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A surcharge pricing policy for supply chain coordination under the just-in-time (JIT) environment in the presence of backordering

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Abstract

In the just-in-time (JIT) manufacturing system, the vendors ship in small batch sizes. Hence the buyers are at an advantageous position as they can reduce their inventory holding costs. The practice of shipping in smaller batch sizes often results in the vendors incurring high costs as the number of production setups and the number of shipments increases for the vendors. The vendors could only switch to JIT mode when there is an assurance from the buyer of providing some compensation for the increased costs. This compensation from the buyer to the vendors is called ‘surcharge price’. In our work, we have studied the surcharge pricing mechanism in the presence of backorder when there is a powerful vendor and have shown that the it can coordinate a decentralized supply chain. Our study incorporates both the cases of the buyer holding full information about the vendors’ costs as well as the case of information asymmetry. In the case of full information, we have shown that the optimal order quantity in the presence of surcharge pricing is less than that of the vendor dominated scenario. At the same time, the optimal order quantity is higher than that in the buyer dominated scenario. Further, the total supply chain costs are also minimized when a surcharge pricing contract is offered in comparison to both the buyer and the vendor dominated scenarios. For the information asymmetry case, we have framed the optimal set of screening contracts and have shown that an optimal surcharge pricing contract to the vendor is free from the probability distribution of the vendor type.

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Notes

  1. Generalized Hazard or failure rate for a non-negative random variable X with probability density function \(\phi \) and cumulative distribution function \(\Phi \) is defined as \(g\left(\xi \right)=\xi \phi \left(\xi \right)/\left(1-\Phi \left(\xi \right)\right)\). The random variable X is said to have an increased generalized hazard rate and \(\Phi \) to have an Increased Generalized Failure Rate (IGFR) distribution if \(g\left(\xi \right)\) is weakly increasing for all \(\xi \) with \(\Phi \left(\xi \right)<1\) (Lariviere 2006).

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  1. XLRI Xavier School of Business, Jamshedpur, 831001, Jharkhand, India

    Abhishek Chakraborty

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Appendices

Appendix A

Proof of Theorem 1

The optimization problem is \(Min \, TCB\left( {Q,Q_{1} } \right) = \frac{{DC_{O} }}{Q} + \frac{{h_{1} Q_{1}^{2} }}{2Q} + \frac{{h_{2} \left( {Q - Q_{1} } \right)^{2} }}{2Q} + \frac{D}{Q}C_{S} - C_{S}\).

First-order conditions for stationarity give

$$ \frac{{\partial TCB\left( {Q,Q_{1} } \right)}}{\partial Q} = - \frac{{DC_{O} }}{{Q^{2} }} - \frac{{h_{1} Q_{1}^{2} }}{{2Q^{2} }} + \frac{{4h_{2} \left( {Q - Q_{1} } \right)Q - 2h_{2} \left( {Q - Q_{1} } \right)^{2} }}{{4Q^{2} }} - \frac{D}{{Q^{2} }}C_{S} = 0 $$
$$ \frac{{\partial TCB\left( {Q,Q_{1} } \right)}}{{\partial Q_{1} }} = \frac{{2h_{1} Q_{1} }}{2Q} - \frac{{2h_{2} \left( {Q - Q_{1} } \right)}}{2Q} = 0 $$

Solving the above system of equations we get \(Q_{1} = \sqrt {\frac{{2Dh_{2} \left( {C_{O} + C_{S} } \right)}}{{h_{1} \left( {h_{1} + h_{2} } \right)}}}\) and \(Q = \sqrt {\frac{{2D\left( {C_{O} + C_{S} } \right)\left( {h_{1} + h_{2} } \right)}}{{h_{1} h_{2} }}}\). Thus, implying \(Q_{2} = \sqrt {\frac{{2D\left( {C_{O} + C_{S} } \right)h_{1} }}{{h_{2} \left( {h_{1} + h_{2} } \right)}}}\).\(\square \)

Proof of Lemma 1

The Hessian matrix for the objective function is given as:

$$ H = \left[ {\begin{array}{*{20}c} {\frac{{\partial^{2} TC}}{{\partial^{2} Q_{1} }}} & {\frac{{\partial^{2} TC}}{{\partial Q\partial Q_{1} }}} \\ {\frac{{\partial^{2} TC}}{{\partial Q_{1} \partial Q}}} & {\frac{{\partial^{2} TC}}{{\partial^{2} Q}}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\frac{{h_{1} + h_{2} }}{Q}} & { - \frac{{Q_{1} \left( {h_{1} + h_{2} } \right)}}{{Q^{2} }}} \\ { - \frac{{Q_{1} \left( {h_{1} + h_{2} } \right)}}{{Q^{2} }}} & {\frac{{Q_{1}^{2} \left( {h_{1} + h_{2} } \right) + 2D\left( {C_{S} + C_{O} } \right)}}{{Q^{3} }}} \\ \end{array} } \right] $$
$$ \det H = \frac{{2D\left( {C_{S} + C_{O} } \right)\left( {h_{1} + h_{2} } \right)}}{{Q^{4} }} > 0 \;{\text{and}}\;\left. {\frac{{\partial^{2} TC}}{{\partial^{2} Q_{1} }}} \right|_{{Q = Q^{*} ,Q_{1} = Q_{1}^{*} }} = \sqrt {\frac{{\left( {h_{1} + h_{2} } \right)h_{1} h_{2} }}{{2D\left( {C_{O} + C_{S} } \right)}}} > 0 $$

This proves the convexity of the objective function.\(\square \)

Proof of Proposition 1

The optimal order quantity under surcharge pricing is \(Q_{SP}^{*} = \sqrt {\frac{{2D\left( {C_{O} + C_{S} } \right)\left( {h_{1} + h_{2} } \right)}}{{h_{1} h_{2} }}}\).

The optimal order quantity under Vendor dominated scenario is Q* = D.

The optimal order quantity under Buyer dominated scenario is \(Q_{BD}^{*} = \sqrt {\frac{{2DC_{O} \left( {h_{1} + h_{2} } \right)}}{{h_{1} h_{2} }}}\).

Clearly, \(Q_{SP}^{*} = \sqrt {\frac{{2D\left( {C_{O} + C_{S} } \right)\left( {h_{1} + h_{2} } \right)}}{{h_{1} h_{2} }}} > Q_{BD}^{*} = \sqrt {\frac{{2DC_{O} \left( {h_{1} + h_{2} } \right)}}{{h_{1} h_{2} }}}\).\(\square \)

Proof of Proposition 2

The total supply chain cost under vendor dominated scenario is TSSCISO = \(C_{S} + C_{O} + \frac{{Dh_{1} h_{2} }}{{2\left( {h_{1} + h_{2} } \right)}}\).

The total supply chain cost under surcharge pricing \(TSCC_{SP} = \sqrt {\frac{{2Dh_{1} h_{2} \left( {Co + C_{S} } \right)}}{{\left( {h_{1} + h_{2} } \right)}}}\)

$$ \left( {TSCC_{ISO} } \right)^{2} - \left( {TSCC_{SP} } \right)^{2} = \left( {C_{S} + C_{O} - \frac{{Dh_{1} h_{2} }}{{2\left( {h_{1} + h_{2} } \right)}}} \right)^{2} \ge 0 $$

Hence we can conclude that the total supply chain cost under surcharge pricing is less than the total supply chain cost under vendor dominated scenario.\(\square \)

Appendix B

Proof of Theorem 2

The optimization problem is

$$ Min \, TCB\left( {Q,Q_{1} } \right) = \frac{{DC_{O} }}{Q} + \frac{{h_{1} Q_{1}^{2} }}{2Q} + \frac{{h_{2} \left( {Q - Q_{1} } \right)_{{}}^{2} }}{2Q} + \frac{D}{{Q_{VEN} }}C_{S} + \frac{{Q_{VEN} }}{2}h_{S} - \frac{Q}{2}h_{S} - C_{S} $$

First-order conditions for stationarity give.

\(\frac{\partial TC}{{\partial Q}} = \frac{{\left( {Q - Q_{1} } \right)h_{2} }}{Q} - \frac{{\left( {Q - Q_{1} } \right)^{2} h_{2} }}{{2Q^{2} }} - \frac{{Q_{1}^{2} h_{1} }}{{2Q^{2} }} - \frac{{DC_{O} }}{{Q^{2} }} - \frac{{h_{s} }}{2} = 0\) and \(\frac{\partial TC}{{\partial Q_{1} }} = \frac{{Q_{1} h_{1} }}{Q} - \frac{{\left( {Q - Q_{1} } \right)h_{2} }}{Q} = 0\).

Solving them, we get \(Q = \sqrt {\frac{{2DC_{O} \left( {h_{1} + h_{2} } \right)}}{{h_{1} h_{2} - \left( {h_{1} + h_{2} } \right)h_{s} }}}\) and \(Q_{2}^{{}} = Q - Q_{1} = h_{1} \sqrt {\frac{{2DC_{O} }}{{\left( {h_{1} h_{2} - \left( {h_{1} + h_{2} } \right)h_{s} } \right)\left( {h_{1} + h_{2} } \right)}}}\).

Proof of Lemma 2

The Hessian matrix for the objective function is given as:

$$ H = \left[ {\begin{array}{*{20}c} {\frac{{\partial^{2} TC}}{{\partial^{2} Q_{1} }}} & {\frac{{\partial^{2} TC}}{{\partial Q\partial Q_{1} }}} \\ {\frac{{\partial^{2} TC}}{{\partial Q_{1} \partial Q}}} & {\frac{{\partial^{2} TC}}{{\partial^{2} Q}}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\frac{{h_{1} + h_{2} }}{Q}} & { - \frac{{Q_{1} \left( {h_{1} + h_{2} } \right)}}{{Q^{2} }}} \\ { - \frac{{Q_{1} \left( {h_{1} + h_{2} } \right)}}{{Q^{2} }}} & {\frac{{Q_{1}^{2} \left( {h_{1} + h_{2} } \right) + 2DC_{O} }}{{Q^{3} }}} \\ \end{array} } \right] $$
$$ \begin{aligned} \det H& = \frac{{2DC_{O} \left( {h_{1} + h_{2} } \right)^{2} }}{{Q^{4} }} > 0\,{\text{and}}\;\left. {\frac{{\partial^{2} TC}}{{\partial^{2} Q_{1} }}} \right|_{{Q = Q^{*} ,Q_{1} = Q_{1}^{*} }}\\ & = \sqrt {\frac{{\left( {h_{1} + h_{2} } \right)\left( {h_{1} h_{2} - \left( {h_{1} + h_{2} } \right)h_{s} } \right)}}{{2D\left( {C_{O} + C_{S} } \right)}}} > 0 \end{aligned} $$

This proves the convexity of the objective function.

Solution to the adverse selection problem

The Lagrangian function for the minimization problem can be written as:

\(\begin{gathered} L = \left[ {\frac{{DC_{O} }}{{Q^{H} }} + \frac{{h_{1} \left( {Q_{1}^{H} } \right)^{2} }}{{2Q^{H} }} + \frac{{h_{2} \left( {Q^{H} - Q_{1}^{H} } \right)_{{}}^{2} }}{{2Q^{H} }} + x^{H} D} \right]\lambda + \left[ {\frac{{DC_{O} }}{{Q^{L} }} + \frac{{h_{1} \left( {Q_{1}^{L} } \right)_{{}}^{2} }}{{2Q^{L} }} + \frac{{h_{2} \left( {Q^{L} - Q_{1}^{L} } \right)_{{}}^{2} }}{{2Q^{L} }} + x^{L} D} \right]\left( {1 - \lambda } \right) \hfill \\ + \mu_{1} \left( {\frac{D}{{Q^{H} }}C_{S}^{H} - C_{S}^{H} - x^{H} D} \right) + \mu_{2} \left( {\frac{D}{{Q^{L} }}C_{S}^{L} - C_{S}^{L} - x^{L} D} \right) + \mu_{3} \left[ {x^{H} D - \frac{D}{{Q^{H} }}C_{S}^{H} - x^{L} D + \frac{D}{{Q^{L} }}C_{S}^{H} } \right] \hfill \\ + \mu_{4} \left[ {x^{L} D - \frac{D}{{Q^{L} }}C_{S}^{L} - x^{H} D + \frac{D}{{Q^{H} }}C_{S}^{L} } \right] \hfill \\ \end{gathered}\) The Karush–Kuhn–Tucker (KKT) conditions can be written as:

Stationarity Conditions:

$$ \begin{aligned} \frac{\partial L}{{\partial Q^{H} }} =& \left( {\frac{{\left( {Q^{H} - Q_{1}^{H} } \right)h_{2} }}{{Q^{H} }} - \frac{{\left( {Q^{H} - Q_{1}^{H} } \right)^{2} h_{2} }}{{2\left( {Q^{H} } \right)^{2} }} - \frac{{\left( {Q_{1}^{H} } \right)^{2} h_{1} }}{{2\left( {Q^{H} } \right)^{2} }} - \frac{{DC_{O} }}{{\left( {Q^{H} } \right)^{2} }}} \right)\lambda\\ & - \frac{{C_{S}^{L} D\mu_{4} }}{{\left( {Q^{H} } \right)^{2} }} + \frac{{C_{S}^{H} D\mu_{3} }}{{\left( {Q^{H} } \right)^{2} }} - \frac{{C_{S}^{H} D\mu_{1} }}{{\left( {Q^{H} } \right)^{2} }} = 0 \end{aligned} $$

\(\frac{\partial L}{{\partial Q^{L} }} = \left( {\frac{{\left( {Q^{L} - Q_{1}^{L} } \right)h_{2} }}{{Q^{L} }} - \frac{{\left( {Q^{L} - Q_{1}^{L} } \right)^{2} h_{2} }}{{2\left( {Q^{L} } \right)^{2} }} - \frac{{\left( {Q_{1}^{L} } \right)^{2} h_{1} }}{{2\left( {Q^{L} } \right)^{2} }} - \frac{{C_{O} D}}{{\left( {Q^{L} } \right)^{2} }}} \right)\left( {1 - \lambda } \right) + \frac{{C_{S}^{L} D\mu_{4} }}{{\left( {Q^{L} } \right)^{2} }} - \frac{{C_{S}^{H} D\mu_{3} }}{{\left( {Q^{L} } \right)^{2} }} - \frac{{C_{S}^{L} D\mu_{2} }}{{\left( {Q^{L} } \right)^{2} }} = 0\)\(\frac{\partial L}{{\partial Q_{1}^{H} }} = \left( {\frac{{Q_{1}^{H} h_{1} }}{{Q^{H} }} - \frac{{\left( {Q^{H} - Q_{1}^{H} } \right)h_{2} }}{{Q^{H} }}} \right)\lambda = 0\),\(\frac{\partial L}{{\partial Q_{1}^{L} }} = \left( {\frac{{Q_{1}^{L} h_{1} }}{{Q^{L} }} - \frac{{\left( {Q^{L} - Q_{1}^{L} } \right)h_{2} }}{{Q^{L} }}} \right)\left( {1 - \lambda } \right) = 0\).

\(\frac{\partial L}{{\partial x^{H} }} = D\left[ {\lambda - \mu_{4} + \mu_{3} - \mu_{1} } \right] = 0\), \(\frac{\partial L}{{\partial x^{L} }} = D\left[ {\left( {1 - \lambda } \right) + \mu_{4} - \mu_{3} - \mu_{2} } \right] = 0\).

Primal Feasibility Conditions: Q.H., Q.L., QH1, QL1, xH, xL ≥ 0.

Dual Feasibility Conditions: μ1, μ2, μ3, μ4 ≥ 0.

Complementary Slackness Conditions:

$$ \begin{aligned} \mu_{1} \left( {\frac{D}{{Q^{H} }}C_{S}^{H} - C_{S}^{H} - x^{H} D} \right) & = 0,\,\mu_{2} \left( {\frac{D}{{Q^{L} }}C_{S}^{L} - C_{S}^{L} - x^{L} D} \right) = 0 \\ \mu_{3} \left[ {x^{H} D - \frac{D}{{Q^{H} }}C_{S}^{H} - x^{L} D + \frac{D}{{Q^{L} }}C_{S}^{H} } \right] & = 0,\,\mu_{4} \left[ {x^{L} D - \frac{D}{{Q^{L} }}C_{S}^{L} - x^{H} D + \frac{D}{{Q^{H} }}C_{S}^{L} } \right] = 0 \\ \end{aligned} $$

The equations \(\lambda - \mu_{4} + \mu_{3} - \mu_{1} = 0\) and \(\left( {1 - \lambda } \right) + \mu_{4} - \mu_{3} - \mu_{2} = 0\) imply \(\mu_{1} + \mu_{2} = 1\).

Further, for λ ≠ 0 or 1 we also have

$$ \begin{aligned} & \frac{{Q_{1}^{H} h_{1} }}{{Q^{H} }} - \frac{{\left( {Q^{H} - Q_{1}^{H} } \right)h_{2} }}{{Q^{H} }} = 0 \;{\text{and}}\;\frac{{Q_{1}^{L} h_{1} }}{{Q^{L} }} - \frac{{\left( {Q^{L} - Q_{1}^{L} } \right)h_{2} }}{{Q^{L} }} = 0\, {\text{implying}}\;\\ &\frac{{Q_{1}^{H} }}{{Q^{H} }} = \frac{{h_{2} }}{{h_{1} + h_{2} }}\; {\text{and}}\;\frac{{Q_{1}^{L} }}{{Q^{L} }} = \frac{{h_{2} }}{{h_{1} + h_{2} }} \end{aligned} $$

The Lagrangian multipliers μi’s can either be equal to 0 or greater than 0 thus yielding 16 possible combinations which are given in the following table (Table 10):

Table 10 Various Combinations of Lagrangian Multipliers
Full size table

Out of these 16 combinations, some combinations will be infeasible. For e.g., the relation \(\mu_{1} + \mu_{2} = 1\) would imply that any combination where μ1 = 0 = μ2 is present is ruled out. Similarly for Case I, i.e., when all μi > 0, the Complementary Slackness Conditions will imply \(\begin{aligned} & \frac{D}{{Q^{H} }}C_{S}^{H} - C_{S}^{H} - x^{H} D = 0 \Rightarrow x^{H} = C_{S}^{H} \left( {\frac{1}{{Q^{H} }} - \frac{1}{D}} \right) \\ & \frac{D}{{Q^{L} }}C_{S}^{L} - C_{S}^{L} - x^{L} D = 0 \Rightarrow x^{L} = C_{S}^{L} \left( {\frac{1}{{Q^{L} }} - \frac{1}{D}} \right) \\ & x^{H} D - \frac{D}{{Q^{H} }}C_{S}^{H} - x^{L} D + \frac{D}{{Q^{L} }}C_{S}^{H} = 0\,{\text{and}}\;x^{H} D - \frac{D}{{Q^{H} }}C_{S}^{H} - x^{L} D + \frac{D}{{Q^{L} }}C_{S}^{H} = 0 \\ \end{aligned}\).

Substituting the values for \(x^{H} = C_{S}^{H} \left( {\frac{1}{{Q^{H} }} - \frac{1}{D}} \right)\) and \(x^{L} = C_{S}^{L} \left( {\frac{1}{{Q^{L} }} - \frac{1}{D}} \right)\) above, we get

$$ C_{S}^{H} \left( {\frac{1}{{Q^{H} }} - \frac{1}{D}} \right)D - \frac{D}{{Q^{H} }}C_{S}^{H} - C_{S}^{L} \left( {\frac{1}{{Q^{L} }} - \frac{1}{D}} \right)D + \frac{D}{{Q^{L} }}C_{S}^{H} = 0 $$
$$ - C_{S}^{H} - C_{S}^{L} \frac{D}{{Q^{L} }} + C_{S}^{L} + \frac{D}{{Q^{L} }}C_{S}^{H} = 0 \Rightarrow \left( {\frac{D}{{Q^{L} }} - 1} \right)\left( {C_{S}^{H} - C_{S}^{L} } \right) = 0 \Rightarrow C_{S}^{H} = C_{S}^{L} {\text{ for }}D \ne Q^{L} $$

which violates the initial assumption made about the different states of the vendor’s setup cost. Hence this case becomes invalid. Under similar arguments, the Cases I, II, III, VII, VIII, IX, X, XI, XII, XIII, and XVI will become infeasible. Hence we are only left with the cases IV, V, VI, XIV, and XV to analyze further.

Case IV μ1 ≠ 0, μ2 ≠ 0, μ3 = 0 = μ4.

From the Complementary Slackness Conditions, we get.

\(\frac{D}{{Q^{H} }}C_{S}^{H} - C_{S}^{H} - x^{H} D = 0 \Rightarrow x^{H} = C_{S}^{H} \left( {\frac{1}{{Q^{H} }} - \frac{1}{D}} \right)\) and \(\frac{D}{{Q^{L} }}C_{S}^{L} - C_{S}^{L} - x^{L} D = 0 \Rightarrow x^{L} = C_{S}^{L} \left( {\frac{1}{{Q^{L} }} - \frac{1}{D}} \right)\).

Since μ3 = 0 = μ4, we must have μ1 = λ and μ2 = 1- λ.

The Stationarity Conditions \(\frac{{Q_{1}^{H} }}{{Q^{H} }} = \frac{{h_{2} }}{{h_{1} + h_{2} }}\) and \(\frac{{Q_{1}^{L} }}{{Q^{L} }} = \frac{{h_{2} }}{{h_{1} + h_{2} }}\) together imply

$$ \begin{aligned} &\frac{{2\left( {Q^{H} - Q_{1}^{H} } \right)h_{2} }}{{Q^{H} }} - \frac{{\left( {Q^{H} - Q_{1}^{H} } \right)^{2} h_{2} }}{{\left( {Q^{H} } \right)^{2} }} - \frac{{\left( {Q_{1}^{H} } \right)^{2} h_{1} }}{{\left( {Q^{H} } \right)^{2} }} = \frac{{2DC_{O} }}{{\left( {Q^{H} } \right)^{2} }} + \frac{{2DC_{S}^{H} }}{{\left( {Q^{H} } \right)^{2} }} \\ &\quad\Rightarrow Q^{H} = \sqrt {\frac{{2D\left( {C_{O} + C_{S}^{H} } \right)\left( {h_{1} + h_{2} } \right)}}{{h_{1} h_{2} }}} \end{aligned} $$

Similarly, \(Q^{L} = \sqrt {\frac{{2D\left( {C_{O} + C_{S}^{L} } \right)\left( {h_{1} + h_{2} } \right)}}{{h_{1} h_{2} }}}\).

We can also find \(Q_{1}^{H} = \sqrt {\frac{{2D\left( {C_{O} + C_{S}^{H} } \right)h_{2} }}{{h_{1} \left( {h_{1} + h_{2} } \right)}}}\) and \(Q_{1}^{L} = \sqrt {\frac{{2D\left( {C_{O} + C_{S}^{L} } \right)h_{2} }}{{h_{1} \left( {h_{1} + h_{2} } \right)}}}\).

Case V μ1 ≠ 0, μ2 = 0, μ3 ≠ 0, μ4 ≠ 0.

From the Complementary Slackness Conditions, we get

$$ \frac{D}{{Q^{H} }}C_{S}^{H} - C_{S}^{H} - x^{H} D = 0 \Rightarrow x^{H} = C_{S}^{H} \left( {\frac{1}{{Q^{H} }} - \frac{1}{D}} \right) $$

\(x^{H} D - \frac{D}{{Q^{H} }}C_{S}^{H} - x^{L} D + \frac{D}{{Q^{L} }}C_{S}^{H} = 0\) and \(x^{L} D - \frac{D}{{Q^{L} }}C_{S}^{L} - x^{H} D + \frac{D}{{Q^{H} }}C_{S}^{L} = 0\).

Substituting the value of \(x^{H} = C_{S}^{H} \left( {\frac{1}{{Q^{H} }} - \frac{1}{D}} \right)\) in \(x^{H} D - \frac{D}{{Q^{H} }}C_{S}^{H} - x^{L} D + \frac{D}{{Q^{L} }}C_{S}^{H} = 0\) and \(x^{L} D - \frac{D}{{Q^{L} }}C_{S}^{L} - x^{H} D + \frac{D}{{Q^{H} }}C_{S}^{L} = 0\) respectively we get.

\(C_{S}^{H} \left( {\frac{D}{{Q^{H} }} - 1} \right) - \frac{D}{{Q^{H} }}C_{S}^{H} - x^{L} D + \frac{D}{{Q^{L} }}C_{S}^{H} = 0\) and \(x^{L} D - \frac{D}{{Q^{L} }}C_{S}^{L} - C_{S}^{H} \left( {\frac{D}{{Q^{H} }} - 1} \right) + \frac{D}{{Q^{H} }}C_{S}^{L} = 0\).

Adding these two equations we get \(\frac{1}{{Q^{H} }}C_{S}^{L} - \frac{1}{{Q^{H} }}C_{S}^{H} = \frac{1}{{Q^{L} }}C_{S}^{L} - \frac{1}{{Q^{L} }}C_{S}^{H} \Rightarrow Q^{H} = Q^{L} {\text{ for }}C_{S}^{L} \ne C_{S}^{H}\).

From the condition, \(x^{H} D - \frac{D}{{Q^{H} }}C_{S}^{H} - x^{L} D + \frac{D}{{Q^{L} }}C_{S}^{H} = 0\) we can also write \(x^{H} = x^{L}\). Also, we have \(Q_{1}^{H} = Q_{1}^{L}\).

Now, for μ2 = 0, we get μ1 = 1. Thus, from the Stationarity conditions we have

$$ \begin{aligned}& \left( {\frac{{\left( {Q^{H} - Q_{1}^{H} } \right)h_{2} }}{{Q^{H} }} - \frac{{\left( {Q^{H} - Q_{1}^{H} } \right)^{2} h_{2} }}{{2\left( {Q^{H} } \right)^{2} }} - \frac{{\left( {Q_{1}^{H} } \right)^{2} h_{1} }}{{2\left( {Q^{H} } \right)^{2} }} - \frac{{DC_{O} }}{{\left( {Q^{H} } \right)^{2} }}} \right)\lambda \\ &- \frac{{C_{S}^{L} D\mu_{4} }}{{\left( {Q^{H} } \right)^{2} }} + \frac{{C_{S}^{H} D\mu_{3} }}{{\left( {Q^{H} } \right)^{2} }} - \frac{{C_{S}^{H} D}}{{\left( {Q^{H} } \right)^{2} }} = 0 \end{aligned} $$
$$ \left( {\frac{{\left( {Q^{L} - Q_{1}^{L} } \right)h_{2} }}{{Q^{L} }} - \frac{{\left( {Q^{L} - Q_{1}^{L} } \right)^{2} h_{2} }}{{2\left( {Q^{L} } \right)^{2} }} - \frac{{\left( {Q_{1}^{L} } \right)^{2} h_{1} }}{{2\left( {Q^{L} } \right)^{2} }} - \frac{{C_{O} D}}{{\left( {Q^{L} } \right)^{2} }}} \right)\left( {1 - \lambda } \right) + \frac{{C_{S}^{L} D\mu_{4} }}{{\left( {Q^{L} } \right)^{2} }} - \frac{{C_{S}^{H} D\mu_{3} }}{{\left( {Q^{L} } \right)^{2} }} = 0 $$

Adding these two equations and substituting QL = QH and QL1 = QH1 we get \(Q^{H} = \sqrt {\frac{{2D\left( {C_{S}^{H} + C_{O} } \right)\left( {h_{1} + h_{2} } \right)}}{{h_{1} h_{2} }}} = Q^{L}\).

Similarly, we can find \(Q_{1}^{H} = \sqrt {\frac{{2D\left( {C_{O} + C_{S}^{H} } \right)h_{2} }}{{h_{1} \left( {h_{1} + h_{2} } \right)}}}\) and \(Q_{1}^{L} = \sqrt {\frac{{2D\left( {C_{O} + C_{S}^{H} } \right)h_{2} }}{{h_{1} \left( {h_{1} + h_{2} } \right)}}}\).

Case VI μ1 = 0, μ2 ≠ 0, μ3 ≠ 0, μ4 ≠ 0.

From the Complementary Slackness Conditions, we get

$$ \frac{D}{{Q^{L} }}C_{S}^{L} - C_{S}^{L} - x^{L} D = 0 \Rightarrow x^{L} = C_{S}^{L} \left( {\frac{1}{{Q^{L} }} - \frac{1}{D}} \right) $$

\(x^{H} D - \frac{D}{{Q^{H} }}C_{S}^{H} - x^{L} D + \frac{D}{{Q^{L} }}C_{S}^{H} = 0\) and \(x^{H} D - \frac{D}{{Q^{H} }}C_{S}^{H} - x^{L} D + \frac{D}{{Q^{L} }}C_{S}^{H} = 0\).

Substituting the value of \(x^{L} = C_{S}^{L} \left( {\frac{1}{{Q^{L} }} - \frac{1}{D}} \right)\) in \(x^{H} D - \frac{D}{{Q^{H} }}C_{S}^{H} - x^{L} D + \frac{D}{{Q^{L} }}C_{S}^{H} = 0\) and \(x^{L} D - \frac{D}{{Q^{L} }}C_{S}^{L} - x^{H} D + \frac{D}{{Q^{H} }}C_{S}^{L} = 0\) respectively we get.

\(x^{H} D - \frac{D}{{Q^{H} }}C_{S}^{H} - C_{S}^{L} \left( {\frac{D}{{Q^{L} }} - 1} \right) + \frac{D}{{Q^{L} }}C_{S}^{H} = 0\) and \(C_{S}^{L} \left( {\frac{D}{{Q^{L} }} - 1} \right) - \frac{D}{{Q^{L} }}C_{S}^{L} - x^{H} D + \frac{D}{{Q^{H} }}C_{S}^{L} = 0\).

Adding the above two equations we get

$$ - \frac{D}{{Q^{H} }}C_{S}^{H} + \frac{D}{{Q^{L} }}C_{S}^{H} - \frac{D}{{Q^{L} }}C_{S}^{L} + \frac{D}{{Q^{H} }}C_{S}^{L} = 0 \Rightarrow Q^{H} = Q^{L} {\text{ for }}C_{S}^{H} \ne C_{S}^{L} $$

From the condition, \(x^{H} D - \frac{D}{{Q^{H} }}C_{S}^{H} - x^{L} D + \frac{D}{{Q^{L} }}C_{S}^{H} = 0\) we can also write \(x^{H} = x^{L}\). Also, we have \(Q_{1}^{H} = Q_{1}^{L}\).

Since μ1 = 0 then μ2 = 1. Thus, from the Stationarity conditions we have

$$ \left( {\frac{{\left( {Q^{H} - Q_{1}^{H} } \right)h_{2} }}{{Q^{H} }} - \frac{{\left( {Q^{H} - Q_{1}^{H} } \right)^{2} h_{2} }}{{2\left( {Q^{H} } \right)^{2} }} - \frac{{\left( {Q_{1}^{H} } \right)^{2} h_{1} }}{{2\left( {Q^{H} } \right)^{2} }} - \frac{{DC_{O} }}{{\left( {Q^{H} } \right)^{2} }}} \right)\lambda - \frac{{C_{S}^{L} D\mu_{4} }}{{\left( {Q^{H} } \right)^{2} }} + \frac{{C_{S}^{H} D\mu_{3} }}{{\left( {Q^{H} } \right)^{2} }} = 0 $$
$$ \begin{aligned}& \left( {\frac{{\left( {Q^{L} - Q_{1}^{L} } \right)h_{2} }}{{Q^{L} }} - \frac{{\left( {Q^{L} - Q_{1}^{L} } \right)^{2} h_{2} }}{{2\left( {Q^{L} } \right)^{2} }} - \frac{{\left( {Q_{1}^{L} } \right)^{2} h_{1} }}{{2\left( {Q^{L} } \right)^{2} }} - \frac{{C_{O} D}}{{\left( {Q^{L} } \right)^{2} }}} \right)\left( {1 - \lambda } \right) \\ &\quad+ \frac{{C_{S}^{L} D\mu_{4} }}{{\left( {Q^{L} } \right)^{2} }} - \frac{{C_{S}^{H} D\mu_{3} }}{{\left( {Q^{L} } \right)^{2} }} - \frac{{C_{S}^{L} D}}{{\left( {Q^{L} } \right)^{2} }} = 0 \end{aligned} $$

Adding these two equations and substituting QL = QH and QL1 = QH1 we get \(\left( {\frac{{h_{1} h_{2} }}{{h_{1} + h_{2} }} - \frac{{2C_{O} D}}{{\left( {Q^{H} } \right)^{2} }}} \right) = \frac{{2C_{S}^{L} D}}{{\left( {Q^{H} } \right)^{2} }}\) implying \(Q^{H} = \sqrt {\frac{{2D\left( {C_{S}^{L} + C_{O} } \right)\left( {h_{1} + h_{2} } \right)}}{{h_{1} h_{2} }}} = Q^{L}\).

Similarly, we can find \(Q_{1}^{H} = \sqrt {\frac{{2D\left( {C_{O} + C_{S}^{L} } \right)h_{2} }}{{h_{1} \left( {h_{1} + h_{2} } \right)}}}\) and \(Q_{1}^{L} = \sqrt {\frac{{2D\left( {C_{O} + C_{S}^{L} } \right)h_{2} }}{{h_{1} \left( {h_{1} + h_{2} } \right)}}}\) respectively.

Case XIV μ1 ≠ 0, μ2 = 0, μ3 ≠ 0, μ4 = 0.

From the Complementary Slackness Conditions, we get

$$ \frac{D}{{Q^{H} }}C_{S}^{H} - C_{S}^{H} - x^{H} D = 0 \Rightarrow x^{H} = C_{S}^{H} \left( {\frac{1}{{Q^{H} }} - \frac{1}{D}} \right) $$
$$ x^{H} D - \frac{D}{{Q^{H} }}C_{S}^{H} - x^{L} D + \frac{D}{{Q^{L} }}C_{S}^{H} = 0 $$

Since μ2 = 0, implies μ1 = 1. Also, μ4 = 0 implies μ3 = 1-λ.

Also from the Stationarity Conditions, we can write

$$ \left( {\frac{{\left( {Q^{H} - Q_{1}^{H} } \right)h_{2} }}{{Q^{H} }} - \frac{{\left( {Q^{H} - Q_{1}^{H} } \right)^{2} h_{2} }}{{2\left( {Q^{H} } \right)^{2} }} - \frac{{\left( {Q_{1}^{H} } \right)^{2} h_{1} }}{{2\left( {Q^{H} } \right)^{2} }} - \frac{{DC_{O} }}{{\left( {Q^{H} } \right)^{2} }}} \right)\lambda = \frac{{C_{S}^{H} D}}{{\left( {Q^{H} } \right)^{2} }} - \frac{{C_{S}^{H} D\left( {1 - \lambda } \right)}}{{\left( {Q^{H} } \right)^{2} }} $$
$$ Q^{H} = \sqrt {\frac{{2D\left( {h_{1} + h_{2} } \right)\left( {C_{S}^{H} + C_{O} } \right)}}{{h_{1} h_{2} }}} $$
$$ \begin{aligned}& \left( {\frac{{\left( {Q^{L} - Q_{1}^{L} } \right)h_{2} }}{{Q^{L} }} - \frac{{\left( {Q^{L} - Q_{1}^{L} } \right)^{2} h_{2} }}{{2\left( {Q^{L} } \right)^{2} }} - \frac{{\left( {Q_{1}^{L} } \right)^{2} h_{1} }}{{2\left( {Q^{L} } \right)^{2} }} - \frac{{C_{O} D}}{{\left( {Q^{L} } \right)^{2} }}} \right) = \frac{{C_{S}^{H} D}}{{\left( {Q^{L} } \right)^{2} }} \\ &\quad\Rightarrow Q^{L} = \sqrt {\frac{{2D\left( {C_{S}^{H} + C_{O} } \right)\left( {h_{1} + h_{2} } \right)}}{{h_{1} h_{2} }}} \end{aligned} $$

Case XV μ1 = 0, μ2 ≠ 0, μ3 = 0, μ4 ≠ 0.

From the Complementary Slackness Conditions, we get.

\(\frac{D}{{Q^{L} }}C_{S}^{L} - C_{S}^{L} - x^{L} D = 0 \Rightarrow x^{L} = C_{S}^{L} \left( {\frac{1}{{Q^{L} }} - \frac{1}{D}} \right)\) and \(x^{H} D - \frac{D}{{Q^{H} }}C_{S}^{H} - x^{L} D + \frac{D}{{Q^{L} }}C_{S}^{H} = 0\).

If μ1 = 0 then μ2 = 1. Also μ4 = λ since μ3 = 0.

Substituting the value of xL in the above equation we get

$$ x^{H} D - \frac{D}{{Q^{H} }}C_{S}^{H} - C_{S}^{L} \left( {\frac{D}{{Q^{L} }} - 1} \right) + \frac{D}{{Q^{L} }}C_{S}^{H} = 0 \Rightarrow $$
$$ \begin{aligned} & \left( {\frac{{\left( {Q^{H} - Q_{1}^{H} } \right)h_{2} }}{{Q^{H} }} - \frac{{\left( {Q^{H} - Q_{1}^{H} } \right)^{2} h_{2} }}{{2\left( {Q^{H} } \right)^{2} }} - \frac{{\left( {Q_{1}^{H} } \right)^{2} h_{1} }}{{2\left( {Q^{H} } \right)^{2} }} - \frac{{DC_{O} }}{{\left( {Q^{H} } \right)^{2} }}} \right)\lambda - \frac{{C_{S}^{L} D\lambda }}{{\left( {Q^{H} } \right)^{2} }} = 0\\ &\quad \Rightarrow Q^{H} = \sqrt {\frac{{2D\left( {C_{O} + C_{S}^{L} } \right)\left( {h_{1} + h_{2} } \right)}}{{h_{1} h_{2} }}} \end{aligned} $$
$$ \begin{gathered} \left( {\frac{{\left( {Q^{L} - Q_{1}^{L} } \right)h_{2} }}{{Q^{L} }} - \frac{{\left( {Q^{L} - Q_{1}^{L} } \right)^{2} h_{2} }}{{2\left( {Q^{L} } \right)^{2} }} - \frac{{\left( {Q_{1}^{L} } \right)^{2} h_{1} }}{{2\left( {Q^{L} } \right)^{2} }} - \frac{{DC_{O} }}{{\left( {Q^{L} } \right)^{2} }}} \right)\left( {1 - \lambda } \right) + \frac{{D\left( {\lambda C_{S}^{L} - C_{S}^{H} } \right)}}{{\left( {Q^{L} } \right)^{2} }} = 0 \hfill \\ \Rightarrow Q^{L} = \sqrt {\frac{{2D\left( {C_{O} + C_{S}^{L} } \right)\left( {h_{1} + h_{2} } \right)}}{{h_{1} h_{2} }}} \hfill \\ \end{gathered} $$

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Chakraborty, A. A surcharge pricing policy for supply chain coordination under the just-in-time (JIT) environment in the presence of backordering. OPSEARCH 58, 1049–1076 (2021). https://doi.org/10.1007/s12597-021-00509-5

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