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A Generalized Production-Inventory Model with Variable Production, Demand, and Cost Rates

  • Research Article-Systems Engineering
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Abstract

The classical economic production quantity model is formulated based on several simplifying assumptions such as constant demand rate, constant holding cost, constant setup cost, and constant production cost. However, these assumptions do not always represent the reality. Therefore, a generalized production-inventory control model is developed in this paper by relaxing the assumption of constant values for all input parameters. In order to represent the true characteristics of practical production environments, the demand rate, the production rate, and all of the cost rates are considered to be either dependent or decision variables. The demand is assumed to be a linear function of both the selling price and the average stock level. The setup cost per production run and the production cost per unit are both assumed to be nonlinear functions of the production rate. The holding cost is assumed to be a linear function of the production cost and the length of the storage time duration. An effective solution method is developed, and an example based on real data is solved. Sensitivity analysis is conducted and used to draw practical managerial insights. The study findings show that demand parameters have the highest impact on the profitability, followed by overhead and holding cost parameters.

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Acknowledgements

The authors acknowledge the support received from the King Fahd University of Petroleum and Minerals (KFUPM).

Author information

Authors and Affiliations

  1. Systems Engineering Department, King Fahd University of Petroleum & Minerals, Dhahran, 31261, Saudi Arabia

    Hesham K. Alfares

  2. Construction Engineering and Management Department, King Fahd University of Petroleum & Minerals, Dhahran, 31261, Saudi Arabia

    Ahmed M. Ghaithan

  3. Interdisciplinary Research Center for Smart Mobility & Logistics, King Fahd University of Petroleum and Minerals, P.O. Box 5067, Dhahran, 31261, Saudi Arabia

    Ahmed M. Ghaithan

Authors
  1. Hesham K. Alfares
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  2. Ahmed M. Ghaithan
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Corresponding author

Correspondence to Ahmed M. Ghaithan.

Appendices

Appendix A. Derivation of (11)

$$ \begin{aligned} \pi \left( {P,S,T} \right) &= S\left( {\alpha I_{\max } - \beta S} \right) - C_{p} \left( {P - \alpha I_{\max } + \beta S } \right)\left( {\frac{{a\left( {\alpha I_{\max } - \beta S} \right)^{2} T}}{{2P^{2} }} + \frac{{b\left( {\alpha I_{\max } - \beta S} \right)^{3} T^{2} }}{{3P^{3} }}} \right) \hfill \\ &\quad- C_{p} \left( {\alpha I_{\max } - \beta S} \right)\frac{T}{P}\left[ {a\left[ {P - \left( {\alpha I_{\max } - \beta S} \right)} \right] + \frac{{\left( {bT - a} \right)}}{2P}\left[ {P^{2} - \left( {\alpha I_{\max } - \beta S} \right)^{2} } \right] - \frac{bT}{{3P^{2} }}\left[ {P^{3} - \left( {\alpha I_{\max } - \beta S} \right)^{3} } \right]} \right] \hfill \\ &\quad- \frac{{KP^{w} }}{T} - C_{p} \left( {\alpha I_{\max } - \beta S} \right) \hfill \\ \end{aligned} $$
$$ \begin{aligned} \pi \left( {P,S,T} \right) &= S\left( {\alpha I_{\max } - \beta S} \right) - C_{p} \left( {P - \left( {\alpha I_{\max } - \beta S} \right) } \right)\left( {\frac{{a\left( {\alpha I_{\max } - \beta S} \right)^{2} T}}{{2P^{2} }} + \frac{{b\left( {\alpha I_{\max } - \beta S} \right)^{3} T^{2} }}{{3P^{3} }}} \right) \hfill \\ &\quad- C_{p} \left( {\alpha I_{\max } - \beta S} \right)\frac{T}{P}\left[ {a\left[ {P - \left( {\alpha I_{\max } - \beta S} \right)} \right] + \frac{{\left( {bT - a} \right)}}{2P}\left[ {P^{2} - \left( {\alpha I_{\max } - \beta S} \right)^{2} } \right] - \frac{bT}{{3P^{2} }}\left[ {P^{3} - \left( {\alpha I_{\max } - \beta S} \right)^{3} } \right]} \right] \hfill \\ &\quad- \frac{{KP^{w} }}{T} - C_{p} \left( {\alpha I_{\max } - \beta S} \right) \hfill \\ \end{aligned} $$
$$ \begin{aligned} \pi \left( {P,S,T} \right) &= S\left( {\alpha I_{\max } - \beta S} \right) - C_{p} \left\{ {\left( {P - \left( {\alpha I_{\max } - \beta S} \right) } \right)\left( {\frac{{a\left( {\alpha I_{\max } - \beta S} \right)^{2} T}}{{2P^{2} }} + \frac{{b\left( {\alpha I_{\max } - \beta S} \right)^{3} T^{2} }}{{3P^{3} }}} \right)} \right\} \hfill \\ &\quad- C_{p} \left( {\alpha I_{\max } - \beta S} \right)\frac{T}{P}\left[ {a\left[ {P - \left( {\alpha I_{\max } - \beta S} \right)} \right] + \frac{{\left( {bT - a} \right)}}{2P}\left[ {P^{2} - \left( {\alpha I_{\max } - \beta S} \right)^{2} } \right] - \frac{bT}{{3P^{2} }}\left[ {P^{3} - \left( {\alpha I_{\max } - \beta S} \right)^{3} } \right]} \right] \hfill \\ &\quad- \frac{{KP^{w} }}{T} - C_{p} \left( {\alpha I_{\max } - \beta S} \right) \hfill \\ \end{aligned} $$
$$ \begin{aligned} \pi \left( {P,S,T} \right) &= S\left( {\alpha I_{{\max }} - \beta S} \right) - C_{p} \left( {\frac{{a\left( {\alpha I_{{\max }} - \beta S} \right)^{2} T}}{{2P}} + \frac{{b\left( {\alpha I_{{\max }} - \beta S} \right)^{3} T^{2} }}{{3P^{2} }} - \frac{{a\left( {\alpha I_{{\max }} - \beta S} \right)^{3} T}}{{2P^{2} }} - \frac{{b\left( {\alpha I_{{\max }} - \beta S} \right)^{4} T^{2} }}{{3P^{3} }}} \right) \hfill \\ &\quad- C_{p} \left[ {aT\left( {\alpha I_{{\max }} - \beta S} \right) - \frac{{aT}}{P}\left( {\alpha I_{{\max }} - \beta S} \right)^{2} + \frac{{\left( {bT - a} \right)}}{{2P}}} \right.\left[ {P\left( {\alpha I_{{\max }} - \beta S} \right)T - \frac{T}{P}\left( {\alpha I_{{\max }} - \beta S} \right)^{3} } \right] \hfill \\ &\quad\left. {\left[ { - \frac{{bT}}{{3P^{2} }}\left[ {\left( {\alpha I_{{\max }} - \beta S} \right)TP^{2} - \frac{T}{P}\left( {\alpha I_{{\max }} - \beta S} \right)^{4} } \right]} \right]} \right] \hfill \\ &\quad- \frac{{KP^{w} }}{T} - C_{p} \left( {\alpha I_{{\max }} - \beta S} \right) \hfill \\ \end{aligned} $$
$$ \begin{aligned} \pi \left( {P,S,T} \right) &= S\left( {\alpha I_{{\max }} - \beta S} \right) - C_{p} \left( {\frac{{a\left( {\alpha I_{{\max }} - \beta S} \right)^{2} T}}{{2P}} + \frac{{b\left( {\alpha I_{{\max }} - \beta S} \right)^{3} T^{2} }}{{3P^{2} }} - \frac{{a\left( {\alpha I_{{\max }} - \beta S} \right)^{3} T}}{{2P^{2} }} - \frac{{b\left( {\alpha I_{{\max }} - \beta S} \right)^{4} T^{2} }}{{3P^{3} }}} \right) \hfill \\ &\quad- C_{p} \left[ {aT\left( {\alpha I_{{\max }} - \beta S} \right) - \frac{{aT}}{P}\left( {\alpha I_{{\max }} - \beta S} \right)^{2} + \frac{{bT^{2} }}{2}\left( {\alpha I_{{\max }} - \beta S} \right) - \frac{{bT^{2} }}{{2P^{2} }}\left( {\alpha I_{{\max }} - \beta S} \right)^{3} } \right. \hfill \\ &\quad\left. { - \frac{{aT}}{2}\left( {\alpha I_{{\max }} - \beta S} \right) + \frac{{aT}}{{2P^{2} }}\left( {\alpha I_{{\max }} - \beta S} \right)^{3} - \frac{{bT^{2} }}{3}\left( {\alpha I_{{\max }} - \beta S} \right) + \frac{{bT^{2} }}{{3P^{3} }}\left( {\alpha I_{{\max }} - \beta S} \right)^{4} } \right] \hfill \\ &\quad- \frac{{KP^{w} }}{T} - C_{p} \left( {\alpha I_{{\max }} - \beta S} \right) \hfill \\ \end{aligned} $$
$$ \begin{gathered} \pi \left( {P,S,T} \right) = S\left( {\alpha I_{{\max }} - \beta S} \right) - C_{p} \left( {\frac{{a\left( {\alpha I_{{\max }} - \beta S} \right)^{2} T}}{{2P}} + \frac{{b\left( {\alpha I_{{\max }} - \beta S} \right)^{3} T^{2} }}{{3P^{2} }} - \frac{{a\left( {\alpha I_{{\max }} - \beta S} \right)^{3} T}}{{2P^{2} }} } \right. \hfill \\ - \frac{{b\left( {\alpha I_{{\max }} - \beta S} \right)^{4} T^{2} }}{{3P^{3} }} + aT\left( {\alpha I_{{\max }} - \beta S} \right) - \frac{{aT}}{P}\left( {\alpha I_{{\max }} - \beta S} \right)^{2} \hfill \\ + \frac{{bT^{2} }}{2}\left( {\alpha I_{{\max }} - \beta S} \right) - \frac{{bT^{2} }}{{2P^{2} }}\left( {\alpha I_{{\max }} - \beta S} \right)^{3} - \frac{{aT}}{2}\left( {\alpha I_{{\max }} - \beta S} \right) \hfill \\ + \left. {\frac{{aT}}{{2P^{2} }}\left( {\alpha I_{{\max }} - \beta S} \right)^{3} - \frac{{bT^{2} }}{3}\left( {\alpha I_{{\max }} - \beta S} \right) + \frac{{bT^{2} }}{{3P^{3} }}\left( {\alpha I_{{\max }} - \beta S} \right)^{4} + \left( {\alpha I_{{\max }} - \beta S} \right)} \right) - \frac{{KP^{w} }}{T} \hfill \\ \end{gathered} $$
$$ \begin{gathered} \pi \left( {P,S,T} \right) = S\left( {\alpha I_{\max } - \beta S} \right) \hfill \\ - C_{p} \left( { - \frac{{aT\left( {\alpha I_{\max } - \beta S} \right)^{2} }}{2P} - \frac{{bT^{2} \left( {\alpha I_{\max } - \beta S} \right)^{3} }}{{6P^{2} }} + \frac{{bT^{2} \left( {\alpha I_{\max } - \beta S} \right)}}{6} + \frac{{aT\left( {\alpha I_{\max } - \beta S} \right)}}{2} + \left( {\alpha I_{\max } - \beta S} \right)} \right) \hfill \\ - \frac{{KP^{w} }}{T} \hfill \\ \end{gathered} $$

Appendix B. Partial derivatives of π(P, S, T)

$$ \begin{gathered} \frac{{\partial \pi \left( {P,S,T} \right)}}{{\partial P}} = - \frac{{KP^{{ - 1 + w}} w}}{T} + \frac{{S\left( { - 1 + T\alpha + \frac{{ - 4ST^{2} \alpha ^{2} \beta + 2\left( {1 - T\alpha } \right)\left( {P - PT\alpha - 2ST\alpha \beta } \right)}}{{2\sqrt { - 4ST^{2} \alpha ^{2} \beta \left( {P + S\beta } \right) + \left( {P - PT\alpha - 2ST\alpha \beta } \right)^{2} } }}} \right)}}{{2T\alpha }} - \hfill \\ (C + \frac{U}{{P^{\theta } }} + VP^{\varepsilon } )\left( { - 1 + T\alpha + \frac{{ - 4ST^{2} \alpha ^{2} \beta + 2\left( {1 - T\alpha } \right)\left( {P - PT\alpha - 2ST\alpha \beta } \right)}}{{2\sqrt { - 4ST^{2} \alpha ^{2} \beta \left( {P + S\beta } \right) + \left( {P - PT\alpha - 2ST\alpha \beta } \right)^{2} } }}} \right) \hfill \\ \left\{ {\frac{a}{{4\alpha }} + \frac{1}{{2T\alpha }} + \frac{{bT}}{{12\alpha }} - \frac{{a\left( { - S\beta + \frac{{ - P + PT\alpha + 2ST\alpha \beta + \sqrt { - 4ST^{2} \alpha ^{2} \beta \left( {P + S\beta } \right) + \left( {P - PT\alpha - 2ST\alpha \beta } \right)^{2} } }}{{2T\alpha }}} \right)}}{{2P\alpha }}} \right\} \hfill \\ - \left( {C + \frac{U}{{P^{\theta } }} + VP^{\varepsilon } } \right)\left( { - S\beta + \frac{{ - P + PT\alpha + 2ST\alpha \beta + \sqrt { - 4ST^{2} \alpha ^{2} \beta \left( {P + S\beta } \right) + \left( {P - PT\alpha - 2ST\alpha \beta } \right)^{2} } }}{{2T\alpha }}} \right)^{2} \hfill \\ \left\{ {\frac{{aT}}{{2P^{2} }} - \frac{{bT\left( { - 1 + T\alpha + \frac{{ - 4ST^{2} \alpha ^{2} \beta + 2\left( {1 - T\alpha } \right)\left( {P - PT\alpha - 2ST\alpha \beta } \right)}}{{2\sqrt { - 4ST^{2} \alpha ^{2} \beta \left( {P + S\beta } \right) + \left( {P - PT\alpha - 2ST\alpha \beta } \right)^{2} } }}} \right)}}{{4P^{2} \alpha }} + } \right. \hfill \\ \left. {\frac{{bT^{2} \left( { - S\beta + \frac{{ - P + PT\alpha + 2ST\alpha \beta + \sqrt { - 4ST^{2} \alpha ^{2} \beta \left( {P + S\beta } \right) + \left( {P - PT\alpha - 2ST\alpha \beta } \right)^{2} } }}{{2T\alpha }}} \right)}}{{3P^{3} }} - } \right\} \hfill \\ \left( { - \theta UP^{{ - 1 - \theta }} + \varepsilon VP^{{ - 1 + \varepsilon }} } \right)\left( { - S\beta + \frac{{ - P + PT\alpha + 2ST\alpha \beta + \sqrt { - 4ST^{2} \alpha ^{2} \beta \left( {P + S\beta } \right) + \left( {P - PT\alpha - 2ST\alpha \beta } \right)^{2} } }}{{2T\alpha }}} \right) \hfill \\ \left\{ {1 + \frac{{aT}}{2} + \frac{{bT^{2} }}{6} - \frac{{aT}}{{2P}}\left( { - S\beta + \frac{{ - P + PT\alpha + 2ST\alpha \beta + \sqrt { - 4ST^{2} \alpha ^{2} \beta \left( {P + S\beta } \right) + \left( {P - PT\alpha - 2ST\alpha \beta } \right)^{2} } }}{{2T\alpha }}} \right)} \right. - \hfill \\ \left. {\frac{{bT^{2} }}{{6P^{2} }}\left( { - S\beta + \frac{{ - P + PT\alpha + 2ST\alpha \beta + \sqrt { - 4ST^{2} \alpha ^{2} \beta \left( {P + S\beta } \right) + \left( {P - PT\alpha - 2ST\alpha \beta } \right)^{2} } }}{{2T\alpha }}} \right)^{2} } \right\} \hfill \\ \end{gathered} $$
(B1)
$$ \begin{gathered} \frac{{\partial \pi \left( {P,S,T} \right)}}{{\partial S}} = - S\beta + \frac{{ - P + PT\alpha + 2ST\alpha \beta + \sqrt { - 4ST^{2} \alpha ^{2} \beta \left( {P + S\beta } \right) + \left( {P - PT\alpha - 2ST\alpha \beta } \right)^{2} } }}{{2T\alpha }} \hfill \\ + S\left( { - \beta + \frac{{2T\alpha \beta + \frac{{ - 4ST^{2} \alpha ^{2} \beta ^{2} - 4T^{2} \alpha ^{2} \beta \left( {P + S\beta } \right) - 4T\alpha \beta \left( {P - PT\alpha - 2ST\alpha \beta } \right)}}{{2\sqrt { - 4ST^{2} \alpha ^{2} \beta \left( {P + S\beta } \right) + \left( {P - PT\alpha - 2ST\alpha \beta } \right)^{2} } }}}}{{2T\alpha }}} \right) \hfill \\ - \left\{ {1 + \frac{{aT}}{2} + \frac{{bT^{2} }}{6} - \frac{{aT}}{P}\left( { - S\beta + \frac{{ - P + PT\alpha + 2ST\alpha \beta + \sqrt { - 4ST^{2} \alpha ^{2} \beta \left( {P + S\beta } \right) + \left( {P - PT\alpha - 2ST\alpha \beta } \right)^{2} } }}{{2T\alpha }}} \right) - } \right. \hfill \\ \left. {\frac{{bT^{2} }}{{2P^{2} }}\left( { - S\beta + \frac{{ - P + PT\alpha + 2ST\alpha \beta + \sqrt { - 4ST^{2} \alpha ^{2} \beta \left( {P + S\beta } \right) + \left( {P - PT\alpha - 2ST\alpha \beta } \right)^{2} } }}{{2T\alpha }}} \right)^{2} } \right\} \hfill \\ \left( { - \beta + \frac{{2T\alpha \beta + \frac{{ - 4ST^{2} \alpha ^{2} \beta ^{2} - 4T^{2} \alpha ^{2} \beta \left( {P + S\beta } \right) - 4T\alpha \beta \left( {P - PT\alpha - 2ST\alpha \beta } \right)}}{{2\sqrt { - 4ST^{2} \alpha ^{2} \beta \left( {P + S\beta } \right) + \left( {P - PT\alpha - 2ST\alpha \beta } \right)^{2} } }}}}{{2T\alpha }}} \right)\left[ {C + \frac{U}{{P^{\theta } }} + VP^{\varepsilon } } \right] \hfill \\ \end{gathered} $$
(B2)
$$ \begin{aligned} & \frac{{\partial \pi \left( {P,S,T} \right)}}{{\partial T}} = \frac{{KP^{w} }}{{T^{2} }} + S\left( {\frac{{P\alpha + 2S\alpha \beta + \frac{{ - 8ST\alpha ^{2} \beta \left( {P + S\beta } \right) + 2\left( { - P\alpha - 2S\alpha \beta } \right)\left( {P - PT\alpha - 2ST\alpha \beta } \right)}}{{2\sqrt { - 4ST^{2} \alpha ^{2} \beta \left( {P + S\beta } \right) + \left( {P - PT\alpha - 2ST\alpha \beta } \right)^{2} } }}}}{{2T\alpha }} - \frac{{ - P + PT\alpha + 2ST\alpha \beta + \sqrt { - 4ST^{2} \alpha ^{2} \beta \left( {P + S\beta } \right) + \left( {P - PT\alpha - 2ST\alpha \beta } \right)^{2} } }}{{2T^{2} \alpha }}} \right) \\ & - \left[ {C + \frac{U}{{P^{\theta } }} + VP^{\varepsilon } } \right] \hfill \\ & \left\{ \begin{gathered} \left( {1 + \frac{1}{2}aT + \frac{1}{6}bT^{2} } \right)\left( {\frac{{P\alpha + 2S\alpha \beta + \frac{{ - 8ST\alpha ^{2} \beta \left( {P + S\beta } \right) + 2\left( { - P\alpha - 2S\alpha \beta } \right)\left( {P - PT\alpha - 2ST\alpha \beta } \right)}}{{2\sqrt { - 4ST^{2} \alpha ^{2} \beta \left( {P + S\beta } \right) + \left( {P - PT\alpha - 2ST\alpha \beta } \right)^{2} } }}}}{{2T\alpha }} - \frac{{ - P + PT\alpha + 2ST\alpha \beta + \sqrt { - 4ST^{2} \alpha ^{2} \beta \left( {P + S\beta } \right) + \left( {P - PT\alpha - 2ST\alpha \beta } \right)^{2} } }}{{2T^{2} \alpha }}} \right) \hfill \\ \end{gathered} \right. \\ & + \left( { - S\beta + \frac{{ - P + PT\alpha + 2ST\alpha \beta + \sqrt { - 4ST^{2} \alpha ^{2} \beta \left( {P + S\beta } \right) + \left( {P - PT\alpha - 2ST\alpha \beta } \right)^{2} } }}{{2T\alpha }}} \right) \hfill \\ & \left\{ {\frac{a}{2} + \frac{{bT}}{3} - \frac{{aT}}{P}\left( {\frac{{P\alpha + 2S\alpha \beta + \frac{{ - 8ST\alpha ^{2} \beta \left( {P + S\beta } \right) + 2\left( { - P\alpha - 2S\alpha \beta } \right)\left( {P - PT\alpha - 2ST\alpha \beta } \right)}}{{2\sqrt { - 4ST^{2} \alpha ^{2} \beta \left( {P + S\beta } \right) + \left( {P - PT\alpha - 2ST\alpha \beta } \right)^{2} } }}}}{{2T\alpha }} - \frac{{ - P + PT\alpha + 2ST\alpha \beta + \sqrt { - 4ST^{2} \alpha ^{2} \beta \left( {P + S\beta } \right) + \left( {P - PT\alpha - 2ST\alpha \beta } \right)^{2} } }}{{2T^{2} \alpha }}} \right) - } \right. \hfill \\ &\frac{a}{{2P}}\left( { - S\beta + \frac{{ - P + PT\alpha + 2ST\alpha \beta + \sqrt { - 4ST^{2} \alpha ^{2} \beta \left( {P + S\beta } \right) + \left( {P - PT\alpha - 2ST\alpha \beta } \right)^{2} } }}{{2T\alpha }}} \right) \\ & - \frac{{bT^{2} }}{{2P^{2} }}\left( \begin{gathered} \frac{{P\alpha + 2S\alpha \beta + \frac{{ - 8ST\alpha ^{2} \beta \left( {P + S\beta } \right) + 2\left( { - P\alpha - 2S\alpha \beta } \right)\left( {P - PT\alpha - 2ST\alpha \beta } \right)}}{{2\sqrt { - 4ST^{2} \alpha ^{2} \beta \left( {P + S\beta } \right) + \left( {P - PT\alpha - 2ST\alpha \beta } \right)^{2} } }}}}{{2T\alpha }} - \frac{{ - P + PT\alpha + 2ST\alpha \beta + \sqrt { - 4ST^{2} \alpha ^{2} \beta \left( {P + S\beta } \right) + \left( {P - PT\alpha - 2ST\alpha \beta } \right)^{2} } }}{{2T^{2} \alpha }} \hfill \\ \end{gathered} \right) \hfill \\ & \left( { - S\beta + \frac{{ - P + PT\alpha + 2ST\alpha \beta + \sqrt { - 4ST^{2} \alpha ^{2} \beta \left( {P + S\beta } \right) + \left( {P - PT\alpha - 2ST\alpha \beta } \right)^{2} } }}{{2T\alpha }}} \right) - \hfill \\ & \left. {\left. {\frac{{bT}}{{3P^{2} }}\left( { - S\beta + \frac{{ - P + PT\alpha + 2ST\alpha \beta + \sqrt { - 4ST^{2} \alpha ^{2} \beta \left( {P + S\beta } \right) + \left( {P - PT\alpha - 2ST\alpha \beta } \right)^{2} } }}{{2T\alpha }}} \right)^{2} } \right\}} \right\} \hfill \\ \end{aligned} $$
(B3)

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Alfares, H.K., Ghaithan, A.M. A Generalized Production-Inventory Model with Variable Production, Demand, and Cost Rates. Arab J Sci Eng 47, 3963–3978 (2022). https://doi.org/10.1007/s13369-021-06516-4

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