Promotional coordination mechanisms with demand dependent on price and sales efforts

ABSTRACT

This paper deals with a channel coordination problem of three-level supply chain comprising of manufacturer, distributor, and retailer while demand is sensitive with price and sales efforts of channel members. The upstream members of the chain offer service level assurance/agreement (SLA), in which fair price, quality assurance, rebate on price, timely delivery, customers’ feedback, are included to the respective downstream members at time of selling their products. In this model, six different types of rebate and sales effort mentioned in SLA are proposed to coordinate the channel. A mathematical model is formulated to quantify the profits of individual members and it is analyzed mathematically as well as numerically to identify the preference of each channel member among proposed contracts. Moreover, an analytical condition is derived to measure the hesitancy the retailer whether he/she would participate in sells effort sharing contract with the manufacturer or the distributor.

KEYWORDS:

• Supply-chain management
• coordination
• price and effort induced demand

Disclosure statement

No potential conflict of interest was reported by the authors.

Case 1

In this case, ranges of profits of the manufacturer and the retailer are $\left[\frac{Z}{Y}{\pi }_c^c,(1-\frac{Z+{\alpha }_2{\gamma }^2}{Y}(2+Z/Y)){\pi }_c^c\right]$ and $\left[\frac{Z(Z+{\alpha }_2{\gamma }^2)}{Y^2}{\pi }_c^c,\frac{(Z+{\alpha }_2{\gamma }^2)}{Y}{\pi }_c^c\right]$, respectively. The distributor profit in this case is $\frac{2(Z+{\alpha }_2{\gamma }^2)}{Y}{\pi }_c^c$. From the above two intervals, it can easily be observed that both the retailer and the manufacturer have equal opportunity to achieve maximum extra profit equal to $\frac{3{(Z+{\alpha }_2{\gamma }^2)}^2}{Y^2}{\pi }_c^c$ and the distributor’s profit is equal to $\frac{6{(Z+{\alpha }_2{\gamma }^2)}^2}{Y^2}{\pi }_c^c$, i.e. the distributor always prefer this situation compared to the manufacturer and the retailer.

Case 2

Since the upper limit of $x$ is one, the situation arises only when

(E.1) $\frac{4(Z+{\alpha }_2{\gamma }^2)}{Y}-\frac{{\alpha }_2{\gamma }^2}{Z}-\frac{Z}{Y}\ge \frac{{\alpha }_1{\eta }^2}{Z}$(E.1)

Here, the manufacturer’s profit lies in the range $\left[\left(\frac{4(Z+{\alpha }_2{\gamma }^2)}{Y}-\frac{{\alpha }_1{\eta }^2+{\alpha }_2{\gamma }^2}{Z}\right){\pi }_c^c,\left(1-\frac{Z+{\alpha }_2{\gamma }^2}{Y}(2+\frac{Z}{Y})\right){\pi }_c^c\right]$. Note that the manufacturer’s profit never attains to his/her non-cooperative profit as the difference of profits between minimum profit obtain in this mechanism and the non-cooperative profit is $\left[\frac{4(Z+{\alpha }_2{\gamma }^2)}{Y}-\frac{{\alpha }_1{\eta }^2+{\alpha }_2{\gamma }^2}{Z}-\frac{Z}{Y}\right]{\pi }_c^c\ge [\frac{{\alpha }_1{\eta }^2}{Z}+\frac{{\alpha }_2{\gamma }^2}{Z}-\frac{{\alpha }_1{\eta }^2+{\alpha }_2{\gamma }^2}{Z}]{\pi }_c^c=0$(by using (I.1)). We can easily observe that the profit of the manufacturer compared to non-cooperative scenario is equal to $\frac{3{(Z+{\alpha }_2{\gamma }^2)}^2}{Y^2}{\pi }_c^c$. Similarly, the retailer’s profit lies in the range $\left[\frac{Z(Z+{\alpha }_2{\gamma }^2)}{Y^2}{\pi }_c^c,\left(\frac{4{\alpha }_1{\alpha }_2\beta }{Z}-\frac{6(Z+{\alpha }_2{\gamma }^2)}{Y}\right){\pi }_c^c\right]$ and the retailer has opportunity to earn maximum extra profit equal to $\left(\frac{4{\alpha }_1{\alpha }_2\beta }{Z}-\frac{(Z+{\alpha }_2{\gamma }^2)}{Y}(6+\frac{Z}{Y})\right){\pi }_c^c$. The distributor’s profit remains constant and is equal to $\frac{6{(Z+{\alpha }_2{\gamma }^2)}^2}{Y^2}{\pi }_c^c$. From condition (I.1), we have $\frac{3Z+4{\alpha }_2{\gamma }^2}{Y}\ge \frac{4{\alpha }_1{\alpha }_2\beta -Z}{Z}$ $\Rightarrow$ $\frac{7(Z+4{\alpha }_2{\gamma }^2)}{Y}\ge \frac{4{\alpha }_1{\alpha }_2\beta }{Z}$. Now $\left[\frac{6{(Z+{\alpha }_2{\gamma }^2)}^2}{Y^2}-\frac{4{\alpha }_1{\alpha }_2\beta }{Z}+\frac{(Z+{\alpha }_2{\gamma }^2)}{Y}(6+\frac{Z}{Y})\right]{\pi }_c^c\ge \left[\frac{3{(Z+{\alpha }_2{\gamma }^2)}^2}{Y^2}\right]{\pi }_c^c\ge 0$, i.e. the distributor gains more compared to the retailer. Comparing profit of the manufacturer with the retailer, we have $\frac{3{(Z+{\alpha }_2{\gamma }^2)}^2}{Y^2}{\pi }_c^c-\left(\frac{4{\alpha }_1{\alpha }_2\beta }{Z}-\frac{(Z+{\alpha }_2{\gamma }^2)}{Y}(6+\frac{Z}{Y})\right){\pi }_c^c=\left(\frac{(3Z+4{\alpha }_2{\gamma }^2)}{Y}-\frac{{\alpha }_1{\eta }^2}{Z}-\frac{{\alpha }_2{\gamma }^2}{Z}\right){\pi }_c^c$. Now, condition (E.1) can be expressed as follows: $\frac{3Z+4{\alpha }_2{\gamma }^2)}{Y}\ge \frac{{\alpha }_1{\eta }^2}{Z}+\frac{{\alpha }_2{\gamma }^2}{Z}$ and, using this condition, we can conclude that the manufacturer maximum profit is always greater compared to the retailer.

Additional information

Notes on contributors

Subrata Saha

Subrata Saha is an Assistant Professor at the Institute of Engineering and Management, Salt Lake, Kolkata, India. Currently, he is doing Post Doc from Department of Materials and Production, Aalborg University, Aalborg East, Denmark.  He obtained his Master of Science in Applied Mathematics and Doctor of Philosophy from University of Kalyani, Kalyani, India. He has published several research papers in the area of inventory and production control in the journals like IJPE, IJPR, AOR, APJOR, CEJOR, Economic Modelling, IJSS, IJMOR, IJOR, IJMSEM, Journal of Water Resource and Protection, Sustainability, and Transportation Research Part E. His current research interests are in the field of production/inventory control, pricing strategies and supply chain management.

Nikunja Mohan Modak

Nikunja Mohan Modak received his Master’s and Doctoral degrees in Mathematics from the University of Kalyani, Kalyani, West Bengal, India. Currently, he is an assistant teacher of mathematics in Palpara Vidyamandir, Chakdaha, Nadia, West Bengal, India. He is a lifetime member of the Indian Science Congress and Operational Research Society of India (ORSI). He won the Prof. B.G. Raghabendra Memorial Award for the best paper at the 45th Annual Convention of ORSI in 2012. He has received the best reviewer award in 2014 from the International Journal of Management Science and Engineering Management. His research interests cover the areas of inventory management, game theory, supply chain management and bibliometric study. He has published more than 38 articles in international journals such as the International Journal of Production Economics, the European Journal of Operational Research, the Journal of Cleaner Production, the Annals of OR, the International Journal of Production Research, Group Decision and Negotiation, Applied Mathematics and Computation, the International Journal of Advanced Manufacturing Technology, the International Journal of Logistics, IJCIM, Control & Cybernetics, the International Journal of Operational Research, the International Journal of Management Science and Engineering Management, Transpotation Research Part A, Safety Science. He is currently interested in Sustainable Supply Chain Management, Bibliometrics and Applications in Business and Economics.

Shibaji Panda

Shibaji Panda received his MS in Applied Mathematics at Jadavpur University, Kolkata, and Ph.D on Inventory and Production Management at Kalyani University, Kalyani, West Bengal, India. Currently, he is an Associate Professor in the Department of Mathematics, Bengal Institute of Technology, Kolkata, India. His main area of research is production and inventory control, supply chain management, multi-objective optimization. He has published several papers in Computers & Industrial Engineering, Transportation Research Part E: Logistics and Transportation Review, Economic Modelling, Applied Mathematics and Computation, Central European Journal of Operational Research, Asia-Pacific Journal of Operational Research, International Journal of Systems Science, Cogent Business & Management,  Yugoslav Journal of Operational Research, International Journal of Management Science and Engineering Management, International Journal of Mathematics in Operational Research etc.

Shib Sankar Sana

Shib Sankar Sana is currently Principal of the Kishore  Bharati Bhagini Nivedita College under the University of Calcutta. He served at the Department of Mathematics as Associate Professor in Bhangar Mahavidyalaya from 2001 to 2018 and Department of Applied Mathematics, Birla Institute of Technology, Mesra, Ranchi as an Associate Lecturer during1999-2001.His main areas of research is the modelling of production planning and inventory control, Biomathematics and Soft Computing. He has published over 130 papers in Journal of cleaner Production, European Journal of Operational Research, International Journal of Production Economics, Computers and Mathematics with Application, Mathematical and Computer Modelling, Applied Mathematics and Computation, Journal of Modelling in Management, American Journal of Mathematical and Management Sciences, International Journal of Systems Science, Far East Journal of Applied Mathematics, Nonlinear Phenomena in Complex Systems, Advanced Modeling and Optimization, Vietnam Journal of Mathematics, Yugoslav Journal of Operational Research, International Journal of Operational Research, International Journal of Services Operations Management, International Journal of Modeling, Identification and Control, International Journal of Innovative Computing and Application, International Journal of Procurement Management, International Journal of Mathematics and Operations Research, IMA Journal of Management Mathematics, International Journal of Information and Decision Science, Indian Journal of Industrial and Applied Mathematics, International Journal of Management Science and Engineering Management, Applied Mathematical Modelling, Decision Support Systems, Service Science, Review in Aquaculture, International Journal of Production Research, Expert Systems with Application, Economic Modelling, Journal of Manufacturing Systems, Journal of Advanced Manufacturing Systems, Journal of Industrial & Management Optimization, Annals of Operations Research, International Journal of Logistics Research and Applications, Computers & Industrial Engineering,  Journal of Intelligent Manufacturing, International Journal of Applied and Computational Mathematics, RAIRO-Operations Research, CIRP Journal of Manufacturing Science and Technology, Control and Cybernetics, Cybernetics and Systems,  etc. He is a member of the editorial board of American Journal of Computational and Applied Mathematics, Universal Journal of Applied Mathematics, International Journal of Advanced Mathematics, American Journal of Industrial and Business Management. My Google Scholar address is : https://scholar.google.co.in/citations?user=WehwoJ0AAAAJ&hl=en