Science Foundation of China (Grant No.71672199).

26

ACCEPTED MANUSCRIPT Appendix A A.1. Proof of Proposition 1 (a) From Eq. (2), for i  1, , n we can have  (q, Q, x)   pi  ei    pi  si  FYi xi  qi  (Qi  xi )   , qi

EYi xi  qi  (Qi  xi )   yi   (q, Q, x)  pi  ci  ( pi  si ) Qi Qi

(A1) 

  (q, Q, x)  pi  ci  ( pi  si ) FYi xi  qi  (Qi  xi )  , if Qi  xi  , Qi if Qi  xi  pi  ci ,

(A2)

where FYi xi () is the probability distribution function of the updated demand forecast

Yi xi . Let fYi xi () be the probability density function of Yi xi , then we have   2 (q, Q, x) ( pi  si ) fYi xi  qi  (Qi  xi )   0, if Qi  xi  , qi2 if Qi  xi ( pi  si ) fYi xi  qi   0,

(A3)

  2 (q, Q, x) ( pi  si ) fYi xi  qi  (Qi  xi )   0, if Qi  xi  , Qi2 if Qi  xi 0,

(A4)

  2 (q, Q, x) ( pi  si ) fYi xi  qi  (Qi  xi )   0, if Qi  xi  , qi Qi if Qi  xi 0,

(A5)

 2 (q, Q, x)  2 (q, Q, x)  2 (q, Q, x)    0, i  k , i, k  1, , n . qi qk Qi Qk qi Qk

(A6)

We can easily check the concavity of  (q, Q, x) by verifying the negative semidefinity of the hessian matrix of  (q, Q, x) . (b) According to the convexity stability under partial minimization (Boyd and Vandenberghe, 2004, pp. 87-88), we know  (Q, x) is jointly concave in Q . Therefore,  (Q)  EX  (Q, x)  is jointly concave in Q because the expectation operation preserves concavity. A.2. Proof of Proposition 2 (a) We have

 p e E X A1 FYi xi (Qi  xi )  E X A2 FYi xi (li  Qi  xi )  E X A3  i i i i i i i i  pi  si

 pi  ci  1bi ,1  w ,  pi  si  27

ACCEPTED MANUSCRIPT i  1, , n , from Eq. (12). Since  (Q) is concave in Q , we know  (Q) Qi is decreasing in Qi , and

 p e  hence E X A1 FYi xi (Qi  xi )  E X A2 FYi xi (li  Qi  xi )  E X A3  i i  is increasing in Qi i i i i i i  pi  si  . If Qi  0 , then Si1 and Si2 are empty sets, which means Ai1 ， Ai2 and Ai3 are empty

sets,

and

hence

 p e E X A1 FYi xi (Qi  xi )  E X A2 FYi xi (li  Qi  xi )  E X A3  i i i i i i i i  pi  si

   0 . If Qi   , then 

 p e E X A1 FYi xi (Qi  xi )  E X A2 FYi xi (li  Qi  xi )  E X A3  i i i i i i i i  pi  si

   1 . If pi  ci  1bi ,1  0 , 

according to pi  ei  ci  si , we have 0 

Qi  0

pi  ci  1bi ,1 pi  si



pi  ci  1 , there must exist pi  si

such

that

 p e E X A1 FYi xi (Qi  xi )  E X A2 FYi xi (li  Qi  xi )  E X A3  i i i i i i i i  pi  si

 pi  ci  1bi ,1 .  pi  si 

The

slackness condition wi Qi  0 in Eq. (13) implies wi  0 . If pi  ci  1bi ,1  0 , then the conditions wi  0 and wi Qi  0 imply Qi  0 , and w  ( pi  ci  1bi ,1 )  0 . These results can be unified as Qi (1 ) , given in Eq. (16). (b)

1  0 or

n

 bi ,1Qi (1 )  B1

i 1

n

implies 1 ( B1   bi ,1Qi )  0 . i 1

If

1  0 or

n

 bi ,1Qi (1 )  B1 , then Q(1 ) will satisfy all KKT conditions given in Eq. (12)-(14).

i 1

Thus Q*  Q(1 ) . (c)

Since

 p e  E X A1 FYi xi (Qi  xi )  E X A2 FYi xi (li  Qi  xi )  E X A3  i i  i i i i i i  pi  si 

is

increasing in Qi , we have i (Qi , 1 )  0, Qi

(A7)

28

ACCEPTED MANUSCRIPT if 1  (pi  ci ) / bi ,1 0, i (Qi , 1 )    bi ,1 . 1  0, if   ( p  c ) / b 1 i i i ,1 p s  i i

(A8)

Using the derivative of implicit function, we have dQi (1 ) i (Qi , 1 ) / 1   0. d 1 i (Qi , 1 ) / Qi

(A9)

Thus Qi (1 ) is nonincreasing in 1 . A.3. Proof of Proposition 3 (a) Using the similar way of proving Proposition 2(a), we know the optimal solution of i (Qi , 1 , 2 )  0 , i  1, , n , for given 1 and 2 , satisfies Eqs. (19)-(20).

ˆ , ˆ ) in Eq. (24), we know Q ˆ is the optimal solution of From the definition of (Q 1 ˆ , ˆ ) also ˆ satisfies Eqs. (19)-(20). The definition of (Q i (Qi , 1 , 2 )  0 , and hence Q 1 n

ˆ , ˆ ) satisfy Eqs. (19)-(21). requires 1 ( B1   bi ,1Qi )  0 , thus (Q 1 i 1

n

n

i 1

i 1

ˆ will (b) If 2  0 or  bi ,2Qˆ i  B2 , then we have 2 ( B2   bi ,2Qˆ i )  0 , and Q ˆ. satisfy all KKT conditions given in Eqs. (19)-(22). Thus, Q*  Q (c) KKT conditions for problem P1 (cˆ (2 )) are as follows

 pi  (ci  2bi ,2 )  ( pi  si ) E X A1 FYi xi (Qi  xi )   i i    0 , i  1, , n  ( pi  si ) E X A2 FY x [(li  (Qi  xi )  )  ]  E X A3 ( pi  ei )  1bi ,1  wi  i i i i i i   ,(A10) n

 wi Qi  0,

i 1

n

1 ( B1   bi ,1Qi )  0. i 1

(A11) (A12)

The conditions in Eqs. (A10)-(A12) are the same as KKT conditions given in

ˆ is the optimal solution of KKT conditions in Eqs. (19)-(21), Eqs. (19)-(21). Since Q ˆ must be the optimal solution to problem P (cˆ ( )) . Q 1 2 A.4. Proof of Proposition 4 We denote by 29

ACCEPTED MANUSCRIPT  p e i (Qi )  E X A1 FYi xi (Qi  xi )  E X A2 (li  Qi  xi )  E X A3  i i i i i i i i  pi  si

 , 

(A13)

From the concavity of  (Q) , we know i (Qi ) is increasing in Qi , i.e., d i (Qi )  0. dQi

(A14)

For any given 2 , we have two cases: (1) ˆ1  0 and (2) ˆ1  0 . We prove these two cases respectively.

ˆ is determined by  (Q , 0,  )  0 , i  1, , n . Thus (1) In the case of ˆ1  0 , Q i i 2 we have

i (Qˆ i , 0, 2 ) d i (Qˆ i )   0, Qˆ i dQˆ i

(A15)

if 2  ( pi  ci ) / bi ,2 0, i (Qˆ i , 0, 2 )    bi ,2 . 2  p  s  0, if 2  ( pi  ci ) / bi ,2  i i

(A16)

Using the derivative of implicit function, we have

dQˆ i  (Qˆ , 0, 2 ) / 2  i i  0. d 2 i (Qˆ i , 0, 2 ) / Qˆ i

(A17)

n

Thus,  bi ,2Qˆ i is nonincreasing in 2 for the case ˆ1  0 . i 1

n

ˆ is determined by  (Q ,  ,  )  0 and  b Q  B . (2) In the case of ˆ1  0 , Q i i 1 2 i ,1 i 1 i 1

For i  1, , n , we denote by i1 (t )  Qi i (Qi )  t  0.

(A18)

From Eq. (A14), we have 1

d i1 (t )  d i (Qi )     0. dt  dQi 

(A19)

Since

( pi  ci  ˆ1bi ,1  2bi ,2 )  ˆ ˆ ˆ i (Qi , 1 , 2 )  i (Qi )   0, pi  si

(A20)

we have

 ( p  c  ˆ b   b )  Qˆ i  i1  i i 1 i ,1 2 i ,2  pi  si 

 .  

(A21) 30

ACCEPTED MANUSCRIPT We define





I (ˆ1 )  i ˆ1  ( pi  ci  2bi ,2 ) / bi ,1 , i  1, , n . Notice

that

(A22)

( pi  ci  ˆ1bi ,1  2bi ,2 )   0

for

i  I (ˆ1 ) ,

and

( pi  ci  ˆ1bi ,1  2bi ,2 )   0 for i  I (ˆ1 ) . Thus we have 1  dQˆ i  d i (Qˆ i )  d  ( pi  ci  ˆ1bi ,1  2bi ,2 )     d 2  dQˆ i  d 2  pi  si

 1    pi  si  0,

 d i (Qˆ i )   ˆ   dQi 

1 We define i  pi  si

1

   

 d ˆ1   bi ,2  , if i  I (ˆ1 )  bi ,1 .  d 2  if i  I (ˆ )

(A23)

1

1

 d i (Qˆ i )   0 , then we have  ˆ  dQ i  

n

d  bi ,2Qˆ i i 1

d 2

 d ˆ    iI ( ˆ1 ) i bi ,2  bi ,1 1  bi ,2 .  d 2 

(A24)

We denote by n

G (ˆ1 , 2 )   bi ,1Qˆ i  B1  0, i 1

(A25)

then we have

G (ˆ1 , 2 )   iI ( ˆ1 ) i bi2,1 , ˆ 1

(A26)

G (ˆ1 , 2 )   iI ( ˆ1 ) i bi ,1bi ,2 . 2

(A27)

Thus we have

 iI ( ˆ1 ) i bi ,1bi ,2 d ˆ1 G (ˆ1 , 2 ) / 2    0. 2 d 2  iI ( ˆ1 ) i bi ,1 G (ˆ1 , 2 ) / ˆ1 Substituting

(A28)

d ˆ1 into Eq. (A24), we have d 2

31

ACCEPTED MANUSCRIPT n

d  bi ,2Qˆ i

  iI ( ˆ1 ) i bi ,1bi ,2    iI ( ˆ1 ) i bi ,2  bi ,2  bi ,1  2  d 2  iI ( ˆ1 ) i bi ,1   ( iI ( ˆ1 ) i bi ,1bi ,2 ) 2   iI ( ˆ1 ) i bi2,2  2  iI ( ˆ1 ) i bi ,1 i 1



( iI ( ˆ1 ) i bi2,2 )( iI ( ˆ1 ) i bi2,1 )  ( iI ( ˆ1 ) i bi ,1bi ,2 ) 2  iI ( ˆ1 ) i bi ,1 2

(A29)

.

Let m  I (ˆ1 ) be the norm of I (ˆ1 ) . If m  1 , we have n

d  bi ,2Qˆ i i 1

d 2



1   b 2  b 2  ( i bi ,1bi ,2 ) 2   0. 2  i i ,2 i i ,1 i bi ,1

(A30)

If m  1 , we re-index i  I (ˆ1 ) as i  1, , m , then Eq. (A29) can be rewritten as m 1

n

d  bi ,2Qˆ i i 1

d 2



m

2   i  j (bi ,1b j ,2  b j ,1bi ,2 )

i 1 j i 1

m

 i bi ,1 2

 0.

(A31)

i 1

n

Therefore,  bi ,2Qˆ i is nonincreasing in 2 for the case ˆ1  0 . i 1

32

ACCEPTED MANUSCRIPT References: Abdel-Malek, L. L., & Montanari, R. (2005). On the multi-product newsboy problem with two constraints. Computers & Operations Research, 32(8), 2095-2116. Araman, V. F., & Caldentey, R. (2009). Dynamic pricing for nonperishable products with demand learning. Operations Research, 57(5), 1169-1188. Biçer, I. (2015). Dual sourcing under heavy-tailed demand: an extreme value theory approach. International Journal of Production Research, 53(16), 4979-4992. Biçer, I. & Seifert, R. W. (2017). Optimal dynamic order scheduling under capacity constraints given demand-forecast evolution. Production and Operations Management, 26(12), 2266-2286. Bickel, P., & Doksum, K. (1977). Mathematical statistics. Holden Day Publisher, San Francisco, CA. Boyd, S., & Vandenberghe, L. (2004). Convex optimization. Cambridge University Press, p.p.101. Cai, W. B., Abdel-Malek, L., Hoseini, B., Rajaei-Dehkordi. S. (2015). Impact of flexible contracts on the performance of both retailer and supplier. Int. J. Production Economics, 170, 429-444. Caro, F., & Gallien, J. (2007). Dynamic assortment with demand learning for seasonal consumer goods. Management Science, 53(2), 276-292. Cheaitou, A., van Delft, C., Jemai, Z., & Dallery, Y. (2014). Optimal policy structure characterization for a two-period dual-sourcing inventory control model with forecast updating. Int. J. Production Economic, 159, 238-249. Cheaitou, A., & Cheaytou, R. (2018). A two-stage capacity reservation supply contract with risky supplier and forecast updating. International Journal of Production Economics, paper in advance. Chen, H., Chen, J., & Chen, Y. F. (2006). A coordination mechanism for a supply chain with demand information updating. International Journal of Production Economics, 103(1), 347-361. Chen, J., & Xu, L. (2001). Coordination of the supply chain of seasonal products. Part A: Systems and Humans, IEEE Transactions on Systems, Man and 33

ACCEPTED MANUSCRIPT Cybernetics, 31(6), 524-532. Chen, M., & Chen, Z. L. (2015). Recent developments in dynamic pricing research: multiple products, competition, and limited demand information. Production and Operations Management, 24(5), 704-731. Choi, T. M., Li, D., & Yan, H. (2003). Optimal two-stage ordering policy with Bayesian information updating. Journal of the Operational Research Society, 54(8), 846-859. Den Bour, A.V. (2015). Dynamic pricing and learning: historical origins, current research, and new directions. Survey in Operations Research and Management Science, 20, 1-18. Dong, L., Rudi, N. (2004). Who benefits from transshipment? exogenous vs. endogenous wholesale prices. Management Science, 50(5), 645-657. Donohue, K. L. (2000). Efficient supply contracts for fashion goods with forecast updating and two production modes. Management Science, 46(11), 1397-1411. Fisher, M., & Raman, A. (1996). Reducing the cost of demand uncertainty through accurate response to early sales. Operations Research, 44(1), 87-99. Gurnani, H., & Tang, C. S. (1999). Note: Optimal ordering decisions with uncertain cost and demand forecast updating. Management Science, 45(10), 1456-1462. Huang, H., Sethi, S. P., & Yan, H. (2005). Purchase contract management with demand forecast updates. IIE Transactions, 37(8), 775-785. Li, Q., Niu, B. Z., & Chu L. (2017). Forward sourcing or spot trading? optimal commodity procurement policy with demand uncertainty risk and forecast update. IEEE System Journal, 11(3), 1526-1536. Ju, W., Gabor, A. F., & Ommeren, J.C.W. (2015). An approximate policy for a dualsourcing inventory model with positive lead times and binomial yield. European Journal of Operational Research, 244(2), 490-497. Kim, C., Klabjan, D., & Simchi-levi, D. (2015). Optimal expediting policies for a serial inventory system with stochastic lead time. Production and Operations Management, 24(10), 1524-1536. Kouvelis, P., & Tang, S. Y. (2012). On optimal expediting policy for supply systems 34

ACCEPTED MANUSCRIPT with uncertain lead-times. Production and Operations Management, 21(2), 309330. Lin, K. Y. (2006). Dynamic pricing with real-time demand learning. European Journal of Operational Research, 174(1), 522-538. Martínez-de-Albéniz, V., & Simchi-Levi, D. (2005). A portfolio approach to procurement contracts. Production and Operations Management, 14(1), 90-114. Miltenburg, J., & Pong, H. C. (2007). Order quantities for style goods with two order opportunities and Bayesian updating of demand. Part II: capacity constraints. International Journal of Production Research, 45(8), 1707-1723. Niederhoff, J. A. (2007). Using separable programming to solve the multi-product multiple ex-ante constraint newsvendor problem and extensions. European Journal of Operational Research, 176(2), 941-955. Nosoohi, I., & Nookabadi, A. S. (2016). Outsource planning through option contracts with demand and cost uncertainty. European Journal of Operational Research, 250, 131-142. Özer, Ö.,Uncu, O., & Wei, W. (2007). Selling to the “newsvendor” with a forecast update: analysis of a dual purchase contract. European Journal of Operational Research, 182(3), 1150-1176. Prasad, A., Stecke, K. E., Zhao X. (2011). Advance selling by a newsvendor retailer. Management Science, 20(1), 129-142. Şen, A., & Zhang, A. X. (2009). Style goods pricing with demand learning. European Journal of Operational Research, 196(3), 1058-1075. Serel, D. A. (2012). Multi-item quick response system with budget constraint. International Journal of Production Economics, 137(2), 235-249. Sethi, S. P., Yan, H., & Zhang, H. (2003). Inventory models with fixed costs, forecast updates, and two delivery modes. Operations Research, 51(2), 321-328. Sethi, S. P., Yan, H., & Zhang, H. (2004). Quantity flexibility contracts: optimal decisions with information updates. Decision Sciences, 35(4), 691-712. Sethi, S., Yan, H., Zhang, H., & Zhou, J. (2005). Information updated supply chain with service-level constraints. Journal of Industrial and Management 35

ACCEPTED MANUSCRIPT Optimization, 1(4), 513-531. Song, H. M., Yang, H., Bensoussan, A., & Zhang, D. (2014). Optimal decision making in multi-product dual sourcing procurement with demand forecast updating. Computers & Operations Research, 41, 299-308. Valencia-Cárdenas, M., Díaz-Serna, F. J., & Correa-Morales, J. C. (2016). Multiproduct inventory modeling with demand forecasting and Bayesian optimization. In: DYNA. Universidad Nacional de Colombia, 83(198), 235-243. Wang, Q. Z., Tsao, D. B. (2006). Supply contract with bidirectional options: The buyer’s perspective. International Journal of Production Economics, 101, 30-52. Yan, H., Liu, K., & Hsu, A. (2003). Optimal ordering in a dual-supplier system with demand forecast updates. Production and Operations Management, 12(1), 30-45. Zhang, B. (2012). Multi-tier binary solution method for multi-product newsvendor problem with multiple constraints. European Journal of Operational Research, 218(2), 426-434. Zhang, B., Hua, Z.S. (2010). A portfolio approach to multi-product newsboy problem with budget constraint. Computers & Industrial Engineering, 58, 759-765. Zhou, S. X., & Chao, X. (2014). Dynamic pricing and inventory management with regular and expedited supplies. Production and Operations Management, 23(1), 65-80.

36

ACCEPTED MANUSCRIPT Figures

1st order received

1st order placed

2nd order placed And Received

Delivery lead time

Demand learning

Pre-season

Period 1

Period 2

Fig. 1. The sequence of events

2

×10 4 2

1.5 Actual sales

Actual sales

1.5

×10 4

1 0.5

1 0.5

0

0.5

1

Initial forecast (a)

1.5

2 ×10 4

0

0.5

1

1.5

Forecast update based on the sales data of first 10 days (b)

2 ×10 4

Fig. 2. The demand forecast precision change based on initial sales data

37

ACCEPTED MANUSCRIPT

Fig. 3. The value of expedited ordering for different B1 in problem P1

Fig. 4. The value of expedited ordering for different B1 in problem P2

38

ACCEPTED MANUSCRIPT

Fig. 5. The impact of the expedited additional costs on the expedited value

0.8 0.6

1

0.4 0.2 0 -0.2 -0.4

0

1

2

3

4

5

6

7

v1

Fig. 6. The expected profit comparison of the two models with different v1

39

ACCEPTED MANUSCRIPT 0.8 0.6

1

0.4 0.2 0 -0.2 -0.4 13

13.5

14

14.5

e1

Fig. 7. The expected profit comparison of the two models with different e1

40

ACCEPTED MANUSCRIPT Tables

Table 1. The notation. n

Number of products

li

The minimum reordering quantity

i

Index for products, i=1,…,n

vi

Unit reservation cost of option contract

j

Index for resources, j=1, 2

ri

Unit execution cost of option contract

pi

Unit selling price

Ki

The reservation quantity

si

Unit salvage value

Xi

The uncertain demand of period 1

ci

Unit cost in the first order

Yi

The uncertain demand of period 2

ei

Unit cost in the expedited order

xi

The realized demand of period 1

bi , j

Coefficient of product i of resource j

yi

The realized demand of period 2

Qi

The initial order quantity

Q =(Q1 , , Qn ) and q =(q1 , , qn )

qi

The expedited order quantity

X  ( X 1 , , X n ) and x =( x1 , , xn )

Bj

The resource limit

Y  (Y1 , , Yn ) and y =( y1 , , yn )

Table 2. The parameters of the illustrative example.

i

pi

ci

ei

si

li

bi ,2

X

i

X

Y

i

Y

i

i

i

1

15

10

11

3

230

22

150

40

450

90

0.9

2

12

8

8.8

4

200

23

200

50

700

100

0.8

3

21

14

15.4

6

250

14

160

40

500

90

0.7

4

18

12

13.2

4

220

8

170

45

600

95

0.9

5

19.5

13

14.3

5

240

9

190

55

550

85

0.8

Table 3. The optimal solutions to the one-constraint and two-constraint problems.

i

One-constraint problem (B1)

Qi*

Qi*, BM Qi*  Qi*, BM

One-constraint problem (B2)

Qi*

Qi*, BM

Qi*  Qi*, BM

Two-constraint Problem

Qi*

Qi*, BM

Qi*  Qi*, BM

1

239

235

4

228

78

150

235

203

32

2

565

496

69

79

208

-129

415

208

207

3

278

315

-37

385

366

19

319

373

-54

4

418

378

40

598

627

-29

480

453

27

5

181

364

-183

555

598

-43

426

434

-8

Profit

16899

9996

6903

16805

11137

5668

17252

9980

7272 41

ACCEPTED MANUSCRIPT

Table 4. Statistical results for randomly generated one-constraint problems. Number of iterations (times)

Computation time (seconds)

n

100

500

1000

100

500

1000

Mean

9.93

8.93

9.30

43.39

186.03

423.79

Std. dev.

1.69

1.86

1.34

6.88

31.27

52.07

95%

Lower

8.70

8.24

8.80

40.82

174.36

404.35

C.I.

Upper

9.96

9.63

9.80

45.96

197.71

443.23

Table 5. Statistical results for randomly generated two-constraint problems. Number of iterations (times)

Computation time (seconds)

n

100

500

1000

100

500

1000

Mean

5

5.53

5.73

104.02

622.72

1318.40

Std. dev.

4.81

5.04

4.73

59.42

385.64

738.21

95%

Lower

3.21

3.65

3.97

81.84

478.72

1042.70

C.I.

Upper

6.79

7.41

7.50

126.21

766.73

1594.00

42