Pricing decisions in dual-channel supply chain including one monopolistic manufacturer and two duopolistic retailers: a game-theoretic approach

Hamed Jafari

Corresponding author

PhD Candidate, Department of Industrial and Systems Engineering, Isfahan University of Technology, Isfahan 84156-83111, Iran

Email address:

Tel: (+9831) 3391-1480; Fax: (+9831) 3391-5526

Seyed Reza Hejazi

Professor, Department of Industrial and Systems Engineering, Isfahan University of Technology, Isfahan 84156-83111, Iran

Email address:

Tel: (+9831) 3391-5506; Fax: (+9831) 3391-5526.

MortezaRasti-Barzoki

Assistant Professor, Department of Industrial and Systems Engineering, Isfahan University of Technology, Isfahan 84156-83111, Iran

Email address:

Tel: (+9831) 3391-1480; Fax: (+9831) 3391-5526.

Appendix A. Notations and proofs in the Bertrand model

Proof of Lemma 1.Combining equations (1) and (2) with (3), (4), and (5), the first and the second order partial deviations of to (i = 1, 2) can be given as:

/ (a1)
(a2)

The second-order partial deviation is negative. Thus, is concave in. □

Proof of Theorem 1.Assume that the wholesale price and the direct price are set by the manufacturer. From Lemma 1, is concave in (i = 1, 2). Thus, in order to maximize the constrained profit function with respect to (, the Lagrange function and the Lagrange multiplier are defined and the constraint given in relation (7) is relaxed. The Lagrange function and the KKT conditions related to retailer-i’s decision problem are:

/ (a3)
(a4)

(1)When, can be derived from. is straightforwardly given from equation (4). Therefore, regarding assumption (6) presented in subsection 2, retailer-i will not do business with the other two players in this case.

(2)When , is gevenfrom . Hence, Theorem 1 is proved by solving (i = 1, 2), simultaneously. □

Proof of Lemma 2.The first order partial deviations of with respect to and can be shown as follows:

/ (a5)
(a6)

Taking the second order partial deviations, Hessian matrix is:

/ (a7)

Regarding assumption (4), i.e.,, , , and the determinate of is . Therefore, the Hessian matrix is negative definite and is jointly concave with respect to and. □

Proof of Theorem 2. From Lemma 2, is jointly concave with respect to and. Thus, in order to maximize with respect to and that hold relations (6) and (7), the Lagrange function and the Lagrange multipliers , , , and are introduced, and the constraints and are also relaxed. The Lagrange function and the KKT conditions are defined as follows:

/ (a8)
(a9)

(1)When, the KKT conditions are reduced to, , and . Therefore, by setting and shown in equations (a5) and (a6) equal to zero, and solving them simultaneously, we have:

By substitutingand into equation (8), (i = 1, 2) is obtained as:

To meet the KKT conditions, the given variables must hold the relations and. After some algebraic manipulations, one can derive that the constraints , , , and are equivalent to , , , and , respectively, i.e., this solution is feasible if .

(2)When and, the KKT conditions are reduced to, , , , , and . Thus, by setting
and solving , we have:

(i = 1, 2) is easily given by substitutingand into equation (8):

Moreover, can be given by substitutingand into equation (a5):

After some algebraic manipulations, the constraints , , , and are equivalent to , , , and , respectively. As a result, this solution is feasible if and.

(3)When and, the KKT conditions are reduced to, , , , and . Hence, setting and solving , one can derive that:

By substitutingand into equation (8), (i = 1, 2) is:

can be shown by substitutingand into equation (a6):

After some algebraic manipulations, the constraints , , , and are equivalent to , , , and , respectively. Thus, this solution is feasible if and.

(4)When, , and , from , can be given. From equation (3), is derived. Hence, regarding assumption (6), the manufacturer will not do business with the retailers in this case.

In other cases, we have either or. Thus, from it is derived that either or , respectively. Using equation (4), one can conclude that or and, regarding assumption (6), at least one of the retailers will not do business with the other two players. This completes the proof of Theorem 2. □

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