Pricing, inventory and production policies in a supply chain of pharmacological products with rework process: a game theoretic approach

Abstract

In this paper, an economic production quantity model in a three levels supply chain including multiple non-competing suppliers, single manufacturer and multiple non-competing retailers for multiple products with rework process under integrated and non-integrated structures is developed. In this chain, suppliers ship raw material to the manufacturer and the manufacturer combines the fixed percentage of different kinds of raw material to produce the finished products and then delivers them to the retailers. The Stackelberg game is established among the members of the chain. Optimizing the total profit of the chain under both structures applying the optimal pricing, inventory and production policies is the purpose of the research. Eventually, a numerical example is presented for comparing the total profit of the integrated and non-integrated chains of proposed model and some sensitivity analyses are performed to study the effects of parameter changes on the decision variables and the chain profit.

Appendix: Proofing the concavity of TP(q k , S r i )

TP(q _k , S ^r _i ) is concave when ${\text{X}} . {\text{H}} . {\text{X}}^{T} < 0,$ where $X = \left[ {\begin{array}{*{20}c} {q_{k} } & {S_{i}^{r} } \\ \end{array} } \right],\quad X^{T} = \left[ {\begin{array}{*{20}c} {q_{k} } \\ {S_{i}^{r} } \\ \end{array} } \right],\quad H = \left[ {\begin{array}{*{20}c} {\frac{{\partial^{2} TP(q_{k} ,S_{i}^{r} )}}{{\partial q_{k}^{2} }}} & {\frac{{\partial^{2} TP(q_{k} ,S_{i}^{r} )}}{{\partial q_{k} \partial S_{i}^{r} }}} \\ {\frac{{\partial^{2} TP(q_{k} ,S_{i}^{r} )}}{{\partial S_{i}^{r} \partial q_{k} }}} & {\frac{{\partial^{2} TP(q_{k} ,S_{i}^{r} )}}{{\partial S_{i}^{r2} }}} \\ \end{array} } \right]$ and

$$ \begin{aligned} TP(q_{k} ,S_{i}^{r} ) = & \sum\limits_{k = 1}^{n} {\left( {S_{k}^{s} - C_{k}^{s} } \right)(a - bS_{i}^{r} ) - \left( {\frac{{h_{k}^{s} q_{k} }}{2} + \frac{{(a - bS_{i}^{r} )A_{k}^{s} }}{{q_{k} }}} \right)} \\ & + \sum\limits_{i = 1}^{m} {\left[ {\left( {\left( {S_{i}^{r} - C(P_{i} ) - \sum\limits_{k = 1}^{n} {S_{k}^{s} } } \right)(a - bS_{i}^{r} )} \right) - \left( {\frac{{h_{i}^{m} \sum\nolimits_{k = 1}^{n} {\lambda_{ik} q_{k} } }}{2}\left[ {1 - \frac{{(1 + \alpha_{i} + \alpha_{i}^{2} )(a - bS_{i}^{r} )}}{{P_{i} }}} \right] + \frac{{(a - bS_{i}^{r} )A_{i}^{m} }}{{\sum\nolimits_{k = 1}^{n} {\lambda_{ik} q_{k} } }}} \right)} \right]} \\ & + \sum\limits_{j = 1}^{b} {\sum\limits_{i = 1}^{m} {\left( {S_{i}^{r} - S_{i}^{r} } \right)(a - bS_{i}^{r} ) - \left( {\frac{{h_{ji}^{r} \rho_{ji} \sum\nolimits_{k = 1}^{n} {\lambda_{ik} q_{k} } }}{2}\left[ {1 - \frac{1}{{\rho_{ji} }}} \right] + \frac{{(a - bS_{i}^{r} )A_{ji}^{r} }}{{\rho_{ji} \sum\nolimits_{k = 1}^{n} {\lambda_{ik} q_{k} } }}} \right)} } \\ \end{aligned} $$

Then, we have:

$$ \left[ {\begin{array}{*{20}c} {q_{k} } & {S_{i}^{r} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\frac{{\partial^{2} TP(q_{k} ,S_{i}^{r} )}}{{\partial q_{k}^{2} }}} & {\frac{{\partial^{2} TP(q_{k} ,S_{i}^{r} )}}{{\partial q_{k} \partial S_{i}^{r} }}} \\ {\frac{{\partial^{2} TP(q_{k} ,S_{i}^{r} )}}{{\partial S_{i}^{r} \partial q_{k} }}} & {\frac{{\partial^{2} TP(q_{k} ,S_{i}^{r} )}}{{\partial S_{i}^{r2} }}} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {q_{k} } \\ {S_{i}^{r} } \\ \end{array} } \right] < 0 $$

(43)

where,

$$ \frac{{\partial^{2} TP(q_{k} ,S_{i}^{r} )}}{{\partial q_{k}^{2} }} = - \sum\limits_{i = 1}^{m} {\left[ {\frac{{2(a - bS_{i}^{r} )A_{k}^{s} }}{{\sum\nolimits_{k = 1}^{n} {\lambda_{ik} q_{k}^{3} } }} + \frac{{2(a - bS_{i}^{r} )A_{i}^{m} }}{{\sum\nolimits_{k = 1}^{n} {\lambda_{ik} q_{k}^{3} } }} + \sum\limits_{j = 1}^{b} {\frac{{2(a - bS_{i}^{r} )A_{ji}^{r} }}{{\rho_{ji} \sum\nolimits_{k = 1}^{n} {\lambda_{ik} q_{k}^{3} } }}} } \right]} < 0 $$

(44)

$$ \frac{{\partial^{2} TP(q_{k} ,S_{i}^{r} )}}{{\partial q_{k} \partial S_{i}^{r} }} = - \frac{{bA_{k}^{s} }}{{\sum\nolimits_{k = 1}^{n} {\lambda_{ik} q_{k}^{2} } }} - \frac{{bh_{i}^{m} \sum\nolimits_{k = 1}^{n} {\lambda_{ik} } }}{{2P_{i} }}(1 + \alpha_{i} + \alpha_{i}^{2} ) - \frac{{bA_{i}^{m} }}{{\sum\nolimits_{k = 1}^{n} {\lambda_{ik} q_{k}^{2} } }} - \sum\limits_{j = 1}^{b} {\frac{{bA_{ji}^{r} }}{{\rho_{ji} \sum\nolimits_{k = 1}^{n} {\lambda_{ik} q_{k}^{2} } }}} = g < 0 $$

(45)

$$ \frac{{\partial^{2} TP(q_{k} ,S_{i}^{r} )}}{{\partial S_{i}^{r} \partial q_{k} }} = - \frac{{bA_{k}^{s} }}{{\sum\limits_{k = 1}^{n} {\lambda_{ik} } q_{k}^{2} }} - \frac{{bh_{i}^{m} \sum\limits_{k = 1}^{n} {\lambda_{ik} } }}{{2P_{i} }}(1 + \alpha_{i} + \alpha_{i}^{2} ) - \frac{{bA_{i}^{m} }}{{\sum\limits_{k = 1}^{n} {\lambda_{ik} q_{k}^{2} } }} - \sum\limits_{j = 1}^{b} {\frac{{bA_{ji}^{r} }}{{\rho_{ji} \sum\limits_{k = 1}^{n} {\lambda_{ik} } q_{k}^{2} }}} = g < 0 $$

(46)

$$ \frac{{\partial^{2} TP(q_{k} ,S_{i}^{r} )}}{{\partial S_{i}^{r2} }} = - 2b < 0 $$

(47)

Therefore, we have:

$$ \left[ {\begin{array}{*{20}c} {q_{k} } & {S_{i}^{r} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}l} { - \sum\limits_{i = 1}^{m} {\left[ {\frac{{2(a - bS_{i}^{r} )A_{k}^{s} }}{{\sum\nolimits_{k = 1}^{n} {\lambda_{ik} q_{k}^{3} } }} + \frac{{2(a - bS_{i}^{r} )A_{i}^{m} }}{{\sum\nolimits_{k = 1}^{n} {\lambda_{ik} q_{k}^{3} } }} + \sum\limits_{j = 1}^{b} {\frac{{2(a - bS_{i}^{r} )A_{ji}^{r} }}{{\rho_{ji} \sum\nolimits_{k = 1}^{n} {\lambda_{ik} q_{k}^{3} } }}} } \right]} } \hfill & g \hfill \\ g \hfill & { - 2b} \hfill \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {q_{k} } \\ {S_{i}^{r} } \\ \end{array} } \right] $$

$$ = \left[ {\begin{array}{*{20}c} { - \sum\limits_{i = 1}^{m} {\left[ {\frac{{2(a - bS_{i}^{r} )A_{k}^{s} }}{{\sum\nolimits_{k = 1}^{n} {\lambda_{ik} q_{k}^{2} } }} + \frac{{2(a - bS_{i}^{r} )A_{i}^{m} }}{{\sum\nolimits_{k = 1}^{n} {\lambda_{ik} q_{k}^{2} } }} + \sum\limits_{j = 1}^{b} {\frac{{2(a - bS_{i}^{r} )A_{ji}^{r} }}{{\rho_{ji} \sum\nolimits_{k = 1}^{n} {\lambda_{ik} q_{k}^{2} } }}} } \right]} } & \begin{aligned} - \frac{{bA_{k}^{s} }}{{\sum\nolimits_{k = 1}^{n} {\lambda_{ik} q_{k} } }} - \frac{{bh_{i}^{m} \sum\nolimits_{k = 1}^{n} {\lambda_{ik} q_{k} } }}{{2P_{i} }}(1 + \alpha_{i} + \alpha_{i}^{2} ) \hfill \\ - \frac{{bA_{i}^{m} }}{{\sum\nolimits_{k = 1}^{n} {\lambda_{ik} q_{k} } }} - \frac{{b\sum\nolimits_{j = 1}^{b} {A_{ji}^{r} } }}{{\sum\nolimits_{j = 1}^{b} {\rho_{ji} } \sum\nolimits_{k = 1}^{n} {\lambda_{ik} q_{k} } }} \hfill \\ \end{aligned} \\ \begin{aligned} - \frac{{bS_{i}^{r} A_{k}^{s} }}{{\sum\nolimits_{k = 1}^{n} {\lambda_{ik} q_{k}^{2} } }} - \frac{{bh_{i}^{m} S_{i}^{r} \sum\nolimits_{k = 1}^{n} {\lambda_{ik} } }}{{2P_{i} }}(1 + \alpha_{i} + \alpha_{i}^{2} ) \hfill \\ - \frac{{bS_{i}^{r} A_{i}^{m} }}{{\sum\nolimits_{k = 1}^{n} {\lambda_{ik} q_{k}^{2} } }} - \frac{{bS_{i}^{r} \sum\nolimits_{j = 1}^{b} {A_{ji}^{r} } }}{{\sum\nolimits_{j = 1}^{b} {\rho_{ji} } \sum\nolimits_{k = 1}^{n} {\lambda_{ik} q_{k}^{2} } }} \hfill \\ \end{aligned} & { - 2bS_{i}^{r} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {q_{k}^{{}} } \\ {S_{i}^{r} } \\ \end{array} } \right] $$

$$ = - \frac{{2aA_{k}^{s} }}{{\sum\nolimits_{k = 1}^{n} {\lambda_{ik} q_{k} } }} - \frac{{2aA_{i}^{m} }}{{\sum\nolimits_{k = 1}^{n} {\lambda_{ik} q_{k} } }} - \frac{{2a\sum\nolimits_{j = 1}^{b} {A_{ji}^{r} } }}{{\sum\nolimits_{k = 1}^{n} {\rho_{ji} } \sum\nolimits_{k = 1}^{n} {\lambda_{ik} q_{k} } }} - \frac{{2bS_{i}^{r} h_{i}^{m} \sum\nolimits_{k = 1}^{n} {\lambda_{ik} q_{k} } }}{{2P_{i} }}(1 + \alpha_{i} + \alpha_{i}^{2} ) - 2bS_{i}^{r} < 0 $$

(48)

Furthermore, the concavityFurthermore, the concavity is obvious.