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March  2017, 7(1): 21-50. doi: 10.3934/naco.2017002

A two-echelon inventory model with stock-dependent demand and variable holding cost for deteriorating items

 1 Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore-721102, West Bengal, India 2 Institute of Applied Mathematics, Middle East Technical University, 06800, Ankara, Turkey

* Corresponding author: sankroy2006@gmail.com.

Received  September 2016 Published  February 2017

In this study, we develop an inventory model for deteriorating items with stock dependent demand rate. Shortages are allowed to this model and when stock on hand is zero, then the retailer offers a price discount to customers who are willing to back-order their demands. Here, the supplier as well as the retailer adopt the trade credit policy for their customers in order to promote the market competition. The retailer can earn revenue and interest after the customer pays for the amount of purchasing cost to the retailer until the end of the trade credit period offered by the supplier. Besides this, we consider variable holding cost due to increase the stock of deteriorating items. Thereafter, we present an easy analytical closed-form solution to find the optimal order quantity so that the total cost per unit time is minimized. The results are discussed with the help of numerical examples to validate the proposed model. A sensitivity analysis of the optimal solutions for the parameters is also provided in order to stabilize our model. The paper ends with a conclusion and an outlook to possible future studies.

Citation: Magfura Pervin, Sankar Kumar Roy, Gerhard Wilhelm Weber. A two-echelon inventory model with stock-dependent demand and variable holding cost for deteriorating items. Numerical Algebra, Control & Optimization, 2017, 7 (1) : 21-50. doi: 10.3934/naco.2017002
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References:
Graphical representation of our proposed Inventory control model
Flowchart of the solution procedure
Graphical representation to show the convexity of total cost. The figure represents $T$, $t_1$ and the total cost $\Pi(T)$, along the x-axis, the y-axis and the z-axis, respectively
Graphical representation to show the convexity of total cost. The figure represents T, t1 and the total cost Π(T), along the x-axis, the y-axis and the z-axis, respectively
Change of total cost with respect to ordering cost, A, of our proposed model
Change of total cost with respect to parameter α of our proposed model
Change of total cost with respect to holding cost, h, of our proposed model
Change of total cost with respect to deteriorating cost, θ, of our proposed model
Research works of various authors related to this area.
 Author(s) Shortages Trade credit policy Stock dependent demand Price discount on backorders Deterio-rations Time varying costs Ghare and Scharder (1963) √ Giri et al. (1996) √ √ Manna and Chaudhuri (2001) √ √ √ Roy (2008) √ √ Min et al. (2010) √ √ Mishra et al. (2013) √ √ Tripathi and Pandey (2013) √ √ Tripathi (2015) √ √ Annadurai and Uthayakumar (2015) √ √ Pervin et al. (2015) √ √ √ Swami et al. (2015) √ √ √ Our paper √ √ √ √ √ √
 Author(s) Shortages Trade credit policy Stock dependent demand Price discount on backorders Deterio-rations Time varying costs Ghare and Scharder (1963) √ Giri et al. (1996) √ √ Manna and Chaudhuri (2001) √ √ √ Roy (2008) √ √ Min et al. (2010) √ √ Mishra et al. (2013) √ √ Tripathi and Pandey (2013) √ √ Tripathi (2015) √ √ Annadurai and Uthayakumar (2015) √ √ Pervin et al. (2015) √ √ √ Swami et al. (2015) √ √ √ Our paper √ √ √ √ √ √
Sensitivity Analysis for different Parameters involved in Example 1.
 Parameter % change value T t1 t2 TC A +50 450 3.9859 0.1978 0.1865 225.4871 +25 375 3.9701 0.1975 0.1841 225.4991 +10 330 3.9621 0.1968 0.1820 225.5123 -10 270 3.9528 0.1042 0.1792 225.5472 -25 225 3.9519 0.1040 0.1763 225.5612 -50 150 3.9482 0.1037 0.1730 225.5860 s +50 30 3.9883 0.1054 0.1579 225.6102 +25 25 3.9865 0.1049 0.1556 225.6372 +10 22 3.9851 0.1046 0.1534 225.6819 -10 18 3.9840 0.1042 0.1518 225.6960 -25 15 3.9832 0.1039 0.1475 225.7542 -50 10 3.9818 0.1034 0.1455 225.7620 c +50 90 3.9864 0.1043 0.1618 226.0171 +25 75 3.9847 0.1039 0.1632 226.1261 +10 66 3.9840 0.1044 0.1659 226.4189 -10 54 3.9834 0.1103 0.1671 226.4703 -25 45 3.9821 0.1111 0.1690 226.5100 -50 30 3.9811 0.1235 0.1724 226.5275 a +50 1.05 4.0854 0.1352 0.1858 225.8906 +25 0.875 3.9981 0.1432 0.1822 225.8940 +10 0.77 3.9702 0.1657 0.1805 225.8976 -10 0.63 3.9453 0.1723 0.1778 225.9121 -25 0.525 3.9321 0.1805 0.1751 225.9407 -50 0.35 3.9161 0.1823 0.1736 225.9522 b +50 1.2 3.9093 0.1834 0.1780 228.3131 +25 1.0 3.924 0.1874 0.1799 228.4309 +10 0.88 3.9398 0.1916 0.1827 228.4971 -10 0.72 3.983 0.1396 0.1848 228.5102 -25 0.60 4.0806 0.1668 0.1864 228.6524 -50 0.40 3.1503 0.1625 0.1882 228.7601
 Parameter % change value T t1 t2 TC A +50 450 3.9859 0.1978 0.1865 225.4871 +25 375 3.9701 0.1975 0.1841 225.4991 +10 330 3.9621 0.1968 0.1820 225.5123 -10 270 3.9528 0.1042 0.1792 225.5472 -25 225 3.9519 0.1040 0.1763 225.5612 -50 150 3.9482 0.1037 0.1730 225.5860 s +50 30 3.9883 0.1054 0.1579 225.6102 +25 25 3.9865 0.1049 0.1556 225.6372 +10 22 3.9851 0.1046 0.1534 225.6819 -10 18 3.9840 0.1042 0.1518 225.6960 -25 15 3.9832 0.1039 0.1475 225.7542 -50 10 3.9818 0.1034 0.1455 225.7620 c +50 90 3.9864 0.1043 0.1618 226.0171 +25 75 3.9847 0.1039 0.1632 226.1261 +10 66 3.9840 0.1044 0.1659 226.4189 -10 54 3.9834 0.1103 0.1671 226.4703 -25 45 3.9821 0.1111 0.1690 226.5100 -50 30 3.9811 0.1235 0.1724 226.5275 a +50 1.05 4.0854 0.1352 0.1858 225.8906 +25 0.875 3.9981 0.1432 0.1822 225.8940 +10 0.77 3.9702 0.1657 0.1805 225.8976 -10 0.63 3.9453 0.1723 0.1778 225.9121 -25 0.525 3.9321 0.1805 0.1751 225.9407 -50 0.35 3.9161 0.1823 0.1736 225.9522 b +50 1.2 3.9093 0.1834 0.1780 228.3131 +25 1.0 3.924 0.1874 0.1799 228.4309 +10 0.88 3.9398 0.1916 0.1827 228.4971 -10 0.72 3.983 0.1396 0.1848 228.5102 -25 0.60 4.0806 0.1668 0.1864 228.6524 -50 0.40 3.1503 0.1625 0.1882 228.7601
Sensitivity Analysis for different Parameters which are involved in Example 1.
 Parameter % change value T t1 t2 TC M +50 0.45 3.9361 0.1790 0.1570 227.1092 +25 0.375 3.9473 0.1796 0.1589 227.4121 +10 0.33 3.9528 0.1853 0.1603 227.6708 -10 0.27 3.9599 0.1854 0.1639 227.8211 -25 0.225 3.9647 0.1861 0.1672 227.8355 -50 0.15 3.9720 0.1864 0.1700 227.8708 N +50 0.75 3.9471 0.1597 0.1968 226.9858 +25 0.625 3.9510 0.1594 0.1940 226.9987 +10 0.55 3.9540 0.1659 0.1903 227.6891 -10 0.45 3.9593 0.1668 0.1881 227.6988 -25 0.375 3.9549 0.1678 0.1854 227.7408 -50 0.25 3.9601 0.1682 0.1826 227.1923 θ +50 0.09 3.9840 0.1604 0.1725 227.83 +25 0.075 3.9844 0.1629 0.1756 227.71 +10 0.066 3.9847 0.1636 0.1791 227.16 -10 0.054 3.9830 0.1646 0.1824 226.88 -25 0.045 3.9871 0.1653 0.1847 226.41 -50 0.03 3.9889 0.1664 0.1879 226.30 δ +50 0.075 3.9840 0.1687 0.1763 225.74 +25 0.0625 3.9844 0.1695 0.1735 225.63 +10 0.055 3.9849 0.1736 0.1712 225.16 -10 0.045 3.9850 0.1741 0.1675 224.98 -25 0.0375 3.9867 0.1750 0.1633 224.47 -50 0.025 3.9893 0.1769 0.1600 224.39
 Parameter % change value T t1 t2 TC M +50 0.45 3.9361 0.1790 0.1570 227.1092 +25 0.375 3.9473 0.1796 0.1589 227.4121 +10 0.33 3.9528 0.1853 0.1603 227.6708 -10 0.27 3.9599 0.1854 0.1639 227.8211 -25 0.225 3.9647 0.1861 0.1672 227.8355 -50 0.15 3.9720 0.1864 0.1700 227.8708 N +50 0.75 3.9471 0.1597 0.1968 226.9858 +25 0.625 3.9510 0.1594 0.1940 226.9987 +10 0.55 3.9540 0.1659 0.1903 227.6891 -10 0.45 3.9593 0.1668 0.1881 227.6988 -25 0.375 3.9549 0.1678 0.1854 227.7408 -50 0.25 3.9601 0.1682 0.1826 227.1923 θ +50 0.09 3.9840 0.1604 0.1725 227.83 +25 0.075 3.9844 0.1629 0.1756 227.71 +10 0.066 3.9847 0.1636 0.1791 227.16 -10 0.054 3.9830 0.1646 0.1824 226.88 -25 0.045 3.9871 0.1653 0.1847 226.41 -50 0.03 3.9889 0.1664 0.1879 226.30 δ +50 0.075 3.9840 0.1687 0.1763 225.74 +25 0.0625 3.9844 0.1695 0.1735 225.63 +10 0.055 3.9849 0.1736 0.1712 225.16 -10 0.045 3.9850 0.1741 0.1675 224.98 -25 0.0375 3.9867 0.1750 0.1633 224.47 -50 0.025 3.9893 0.1769 0.1600 224.39

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