optimal pricing in the presence of advance booking strategies for complementary supply...

Scientia Iranica E (2020) 27(3), 1616{1633

Sharif University of TechnologyScientia Iranica

Transactions E: Industrial Engineeringhttp://scientiairanica.sharif.edu

Optimal pricing in the presence of advance bookingstrategies for complementary supply chains

M. Noori-daryan, A.A. Taleizadeh�, and M. RabbaniSchool of Industrial Engineering, College of Engineering, University of Tehran, Iran.

Received 15 March 2018; received in revised form 19 October 2018; accepted 7 January 2019

KEYWORDSPricing;Selling;O2O commerce;Advance booking;Game theory;Supply chain.

Abstract. With the high competition among companies, they tend to improve theirmarket share by applying di�erent selling mechanisms such as online to o�ine commerceas an e�cient selling mechanism in which they sell their products via both online andreal stores. This study deals with a selling problem for two complementary supply chainsincluding a supplier and a shopping center where the commodities are sold by a virtualshopping center and a traditional one that present items as complementary shoppingcenters. It was assumed that market demand depended on price and service level so thatthe customers could purchase items via both shopping centers based on their priorities.Also, to analyze the reactions of the partners of the chain, di�erent games were deemed.The aim was to obtain the closed-form solutions for the decision variables of the membersof the network in order to maximize their pro�ts. Prices of di�erent selling periods ateach echelon of the chains were decision variables of the model. The closed-form solutionsfor the decision variables were derived and the solutions were examined by a numericalexample. Several sensitivity analyses of the key factors were performed to determine thee�cient ones for the variables and pro�ts.© 2020 Sharif University of Technology. All rights reserved.

1. Introduction and literature review

Recently, pricing policies have become a major concernfor industries and corporates as an e�cient tool for im-proving pro�tability, since one of the factors inuencingdecisions of the customers is the price of products.Companies strive to apply optimal pricing decisionsto the management and control of the market demandin addition to raising their pro�ts. Thus, competitionamong companies as well as supply chains increasesto satisfy demands of the customers and subsequently,enhance pro�ts through attracting the market demand.

*. Corresponding author.E-mail addresses: [email protected] (M.Noori-daryan); [email protected] (A.A. Taleizadeh);[email protected] (M. Rabbani)

doi: 10.24200/sci.2019.50604.1784

Some authors have assumed pricing policies in their re-search. Monahan (1984) [1] reviewed a discount schemeproposed by a seller to buyers. The seller sought toobtain the optimal price of the commodity and theiraim was to encourage customers to buy more amountsof the desired product at reasonable prices, therebyincreasing pro�ts and reducing operational costs.

Whitin (1969) [2] examined pricing and inventorypolicies to minimize the cost of inventory systems.Benerjee (1986) [3] developed a model considering theoptimal discount from the point of view of the seller.In this model, which was an extension of Monahan'smodel, the seller bought their goods from anothersupplier. Ardalan (1991) [4] jointly studied optimalinventory and pricing policies for various combinationsof sales periods and response times. Also, an algorithmwas proposed to determine the combination of optimalordering and pricing strategies.

Boyaci and Gallego (2002) [5] analyzed coordi-

M. Noori-daryan et al./Scientia Iranica, Transactions E: Industrial Engineering 27 (2020) 1616{1633 1617

nation issues in a two-tier supply chain including awholesaler and one or more retailers. In this chain,the demand of the customers was a function of theprice of goods and the operating costs included thoseof purchasing, commissioning, ordering, and keepinginventory. Elmaghraby and Keskinocak (2003) [6]performed a review of dynamic pricing literature as aninuential �eld in business. They focused on dynamicpricing with inventory issues. Agrawal et al. (2004) [7]presented a dynamic model for the allocation of inven-tory to a group of retailers to address the imbalanceamong inventory problems. They examined demandand net income information for inventory planning andbalancing in a retail network and addressed the issuewith the aim of determining delivery time and inven-tory levels for retailers under a dynamic planning. Huaet al. (2010) [8] reviewed optimal pricing and deliverydecisions in centralized and decentralized supply chainsinvolving a producer and a retailer with two channels(online sales and direct sales). Using a Stackelberggame, the impact of delivery time on the selling pricesof producer and retailer was evaluated and the resultsshowed that delivery time had a considerable e�ect onthe retail prices.

Chen et al. (2013) [9] examined a competitivedecision-making model for two companies by a Stack-elberg cooperative method and found that a non-cooperative approach would lead to a higher incomeand lower cost to the retailer and consequently, itbrings about lower pro�t for the producer. Chenand Hu (2012) [10] introduced a single-product pricingmodel with a common inventory under a �nite planninghorizon with a deterministic demand. In this model,the ordering and pricing decisions were determinedsimultaneously at the beginning of each decision periodwhere demand depended on the price. Taleizadehand Noori-daryan (2015a) [11] developed a non-linearprogramming model to examine the reactions of themembers of a three-echelon supply chain in terms ofpricing, manufacturing, and ordering decisions.

Saha et al. (2017) [12], Yan et al. (2018) [13],and Jamali and Rasti-Barzoki (2018) [14] discusseddi�erent mathematical revenue management problemsto model and study the behavior of the partners of asupply chain under various decision making processesusing Stackelberg and wide optimal methods. Noori-daryan et al. (2017) [15] considered a multi-nationalsupply chain including some capacity-constraint pro-ducers and a retailer in which the members madedecisions about their priorities such as price, orderquantity, lead-time, and selection of supplier underdi�erent games. In addition, some investigationshave been carried out in revenue management areato optimize price decisions under various conditions,e.g. [16{31].

Booking strategies as incentive mechanisms are

applied by companies to the management of sellingcapacities. Under these types of policies, companieslaunch their items at a lower price than their real pricesunder capacity restriction to encourage customers andabsorb the demand of the market. Some researchershave studied pricing policies under booking strategiesin di�erent industries, e.g. [32{42]. In 1992 and 1993,Gale and Holmes evaluated booking strategies andfound out that pre-sales strategies o�ered to passengersfor pick ights were pro�table for the airlines. Li(2001) [43] discussed a pricing issue for depreciablecommodities or services such as tickets or rooms ofa hotel. The results of this study showed that, asthe airlines expected, leisure travelers were more price-sensitive than business travelers. Then, in 1998,Dana investigated an advance booking policy in acompetitive environment.

Chen and Schwartz (2006) [44] considered a reser-vation strategy for studying the behavior of customersand how much they bought from a hotel and anairline. Cho and Tang (2010) developed a Stackelberggame model to examine the interactions between amanufacturer and a retailer where each company set itssales prices under reservation policies. They found thatthe policy was always pro�table for the producer andthe retailer bene�ted only when there was a probabilityof shortage. Acciaro (2011) [45] o�ered a service-based model based on reservation strategy of the carrierwith the assumption that the service-level customerswere di�erent depending on the types of customer,cargo, and route. Mei and Zhang (2013) [46] proposeda pricing issue with a reserved discount policy andpresented a stochastic programming model. In thisresearch, discount rates, prices, and optimal orderingwere determined and numerical results indicated highersensitivity and pro�tability of the sales price. Theyconcluded that reasonable prices would lead to thegreatest pro�t for the companies.

Ata and Dana Jr. (2004) [47] developed a mathe-matical model to examine a price di�erentiation strat-egy regarding di�erent reservation times of customers.Ling et al. (2015) [48] proposed an approach toenhancing coordination between a hotel and an onlinetravel agency to book rooms of the hotel. In thisapproach, customers were able to book their roomsthrough the sales system of the hotel or an onlinetravel agency. Furthermore, the hotel anticipated thepossible market demands with regard to the bookinginformation and the optimal availability approach tobooking rooms, taking into account the maximumrevenue through both methods (online and o�ine),was identi�ed. Bilotkach et al. (2015) [49] examinedthe impact of reducing the price of ight ticket onthe revenue of two airlines and found that reducingthe standard rate of fares would increase the averagedemand of consumers by an average of 2.7 percent.

1618 M. Noori-daryan et al./Scientia Iranica, Transactions E: Industrial Engineering 27 (2020) 1616{1633

It should be noted that this fall in prices did nothave considerable e�ect on the demand of customersfor tourist trips. Guizzardi et al. (2017) [50] studiedthe impact of booking strategies, service quality, andservice levels on purchasing decisions of customers for ahotel. Zhao et al. (2017) [51] investigated the behaviorsof �rms in terms of capacity allocation and pricingin full-fare and discount market. Ben��tez-Aurioles(2018) [52] studied the relation between reservation andprice in the tourist industry.

Game theory is a methodology for surveying thebehavior of players under competition and collabora-tion in a supply chain. Several authors have employedthis methodology to identify the decision variables.For instance, Dai et al. (2005) [53] analyzed pricingstrategies for several competing companies o�ering thesame services to a group of customers to improve theirrevenue management. Companies had limited capacityand the market demand to each company depended onits sales price. They used game theory approaches tochecking the systems when companies were faced withdeterministic and probabilistic demand. They applieda Nash equilibrium point to the case in which thedemand to each company was a linear function of price.Liu et al. (2007) [54] considered a decentralized supplychain consisting of a retailer and a provider in whichdemand depended on price and service level and therewas a Stackelberg game among the partners. Theyconcluded that decentralized decision-making resultedin lower performance and the supplier should improveits internal operations before pursuing a coordinatingstrategy with retailers.

Szemerekovsky and Zhang (2009) [55] reviewedpricing and two-level advertising decisions in a single-producer-single-retailer supply chain. They assumedthat the demand of the customers was a function ofretail price and advertising cost paid by the manufac-turer and retailer, and considered a Stackelberg gamein which the producer was the leader and the retailerwas the follower. They found that better performancewould be achieved if the producer incurred the adver-tising costs and o�ered a lower wholesale price to theretailer. Grauberger and Kimms (2016) [56] presenteda non-linear programming model to determine optimalpricing and reservation constraints under Nash gamesand found that revenues of the �rms in an exclusivemarket were lower than those in a non-exclusive one.Shah et al. (2014) [20], Taleizadeh et al. (2015b) [57],and, Noori-daryan and Taleizadeh (2016) [58] alsoconsidered game theory approaches to solving theirpresented models.

A look at the literature review given above indi-cates that there is no study investigating pricing deci-sions in two dual-channel supply chains under an ad-vance booking strategy in which demand is service-leveland price-sensitive and two complementary commodi-

ties are sold. In addition, upstream members o�er anadvance booking policy to downstream members as wellas customers. All members determine their optimalpricing decisions under a non-cooperative game so thattheir pro�ts are optimized. The rest of the paper is or-ganized as follows. The problem is described in Section2. Section 3 formulates the introduced model. In Sec-tion 4, a solution procedure is proposed and in Section5, a real case and the results are given. Finally, Section6 presents the conclusion and �ndings of the study.

2. Problem statement

Here, a pricing problem for complementary items underan advance booking mechanism is studied in whichcommodities are launched by the sellers at a lowerprice in advance and then, they are priced higherthan before. The commodities are provided by twocomplementary supply networks in which the suppliersells them through a virtual and a real shopping centerin each chain. Under this strategy, the networks aim toencourage the customers to order in advance in orderto manage the market demand. The structure of thenetworks is presented in Figure 1.

The purpose of this study is to �nd the optimaldecisions of the members of the networks, the suppliers,and the retailer on the selling prices of commoditieswithin pre-selling and selling periods so that the pro�tof the members and the networks is optimized. Inaddition, the reactions of the suppliers who supplythe complementary products are surveyed by applyinga Nash game. A leader-follower game is assumedbetween the suppliers and its shopping center in whichthe role of the follower is assigned to the retailerand the real store, and the suppliers as the leadersoptimally determine their decision policies accordingto the reaction of the retailer.

The problem is modeled by utilizing the followingassumptions:1. Price and service level a�ect the demand of the

customers;2. Each network is composed of a supplier and a

shopping center;3. Suppliers present complementary items to the cus-

tomers;

Figure 1. Structure of supply chains.


4. Customers strive for purchasing both commoditiesas complementary ones;

5. Suppliers launch their commodities through anonline and a traditional shopping center;

6. A common shopping center is considered to sellboth commodities for attracting the customers ofone item to the other complementary item;

7. Service level of the real shopping center is higherthan that of the online stores;

8. An advance booking strategy is applied to sellingthe commodities by online and o�ine shoppingcenters.

The problem in hand is formulated by utilizingthe following notation:Parameters:q11(t) Booking level of the �rst supplier

within [0; t1]

q21(t) Booking level of the �rst supplierwithin [t1; T ]

q12(t) Booking level of the second supplierwithin [0; t1]

q22(t) Booking level of the second supplierwithin [t1; T ]

q11R (t) Booking level of the �rst item in thereal store within [0; t1]

q12R (t) Booking level of the �rst item in thereal store within [t1; T ]

q21R (t) Booking level of the second item in thereal store within [0; t1]

q22R (t) Booking level of the second item in thereal store within [t1; T ]

d11(p11) Demand rate of customers from the

�rst supplier at selling price p11 within[0; t1] per unit time


�rst supplier at selling price p21 within[t1; T ] per unit time


second supplier at selling price p12within [0; t1] per unit time


second supplier at selling price p22within [t1; T ] per unit time

d11R (p11R ) Demand rate of customers from the

real store for the �rst item at sellingprice p11R within [0; t1] per unit time


real store for the �rst item at sellingprice p12R within [t1; T ] per unit time


real store for the second item at sellingprice p21R within [0; t1] per unit time


real store for the second item at sellingprice p22R within [t1; T ] per unit time

v11 Wholesale price of the �rst supplier forthe �rst item within [0; t1] per item

v21 Wholesale price of the �rst supplier forthe �rst item within [t1; T ] per item

v12 Wholesale price of the second supplierfor the second item within [0; t1] peritem

v22 Wholesale price of the second supplierfor the second item within [t1; T ] peritem

k11 Unit cost of the �rst supplier for the�rst item within [0; t1]

k21 Unit cost of the �rst supplier for the�rst item within [t1; T ]

k12 Unit cost of the second supplier for thesecond item within [0; t1]

k22 Unit cost of the second supplier for thesecond item within [t1; T ]

t1 Time of price-changeT Length of the pre-selling period 1 Total pro�t of the �rst supplier per

unit time 2 Total pro�t of the second supplier per

unit time R Total pro�t of the real store per unit

timeDecision variables:p11 Selling price of the �rst supplier within

[0; t1] per item per unit time

p21 Selling price of the �rst supplier within[t1; T ] per item per unit time

p12 Selling price of the second supplierwithin [0; t1] per item per unit time

p22 Selling price of the second supplierwithin [t1; T ] per item per unit time

p11R Selling price of the real store for the�rst item within [0; t1] per item perunit time

p12R Selling price of the real store for the�rst item within [t1; T ] per item perunit time

p21R Selling price of the real store for thesecond item within [0; t1] per item perunit time


p22R Selling price of the real store for thesecond item within [t1; T ] per item perunit time

3. Mathematical model and analysis

In this section, a pricing model is introduced for twocomplementary supply networks launching two com-plementary commodities under advance booking policy.Each network involves a supplier and a retailer in whichthe suppliers sell their commodities through an onlineand a real shopping center. The formulation of themodel for the members of the networks is presented inthe following.

3.1. Model for the �rst supplierIn this case, the �rst commodity is presented by the�rst supplier via its virtual shopping center and a realshopping center where it is priced in the virtual storelower than the real store, i.e., p11 < p11R and p21 < p12R .Buyers decide about purchasing based on the price andservice level presented by the sellers as their priorities.The pro�t of the �rst supplier in virtual and realshopping centers is modeled as given.

3.1.1. Model for the �rst supplier: Selling via anonline shopping center

An advance booking strategy is employed by thesupplier to persuade the buyers to order in advance.In fact, the supplier strives to manage market demandby launching the item in two certain periods. The itemis cheaper within [0; t1] than in the selling period [t1; T ],i.e., p11 < p21. Additionally, since the buyers who needmore guidance tend to purchase from real stores, it issupposed that service level of the real shopping centeris higher than that of online stores, i.e., s11 < s11R ands21 < s12R .

According to the assumptions, the demand of thebuyers depends on the price and service level and theywant to purchase both commodities. Thus, their de-mands from the online store of the �rst supplier during[0; t1] and [t1; T ] are respectively modeled as follows:

d11(p11)=�1�1��1p11��2p12+�Rp11R +1s11�Rs11R ; (1)

d21(p21)=�1�1��1p21��2p22+�Rp12R +1s21�Rs12R : (2)

Then, the booking levels of the commodity within[0; t1] and [t1; T ] are modeled di�erentially as follows:

dq11(t)dt

= d11(p11); 0 � t � t1; (3)

dq21(t)dt

= d21(p21); t1 � t � T: (4)

Since q11(0) = 0 and q21(T ) = Z1, booking levels of the�rst commodity at time t are as follows:

q11(t) =d11(p

11)t =

��1�1 � �1p11 � �2p12 + �RpA1R

+1s11 � Rs11R � t; (5)q21(t) =d

21(p

21)(T � t) = ��1�1 � �1p21 � �2p22

+�Rp12R + 1s21 � Rs12R � (T � t): (6)

Thereupon, the number of items sold within [0; t1] canbe calculated as follows:

z11 =t1Z

0

d11(p11)dt = d

11(p

11)t1 =

��1�1 � �1p11

��2p12 + �RpA1R + 1s11 � Rs11R � t1; (7)and within [t1; T ] can be identi�ed as follows:

z21 =TZt1

d21(p21)dt = d

21(p

21) (T � t1) = ��1�1 � �1p21

��2p22 + �Rp12R + 1s21 � Rs12R � (T � t1): (8)Therefore, the pro�t earned by the supplier throughan online shopping center is as follows:

�O1 =(p11 � k11)z11 + (p21 � k21)z21 = (p1A � k1A) [�1�1��1p11 � �2p12 + �Rp11R + 1s11 � Rs11R � t1+ (p21 � k21) ��1�1 � �1P 21 � �2P 22 + �RP 12R+1S21 � RS12R � (T � t1): (9)

3.1.2. Model for the �rst supplier: Selling via a realshopping center

The real common retailer orders the needed commodi-ties based on the expected demand at its wholesaleprices. Then, it sells them for a pro�t under a merchantmodel achieved by an agreement between the retailerand the supplier. Note that the supplier presentsits commodities to the real retailer under an advancebooking policy by which v1A < v2A. Therefore, thedemands of the retailer from the �rst supplier during[0; t1] and [t1; T ] are respectively modeled as follows:

d11R (p11R ) =�R(1� �1)�1 � �Rp11R � �Rp21R + �1p11

+ Rs11R � 1s11; (10)d12R (p

12R ) =�R(1� �1)�1 � �Rp12R � �Rp22R + �1p21

+ Rs12R � 1s21: (11)The booking levels of the items within [0; t1] and [t1; T ]are di�erentially formulated as follows:


dqR11 (t)dt

= d11R (p11R ); 0 � t � t1; (12)

dqR21 (t)dt

= d12R (p12R ); t1 � t � T: (13)

Since qR11 (0) = 0 and qR21 (T ) = Z1R, booking levels ofthe �rst commodity at time t are as follows:

qR11 (t) =d11R�p11R�t =

��R(1� �1)�1 � �Rp11R

��Rp21R + �1p11 + Rs11R � 1s11� t; (14)qR21 (t)=d

12R�p12R�

(T�t)=��R(1��1)�1��Rp12R��Rp22R +�1p21+Rs12R �1s21� (T�t): (15)

Thus, the quantity of the �rst item during [0; t1] iscalculated as follows:

zR11 =t1Z

0

d11R (p11R )dt = d

11R�p11R�t1 = [�R(1� �1)�1

��Rp11R ��Rp21R +�1p11+Rs11R �As11� t1; (16)and during [t1; T ] is identi�ed as follows:

zR21 =TZt1

d12R (p12R )dt=d

12R (p

12R )(T�t)=[�R(1��1)�!1

��Rp12R��Rp22R +�1p21+Rs12R �1s21� (T�t1): (17)Then, the pro�t earned by the �rst supplier through areal selling channel is as follows:

R1 =(v11 � k11)zR11 + (v21 � k21)zR21

=(v11 � k11) ��R(1� �1)�1 � �RP 11R � �RP 21R+�1P 11 + RS

11R � 1S11� t1

+ (v21 � k21) ��R(1� �1)�1 � �RP 12R � �RP 22R+�1P 21 + RS

A2R � 1S21� (T � t1): (18)

Eventually, pro�t of the �rst supplier is as follows:

1 = o1 + R1 = (p

11 � k11)z11 + (p21 � k21)z21

+ (v11 � k11)zR11 + (v21 � k21)zR21 : (19)3.2. Model for the second supplierSimilarly, the second commodity is presented by thesecond supplier via its virtual shopping center and areal shopping center and it is priced in the virtual storelower than in the real store, i.e., p12 < p21R and p22 < p22R .Buyers decide about purchasing based on the price andservice level o�ered by the sellers as their priorities.Therefore, the pro�t of the second supplier in virtualand real shopping centers is modeled as given.

3.2.1. Model for the second supplier: Selling via anonline shopping center

An advance booking strategy is employed by thesupplier to persuade the buyers to order in advance.Indeed, the supplier strives to manage the marketdemand by launching the item in two certain periodsin which it is cheaper within [0; t1] than in the sellingperiod [t1; T ], i.e., p12 < p22. Additionally, since thebuyers who need more guidance tend to purchase fromreal stores, it is supposed that service level of the realshopping center is higher than that of the online stores,i.e., s12 < s21R and s22 < s22R .

According to the assumptions, the demand of thebuyers depends on the price and service level and theywant to purchase both commodities. Therefore, theirdemands from the online store of the second supplierduring [0; t1] and [t1; T ] are respectively modeled asfollows:d12(p

12)=�2�2��2P 12 ��1P 11 +�RP 21R +2S12�RS21R ;

(20)

d22(p22)=�2�2��2p22��1p21+�Rp22R +2s22�Rs22R : (21)

Thus, the booking levels of the items duringperiods [0; t1] and [t1; T ] are modeled di�erentially asfollows:dq12(t)dt

= d12(p12); 0 � t � t1; (22)

dq22(t)dt

= d22(p22); t1 � t � T: (23)

Since q12(0) = 0 and q22(T ) = Z2, booking levels of thesecond items at time t are as follows:q12(t) =d

12(p

12)t =

��2�2 � �2p12 � �1p11 + �Rp21R

+2s12 � Rs21R � t; (24)q22(t) =d

22(p

22)(T � t) = ��2�2 � �2p22 � �1p21

+�Rp22R + 2s22 � Rs22R � (T � t): (25)

Hence, the number of commodities within [0; t1] iscalculated as follows:

z12 =t1Z

0

d12(p12)dt = d

12(p

12)t1 =

��2�2 � �2p12

��1p11 + �Rp21R + 2s12 � Rs21R � t1; (26)and the number of items during period [t1; T ] isidenti�ed as follows:

z2H2 =TZt1

d22(p22)dt = d

22(p

22)(T � t) = ��2�2 � �2p22

��1p21 + �Rp22R + 2s22 � Rs22R � (T � t1): (27)


Furthermore, the pro�t earned by the second supplierthrough its online shopping center is as follows: o2 =(p

12 � k12)z12 + (p22 � k22)z22 = (p12 � k12) [�2�2��2s12 � �1s11 + �Rs21R + 2s12� t1+ (p22 � k22) ��2�2 � �2p22 � �1p21 + �Rp22R+2s22

�(T � t1): (28)

3.2.2. Model for the second supplier: Selling via a realshopping center

In this case, similarly to the previous one, the realstore buys the item under an advance booking policyfrom the second supplier at the o�ered wholesale priceswhere v12 < v22 and p21R < p22R . A merchant contract isconsidered between the retailer and the supplier so thatthe retailer sells its ordered items for a pro�t based onits own opinion about the price of the item.

Therefore, the demands of the retailer from thesecond supplier during [0; t1] and [t1; T ] are respectivelymodeled as follows:d21R (p

21R ) =�R(1� �2)�2 � �Rp21R � �Rp11R

+ �2p12 + Rs21R � 2s12; (29)

d22R (p22R ) =�R(1� �2)�2 � �Rp22R � �Rp12R

+ �2p22 + Rs22R � 2s22: (30)

Thus, the booking levels of the items within [0; t1] and[t1; T ] are di�erentially indicated as follows:

dqR12 (t)dt

= d21R (p21R ); 0 � t � t1; (31)

dqR22 (t)dt

= d22R (p22R ); t1 � t � T: (32)

Since qR12 (0) = 0 and qR22 (T ) = Z2R, booking levels ofthe items at time t are as follows:qR12 (t) =d

21R (p

21R )t=

��R(1��2)�2��Rp21R

��RpA1R +�2p12 + Rs21R � 2s12� t; (33)qR22 (t) =d

22R (p

22R )(T � t1) = ��R(1��2)�2 � �Rp22R

��Rp12R + �2p22 + Rs22R �2s22� (T�t): (34)Then, the number of items within [0; t1] is demon-strated as follows:

zR12 =t1Z

0

d21R (p21R )dt=d

21R (p

21R )t1 = [�R(1��2)�2

��Rp21R ��Rp11R + �2p12+Rs21R �2s12� t1: (35)Hence, the quantity of the second item during [t1; T ] iscalculated as follows:

zR22 =TZt1

d22R (p22R )dt = d

22R (p

22R )(T � t1)

=��R(1� �2)�2 � �TPH2T � �TPA2T + �HP 2H

+TSH2T � HS2H� (T � t1): (36)Therefore, the pro�t earned by the second supplierthrough the real selling channel is as follows:

�R2 =(p21R � v12)zR12 + (p22R � v22)zR22

=(p21T � v12) ��T (1� �2)�2 � �Rp21R � �Rp11R+�2p12 + Rs

21R � 2s12� t1

+ (p22T � v22) ��T (1� �2)�2 � �Rp22R � �Rp12R+�2p22 + Rs

22R � 2s22� (T � t1): (37)

Finally, the pro�t of the second supplier is as follows:

�2 =�O2 + �R2 = (p

12 � k12)z12 + (p22 � k22)z22

+ (v12 � k12)zR12 + (v22 � k22)zR22 : (38)3.3. Model for the real retailerBesides the online stores, the real shopping centerlaunches the �rst and the second commodities, simulta-neously, as complementary items. It is supposed thatthe suppliers commonly sell the items through a realshopping center, along with online stores, to attractthe buyers who need more guidance or prefer to touchthe purchased commodities. Thus, the pro�t comprisestwo components for selling the �rst and the secondcommodities. The booking level of the �rst item isformulated di�erentially as:

dq11R (t)dt

= d11R (p11R ); 0 � t � t1; (39)

dq12R (t)dt

= d12R (p12R ); t1 � t � T: (40)

Since q11R (0) = 0 and q12R (T ) = Z1R, booking levels forthe �rst item at time t are as follows:

q11R (t) =d11R (p

11R )t =

��R(1� �1)�1 � �Rp11R

��2p21R + �1p11 + Rs11R � 1s11� t; (41)q12R (t) =d

12R (p

12R )(T � t) = ��R(1� �1)�1��Rp12R

��2p22R + �1p21 + Rs12R �1s21� (T�t): (42)Furthermore, the number of the �rst commodity within[0; t1] is calculated as follows:


z11R =t1Z

0

d11R (p11R )dt=d

11R (p

11R )t1 =[�R(1��1)�1

��Rp11R � �2p21R + �1p11+Rs11R �1s11� t1; (43)and within [t1; T ] is identi�ed as follows:

z12R =TZt1

d12R (p12R )(T�t)dt = [�R(1��1)�1

��Rp12R ��2p22R +�1p21+Rs12R �1s21�(T�t1):(44)

The pro�t earned by the retailer through selling the�rst item is as follows:

1R =(p11R � v11)z11R + (p12R � v21)z12R

=(p11R � v11) ��R(1� �1)�1 � �Rp11R � �2p21R+�1p11 + Rs

11R � 1s11� t1

+ (p12R � v21) ��R(1� �1)�1 � �Rp12R � �2p22R+�1p21 + Rs

12R � 1s21� (T � t1): (45)

In addition, the booking level of the second item is:

dq21R (t)dt

= d21R (p21R ); 0 � t � t1; (46)

dq22R (t)dt

= d22R (p22R ); t1 � t � T: (47)

Since q21R (0) = 0 and q22R (T ) = Z2R, booking levels ofthe second item at time t are as follows:

q21R (t) =d21R (p

21R )t =

��R(1� �2)�2 � �Rp21R

��Rp11R + �2p12 + Rs21R � 2s12� t; (48)q22R (t) =d

22R (p

22R )(T � t) = ��R(1� �2)�2 � �Rp22R

��Rp12R + �2p22 + Rs22R � 2s22� (T�t): (49)Thus, the quantity of the second item within [0; t1] iscalculated as follows:

z21R =t1Z

0

d21R (p21R )dt=d

21R (p

21R )t1 = [�R(1��2)�2

��Rp21R ��Rp11R +�2p12+Rs21R �2s12� t1; (50)and within [t1; T ] is:

z22R =Z Tt1d22R (p

22R )dt = d

22R (p

22R )(T � t1)

=��R(1� �2)�2 � �Rp22R � �Rp12R + �2p22

+Rs22R � 2s22� (T � t1): (51)Therefore, the pro�t earned by the retailer throughselling the second item is equal to:

2R =(p21R �v12)z21R + (p22R �v22)z22R

=(p21T �v12) ��R(1��2)�2��Rp21R ��Rp11R+�Hp1H+ T s

H1T � Hs1H� t1

+(pH2T �v2H) ��T (1� �H)�H��T pH2T��T pA2T +�Hp2H+T sH2T �Hs2H� (T�t1): (52)

Gradually, the total pro�t of the retailer can be statedas follows:

R =(p11R � v11)z11R + (p12R � v21)z12R + (p21R � v11)z21T+ (p22R � v22)z22R : (53)

4. Solution Procedure

In this section, a solution procedure to �nd the closed-form solutions for the decision variables is examined inorder to analyze the reactions of the members of supplychains by considering game theory approaches such asNash and Stackelberg game. In this case, it is assumedthat there is a Nash game between the suppliers whosupply the commodities independently while there is aStackelberg game between the members of each chainwhere the real shopping center is the follower and thesupplier is the leader of the market.

In the Stackelberg game, the optimal decisionpolicies of the leaders depend on the optimal decisionsof the followers and the best reaction of the retailer isconsidered by the leaders. The prices of commoditieswithin [0; t1] and [t1; T ] are the decision variables ofthe retailer, as the follower, while the prices of theitems within time [0; t1] and [t1; T ] in online storesare determined by the suppliers, as the leaders. Inaddition, the optimality of the objective functionsregarding the decision variables should be proven.

Theorem 1. The pro�t of the real retailer R(p11R ; p12R ; p21R ; p22R ) is concave.

Proof. If Eqs. (B.12) to (B.15) are met (see Ap-pendix B), concavity of the pro�t function for theretailer is proven. Thus, by taking the �rst-order


derivatives of the pro�t function in Eq. (53), theoptimal values of p11R , p12R , p21R , p22R are obtained asfollows:

@ R@p11R

=��R(1� �1)�1 � �Rp11R � �2p21R + �1p11

+Rs11R � 1s11� t1 � �(p11R � v11)+(p21R � v12)��Rt1 = 0; (54)

@ R@p12R

=��R(1� �1)�1 � �Rp12R � �2p22R + �1p21

+Rs12R � 1s21� (T � t1)� (P 12R � v21)�R(T � t1)� (P 22R � v22)�R(T � t1) = 0; (55)

@ R@p21R

=��R(1� �2)�2 � �Rp21R � �Rp11R + �2p12

+Rs21R � 2s12� t1 � �(p21R � v12)+(p11R � v11)��Rt1 = 0; (56)

@ R@p22R

=��R(1� �2)�2 � �Rp22R � �Rp12R + �2p22

+Rs22R � 2s22� (T � t1)� (p22R � v22)�R(T � t1)� (p12R � v21)�R(T � t1) = 0: (57)

Then, the optimal decisions of the retailer with regardto the policies of the suppliers after simplifying theabove equations are as follows:

p11�R = Y1 +�1p1�12�R

; (58)

p11�R = Y2 +�1p2�12�R

; (59)

p21�R = Y3 +�2p1�22�R

; (60)

p21�R = Y4 +�2p2�22�R

; (61)

for which the equations utilized to simplify the relationsare illustrated in Appendix A. Besides, based on thehypothesis, a Stackelberg game is considered betweenmembers of the networks, where a real retailer decidesabout the optimal policies and the suppliers as theleaders characterize the variables regarding the optimal

reactions of the retailer. A Nash game is assumedbetween suppliers in order to obtain their decisionvariables, simultaneously.

Theorem 2. The pro�t of the �rst supplier, 1(p11; p21), is concave.

Proof. Having the concavity of the pro�t function forthe �rst supplier proven (see Appendix C), the optimalvalues of p11R , p12R , p21R , p22R are replaced into Eq. (19),which becomes: 1 =(p11 � k11) ��1�1 � �1p11 � �2p12 + �RpA1�R

+1s11 � Rs11R � t1 + (p21 � k21) ��1�1 � �1p21��2p22 + �Rp12�R + 1s21 � Rs12R � (T � t1)+ (v11 � k11) ��R(1� �1)�1 � �Rp11�R��Rp21�R + �1p11 + Rs11R � 1s11� t1+ (v21 � k21) ��R(1� �1)�1 � �Rp12�R��Rp22�R + �1p21 + Rs12R � 1s21� (T � t1): (62)

Then, the optimal values of p11 and p21 are obtained bytaking into account the �rst-order derivatives of theobjective function, Eq. (62), with regard to p11 and p21as follows:@ 1@p11

=��1�1��1p11��2p12+�Rp11R +1s11�Rs11R � t1� (p11 � k11)�1t1 + (v11 � k11)�1 = 0; (63)

@ 1@p21

=��1�1 � �1p21 � �2p22 + �Rp12R + 1s21�Rs12R � (T � t1)� (p21 � k21)�1(T � t1)+ (v21 � k21)�1(T � t1) = 0: (64)

Hence, the closed-form solutions for the variables of the�rst supplier are as follows:

p1�1 =�Y7 + Y11Y12 � �1 ; (65)

p2�1 =�2�R + Y13Y12 � �1 : (66)

Theorem 3. The pro�t of the second supplier 2(p12; p22) is concave.

Proof. Having concavity of the pro�t function for thesecond supplier proven (see Appendix D), the optimalvalues of P 11R , P 12R , P 21R , P 22R are replaced into Eq. (38),which becomes:


2 =(p12 � k12) ��2�2 � �2p12 � �1p11 + �Rp21�R+2s12 � Rs21R � t1 + (p22 � k22) ��2�2 � �2p22��1p21 + �Rp22�R + 2s22 � Rs22R � (T � t1)+ (P 21R � v12) ��R(1� �2)�2 � �Rp21�R��Rp11�R + �2p12 + Rs21R � 2s12� t1+ (P 22R � v22) ��R(1� �2)�2 � �Rp22�R��Rp12�R + �2p22 + Rs22R � 2s22� (T � t1): (67)

The optimal values of p12 and p22 are derived by takingthe �rst-order derivatives of the objective function inEq. (67) with regard to p12 and p22 as:

@ 2@p12

=��2�2��2p12��1p11+�Rp21R +2s12�Rs21R � t1� (p12 � k12)�2t1 + (v12 � k12)�2t1 = 0; (68)

@ 2@p22

=��2�2 � �2p22 � �1p21 + �Rp22R + 2s22�Rs22R � (T � t1)� (p22 � k22)�2(T � t1)+ (v22 � k22)�2(T � t1) = 0: (69)

The closed-form solutions for the variables of thesecond supplier are as follows:

p1�2 =��1Y11 + Y14Y12�2Y12 � �1�2 ; (70)

p2�2 =��1Y13 + Y15Y12�2Y12 � �2�1 : (71)

By replacing the closed-form solutions using Eqs. (65),(66), (70), and (71) into Eqs. (58){(61), the closed-formsolutions for the variables are as follows:

p11�R = Y1 + Y7�Y11 � Y14Y12 � �1

�; (72)

p12�R = Y2 + Y7�Y13 � Y15Y12 � �1

�; (73)

p21�R =Y9

2�R+

�22�R

��1Y11 + Y14Y12�2Y12 � �1�2

�; (74)

p22�R =Y4

2�R+

�22�R

��1Y13 + Y15Y12�2Y12 � �1�2

�: (75)

Note that the above-used relations are illustrated inAppendix A.

5. Numerical example and sensitivity analysis

5.1. Real caseAs a real case, we set the introduced model for twotourism supply chains with customers who wanted tobook the tickets of an airline and the rooms of a hotelfor a travel. It was supposed that the �rst chainlaunched the tickets and the second one provided therooms for the buyers and their orders were sensitive toprice and service level. In the �rst chain, an airline anda travel agency were the members and the second chainincluded a hotel and the same travel agency.

As it is common now, the airline and the hotelpresented their commodities through virtual stores anda common travel agency. Since the complementaryitems were presented by the airline and the hotel,companies preferred to sell their items through acommon travel agency to increase their market shares.Thus, this section numerically indicates applicabilityof the proposed model. Market demand depended onservice level and selling price. The impacts of keyparameters on the optimal values of decision variablesand the pro�ts of the supply networks were analyzed.

5.2. Numerical exampleSuppose that k11 = 10, k21 = 12, �1 = 40000, �1 = 0:4,�1 = 21, �2 = 21:5, �R = 20, 1 = 10, R = 15, s11 = 1,s11R = 2, s21 = 1, s12R = 2, v11 = 20, v21 = 25, �R = 0:5,v12 = 23, v22 = 28, s12 = 1, s21R = 2, s22 = 1, s22R = 2,

2 = 19, �2 = 0:4, k12 = 12, and k22 = 17. The resultsare indicated in Tables 1 and 2.

According to Tables 1 and 2, when the airlineand the hotel sell their commodities under an advancebooking policy, their pro�ts would be higher than whenthey sell them at their real prices. Therefore, theytend to apply an ADB selling strategy to improvetheir bene�ts from attracting the market demand bythe proposed selling prices in di�erent selling periods.In addition, the online stores launch the items at alower price than the travel agency due to direct selling.Therefore, the �rms succeed to improve their marketshares using di�erent selling channels in addition toconsidering an ADB policy.

5.3. Sensitivity analysisIn this section, to analyze the decisions of the membersand their pro�ts with the variations of parameters,the optimal values of the variables and the pro�tsof the partners with respect to the changes in someparameters are surveyed. The related results areshown in Tables 3{6 and the diagrams are shownin Figures 2{5. Examining the results leads to thefollowing managerial insights:

Table 3 shows that increase in sensitivity of cus-tomers to the price of the �rst item (ticket) results indecrease in the demand of buyers and diminishes the


Table 1. Results for the example with � = 0:5.

Member p11 p21 1 p

12 p

22 2 p

11R p

12R p

21R p

22R R

First supplier 374.13 377 2658662.86 | | | | | | | |Second supplier | | | 631.38 634.40 8110090.40 | | | | |Real shopping center | | | | | | 544.42 548.42 726.14 730.27 100544.59

Table 2. Results for the example with � = 0.

Member p11 p21 1 p12 p22 2 p

11R p12R p21R p22R R

First supplier | 377 2649773.95 | | | | | | | |Second supplier | | | | 634.40 8083506 | | | | |Real shopping center | | | | | | | 548.42 | 730.27 89447.64

Table 3. E�ect of sensitivity of the prices of the �rst supplier to demand changes on the optimal values of the variablesand pro�ts.Parameter Change% p11 p

21 1 p

12 p

22 2 p

11R p

12R p

21R p

22R R

�1

+0.75 224.07 228.28 1545943.45 619.66 619.75 7762525.95 553.87 560.23 719.84 722.39 Infeasible+0.50 257.42 261.32 1792968.55 623.57 624.64 7877539.79 550.72 556.29 721.94 725.02 Infeasible+0.25 304.10 307.59 2139098.75 627.47 629.52 7993394.61 547.57 552.35 724.04 727.64 Infeasible

0 374.13 377 2658662.86 631.38 634.40 8110090.40 544.42 548.42 726.14 730.27 100544.59{0.25 490.84 492.65 3525094.79 635.29 639.29 8227627.17 541.27 544.48 728.24 732.89 830727.32{0.50 724.26 723.98 5258696.27 639.2 644.17 8346004.91 538.12 540.54 730.34 735.52 2192930.50{0.75 Infeasible Infeasible Infeasible 643.10 649.06 8465223.63 534.97 536.60 732.44 738.14 6080838.15

Table 4. E�ect of sensitivity of the prices of the second supplier to demand changes on the optimal values of the variablesand pro�ts.

Parameter Change% p11 p21 1 p

12 p

22 2 p

11R p

12R p

21R p

22R R

�2

+0.75 Inf.� Inf. Inf. 372.62 376.91 4773323.89 537 539.39 737.21 743.81 1743897.04+0.50 364.71 365.52 2525873.27 430.12 434.13 5514177.43 539.47 542.4 733.56 739.3 1215973.09+0.25 369.42 371.25 2591890.34 510.62 514.24 6552152.54 541.94 545.41 729.85 734.78 668281.70

0 374.13 377 2658662.86 631.38 634.40 8110090.40 544.42 548.42 726.14 730.27 100544.59{0.25 378.84 382.72 2726190.83 Inf. Inf. Inf. 546.89 551.43 722.43 725.75 Inf.{0.50 383.55 388.45 2794474.24 Inf. Inf. Inf. 549.36 554.44 718.72 721.24 Inf.{0.75 388.26 394.19 2863513.11 Inf. Inf. Inf. 551.83 557.45 715.72 716.72 Inf.

�: Inf.: Infeasible

Table 5. E�ect of sensitivity of the prices of the real shopping center to demand changes on the optimal values of thevariables and pro�ts.

Parameter Change% p11 p21 1 p

12 p

22 2 p

11R p

12R p

21R p

22R R

�R

+0.75 376.13 379.70 2689865.81 635.43 639.15 8222618.71 315.98 319.55 421.11 424.75 -185766.21+0.50 375.46 378.8 2679438.30 634.08 637.57 8185016.39 366.74 370.41 488.89 492.64 -122917.12+0.25 374.8 377.89 2669037.32 632.73 635.99 8147506.95 437.81 441.61 583.79 587.69 -33997.75

0 374.13 377 2658662.86 631.38 634.40 8110090.40 544.42 548.42 726.14 730.27 100544.59{0.25 373.46 376.08 2648314.92 630.03 632.82 8072766.73 722.09 726.42 963.39 967.89 326332.90{0.50 372.8 375.18 2637993.51 628.68 631.24 8035535.94 1077.44 1082.44 1437.89 1443.14 780236.13{0.75 372.13 374.27 2627698.62 627.33 629.66 7998398.04 2143.48 2150.48 2861.38 2868.88 2146599.04

pro�t of the �rst supplier (airline). Consequently,demand for the second commodity (room) as thecomplementary item is reduced and the pro�t ofthe second supplier (hotel) is also diminished. Onthe other hand, selling prices of the retailer (travel

agency) are reduced, because they present substi-tutable items and should sell items at lower pricesto attract customers and manage their market share.Nonetheless, their pro�t is decreased because of thereduction in prices. The diagrams of the pro�ts


Figure 2. Sensitivity of the customers of the �rst supplierto changes in prices versus pro�ts of the �rms.

Figure 3. Sensitivity of the customers of the secondsupplier to changes in prices versus pro�ts of the �rms.

Figure 4. Sensitivity of the customers of the realshopping center to changes in prices versus pro�ts of the�rms.

Table 6. E�ect of changes in the pre-selling period on theoptimal values of pro�ts.

Parameter Value 1 2 RValue Value Value

�

0 2649773.95 8083506 89447.640.10 2651551.73 8088822.88 91667.030.20 2653329.51 8094139.76 93886.420.30 2655107.3 8099456 96105.810.40 2656885.08 8104773.52 98325.200.50 2658662.86 8110090.40 100544.590.60 2660440.64 8115407.28 102763.970.70 2662218.42 8120724.16 104983.360.80 2663996.20 8126041.04 107202.750.90 2665774 8131357.92 109422.14

Figure 5. Changes in booking period versus pro�ts of the�rms.

of the �rms with regard to sensitivity of marketdemand to the changes in prices for the �rst itemare indicated in Figure 2;

Based on Table 4, it is found that reducing sensi-tivity of customers to the price of the second item(room) leads to increase in the pro�t of the secondsupplier with higher prices. In addition, if priceelasticity of the demand of customers decreases,their demand is slightly inuenced by price changesand, in turn, the pro�t of the suppliers and theretailer will increase. Therefore, they price theircommodities in a way to maximize their pro�ts. Inother word, when sensitivity of customers to theprice of room decreases, the hotel tends to increaseits booking prices and the airline as well increasesits price and, subsequently, pro�t. In addition,the travel agency, as a competitive selling channel,increases its selling prices with consideration of thesensitivity of customers. The diagrams of the pro�tsof the �rms with regard to sensitivity of market


demand to the changes in prices for the second itemare presented in Figure 3;

The results in Table 5 state that the retailer launchesthe items at lower prices if price elasticity of demandincreases, as a result of which the pro�t of theretailer (travel agency) decreases. The customersdo not want to purchase the commodities at higherprices. Hence, the suppliers should present items viavirtual stores at lower prices to absorb the highlyprice-dependent demands. The diagrams of thepro�ts of the �rms with regard to sensitivity ofcustomers to the changes in prices of the tickets andthe rooms are given in Figure 4;

The e�ects of changes in booking periods on theoptimal values of decision variables and pro�ts areillustrated in Table 6. From the results, it canbe concluded that the suppliers and the retailertend to employ an advance booking strategy to sellcommodities for improving their pro�ts. Also, theygain higher pro�t if they sell items during a longerpre-selling period. The diagrams of the pro�ts of the�rms with regard to the changes in the pre-sellingperiod are provided in Figure 5;

According to the above analysis, it is concluded thatprice is a key decision variable and it has a signi�-cant impact on purchasing decisions the customers.Increasing/decreasing sensitivity of the customers tothe price changes leads to decrease/increase in thepro�t of the members of a chain.

6. Conclusion

A mathematical model was developed for a sellingproblem under an advance booking policy in whichmarket demand depended on service level and price ofcommodities with two complementary supply chains.It was supposed that each chain comprised a supplierand a retailer; the supplier presented their own com-modities both online and through a retailer who soldboth commodities to buyers as a common member ofboth chains. The purpose was to �nd closed-formsolutions for the decision variables of the members ofthe networks in order to maximize pro�t. Selling pricesof items at each echelon of a chain in di�erent selling pe-riods were the decision variables of the proposed model.

A real case was presented for the introducedmodel in which two tourism supply chains were con-sidered. They launched the complementary items ofthickets and rooms via a website and a travel agency. Inthis case, an airline and a hotel, as the suppliers of thechains, sold their commodities by applying an advancebooking selling strategy. A Nash game was assumedbetween the suppliers and a Stackelberg game betweenthe members of each chain, where the real shopping

center was the follower and the supplier was the leaderof the market.

In addition, there was a merchant agreementbetween the suppliers and a common travel agency tosell their items. The model was numerically analyzedand the e�ects of changes in some parameters on thedecision variables and the members were evaluated.It was found that the members preferred to employan advance booking policy to sell the commoditiesand manage the market demand and the policy wasbene�cial to all the members of the supply chains.Also, suppliers who sold their items via both channelsearned higher pro�t than those who performed onlyin the traditional way, because they attracted morecustomers and were price-and service-oriented. For thefuture research, the following issues are recommended:

Considering the proposed model under variousbooking strategies;

Considering the proposed model under a stochasticsetting;

Considering the proposed model under competitiveconditions; and

Considering the proposed model in multi-echelonsupply chains.

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Appendix A

Finding the closed-form solutionsFor �nding the closed-form solutions for the decisionvariables, the following parameters are utilized:

Y1 =[�R(1� �1)�1) + Rs11R � 1s11 + v11�R]

2�R; (A.1)

Y2 =[�R(1� �1)�1) + Rs12R � 1s21 + v21�R]

2�R; (A.2)

Y3 =[�R(1� �2)�2) + Rs21R � 2s12 + v12�R]

2�R; (A.3)

Y4 =[�R(1� �2)�2) + Rs22R � 2s22 + v22�R]

2�R; (A.4)

Y5 =[1��R)[(1��1)�1+(1��2)�2]+(v11 +v12)�R]

2�R;

(A.5)

Y6 =(1��R)[(1��1)�1+(1��2)�2]+(v21 +v22)�R

2�R;

(A.6)

Y7 =�1

2�R; (A.7)

Y8 = 2�2�R; (A.8)

Y9 = 2�RY4; (A.9)

Y10 =Y8

2�R; (A.10)

Y11 =�1�1 + Y1�R + 1s11 � Rs11R+ (v11 � 2k11)(��1 + �RY7); (A.11)

Y12 = 2(�1 � �RY6); (A.12)Y13 =�1�1 + Y2�T + 1s21 � Rs12R

+ (v21 � 2k21)(��1 + �RY7) + (v21 � k21);(A.13)

Y14 = �2�2 + 2s12 � Rs21R +��22

�(v12 � 2k12); (A.14)

Y15 = �2�2+�RY4+2s22�Rs22R ��22

�(v22�2k22):

(A.15)

Appendix B

Proving concavity of the pro�t of the realretailerTo prove concavity of the pro�t of the real retailer,�rst, the Hessian matrix should be formed as follows:

H =

2666666666664

@2 R@p112R

@2 R@p11R @p

12R

@2 R@p11R @p

21R

@2 R@p11R @p

22R

@2 R@p12R @p

11R

@2 R@pA22R

@2 R@p12R @p

21R

@2 R@p12R @p

22R

@2 R@p21R @p

11R

@2 R@p21R @p

12R

@2 R@p212R

@2 R@p21R @p

22R

@2 R@p22R @p

11R

@2 R@p22R @p

12R

@2 R@p22R @p

21R

@2 R@p222R

3777777777775;(B.1)

where:

@2 R@p112R

= �2�Rt1; (B.2)

@2 R@p11R @p12R

=@2 R

@p12R @p11R= 0; (B.3)

@2 R@p11R @p21R

=@2 R

@p21R @p11R= �2�2t1; (B.4)

@2 R@p11T @p22T

=@2 R

@p22R @p11R= 0; (B.5)

@2 R@p122R

= �2�R(T � t1); (B.6)

@2 R@p12R @p21R

=@2 R

@p21R @p12R= 0; (B.7)

@2 R@p12R @p22R

=@2 R

@p22R @p12R= �2�2(T � t1); (B.8)

@2 R@p212R

= �2�Rt1; (B.9)

@2 R@p21R @p22R

=@2 R

@p22R @p21R= 0; (B.10)

@2 R@p222R

= ��R(T � t1): (B.11)

Then, it is found that �R is concave if Eqs. (B.12){(B.15) are met:


jH1j =��@2 R@p112R

�� = �2�Rt1 < 0; (B.12)jH2j =

��2�Rt1 00 �2�R(T � t1)��= 4�2T t1(T � t1) > 0; (B.13)

jH3j =��2�Rt1 0 �2�2t10 �2�R(T � t1) 0�2�2t1 0 �2�Rt1

��= 8

��22�R � �3R� t21(T � t1) < 0; (B.14)

jH4j =��2�Rt1 0 �2�2t1 0

0 �2�R(T � t1) 0 �2�2(T � t1)�2�2t1 0 �2�Rt1 00 �2�2(T � t1) 0 �2�R(T � t1)

��= 16

��4R + �

42�t21(T � t1)2 > 0: (B.15)

Appendix C

Proving concavity of the pro�t of the airlineTo prove concavity of the pro�t of the �rst supplier,�rst, the Hessian matrix should be formed as follows:

H =

264 @2 1@p121 @2 1@p11@p21@2 1@p21@p

11

@2 1@p221

375 ; (C.1)where:

@2 A@p121

= �2�1t1; (C.2)

@2 1@p11@p21

=@2 1@p21@p11

= 0; (C.3)

@2 A@p221

= �2�1(T � t1) < 0: (C.4)

Hence, it is found that 1 is concave if conditions (C.5)and (C.6) are met:

jHj1 = @2 1@p121

= �2�1t1 < 0; (C.5)

jHj2 =@2 1@p121

:@2 1@p221

� @2 1@p11@p21

:@2 1@p21@p11

=4�21t1(T � t1) > 0: (C.6)

Appendix D

Proving concavity of the pro�t of the secondsupplierTo prove concavity of the pro�t of the second supplier,�rst, the Hessian matrix should be formed as follows:

H =

264 @2 2@p122 @2 2@p12@p22@2 2@p22@p

12

@2 2@p222

375 ; (D.1)where:

@2 2@p122

= �2�2t1; (D.2)

@2 2@p12@p22

=@2 2@p22@p12

= 0; (D.3)

@2 2@p222

= �2�2(T � t1) < 0: (D.4)

Hence, it is found that �2 is concave if Conditions (D.5)and (D.6) are met:

jHj1 = @2 2@p122

= �2�2t1 < 0; (D.5)

jHj2 =@2 2@p122

:@2 2@p222

� @2 2@p12@p22

:@2 2@p22@p12

=4�22t1(T � t1) > 0: (D.6)

Biographies

Mahsa Noori-daryan received her BSc degree inPhysics from Science and Research Branch of IslamicAzad University, Tehran, Iran, and her MSc degreein Industrial Engineering from South Tehran Branchof Islamic Azad University, Tehran, Iran. She is aPhD candidate in Industrial Engineering at Univer-sity of Tehran. Her research interests are in supplychain management, inventory control, and operationresearch.

Ata Allah Taleizadeh is an Associate Professorin the School of Industrial Engineering at Universityof Tehran in Iran. He received his PhD degree inIndustrial Engineering from Iran University of Scienceand Technology. Moreover, he received his BSc andMSc degrees both in Industrial Engineering from AzadUniversity of Qazvin and Iran University of Science andTechnology, respectively. His areas of research interestinclude inventory control and production planning,pricing and revenue optimization, game theory, anduncertain programming. He has published several


books and papers in reputable journals and serves asthe editor/editorial board member for a number ofinternational journals.

Masoud Rabbani is a Professor in the School ofIndustrial Engineering at University of Tehran, Iran.He received his PhD in Industrial Engineering fromAmirkabir University of Technology, Iran. His researchinterests include manufacturing and production sys-tems, soft computing techniques, Multi-Criteria Deci-sion Making (MCDM), transportation, and scheduling.He has published papers in journals such as the Interna-

tional Journal of Advanced Manufacturing Technology,Applied Mathematics and Computation, InternationalJournal of Production Research, Soft Computing, Ad-vances in Engineering Software, Applied Mathemati-cal Modeling, Engineering Optimization, InternationalJournal of Manufacturing Technology and Manage-ment, Advanced Engineering Informatics, EuropeanJournal of Operational Research, Expert Systems withApplications, International Journal of Computer Math-ematics, Applied Soft Computing, Computers and Op-erations Research, Journal of Manufacturing Systems,and Journal of Applied Sciences among others.

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