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# An EOQ inventory model for deteriorating items with controllable deterioration rate under stock-dependent demand rate and non-linear holding cost

• * Corresponding author: Longzhou Cao

This work is supported by the Major Research Plan of the National Natural Science Foundation of China (91430108), the National Natural Science Foundation of China (11771322), the National Social Science Foundation of China (19BGL190) and the Major Program of Tianjin University of Finance and Economics (ZD1302)

• The main purpose of this paper is to investigate the retailer's strategy in selecting the order-up-to level, the reorder point and the preservation technology investment for deteriorating items, aiming to maximize his total profit per unit time. We formulate the problem into a mathematical model that takes into account stock-dependent demand rate, stock-dependent holding cost. The terminal conditions are relaxed to allow that the reorder point can be one of the following two cases: (1) $N\leq0$, i.e., the reorder point may be negative or zero. When the reorder point is negative, the shortage is allowed and partial backlogged. (2) $N\geq0$, i.e., the reorder point may be without shortage or zero. We prove the existence and uniqueness of the optimal order-up-to level, the reorder point and the preservation technology investment under any given two of them for both the two cases. We then present an algorithm to search for decision variables such that the total profit per unit time is maximized. Finally, numerical examples, comparisons in performance and sensitivity analysis are carried out to examine the results obtained. On the basis of the above results, some useful managerial insights are revealed.

Mathematics Subject Classification: Primary: 90B05; Secondary: 91B06.

 Citation: • • Figure 1.  Graphical representation for the case of $N\leq0$

Figure 2.  Graphical representation for the case of $N\geq0$

Figure 3.  The effect of the deterioration rate on $Q$, $N$, $\xi$, $S$, $U$, $AR$, $\theta_1$, $T$, and $t_1$ for the cases with and without preservation technology investment

Table 1.  Notation

 Decision variables $Q$ the order-up-to level at time $t = 0$. $N$ the reorder point at time $T$, i.e., $I(T) = N$. $\xi$ the preservation technology cost per unit time for reducing the deterioration rate. auxiliary variables $S(Q, \xi, N)$ the order quantity per cycle. $T(Q, \xi, N)$ the inventory cycle length. $t_1(Q, \xi)$ the time when the inventory level comes down to zero. Constant parameters $p$ the purchasing cost per unit item. $s$ the selling price per unit item. $h$ the holding cost per unit per unit time. $I(t)$ the inventory level at time $t$. $D(t)$ the demand rate at time $t$. $\gamma$ the elasticity of holding cost. $\beta$ the demand elasticity. $c_s$ the shortage cost per unit per unit time. $c_l$ the lost sale cost per unit. $\lambda$ the scale parameter of the demand rate. $\delta$ the partial backlogging parameter, $\delta\in [0, 1]$. $\theta$ the deteriorating rate. $K$ the replenishment cost per order. $m(\xi)$ the proportion of the reduced deterioration rate, $0\leq m(\xi)\leq 1$. $AR(Q, \xi, N)$ the total profit per unit time. $TR(Q, \xi, N)$ the total profit per inventory cycle.

Table 2.  Sensitivity analysis for Example 3

 parameter value $AR^*$ $Q^*$ $N^*$ $\xi^*$ $T^*$ $S^*$ $\theta$ 0.06 24.1103 3.0660 0.4157 0 2.1865 2.6503 0.13 21.5417 3.0618 0.4074 2.3451 2.1880 2.6544 0.2 20.1058 3.0618 0.4074 3.7811 2.1880 2.6544 0.27 19.1054 3.0618 0.4074 4.7814 2.1880 2.6544 0.34 18.3370 3.0618 0.4074 5.5498 2.1880 2.6544 $K$ 10 25.4420 2.8487 0.6882 3.6616 1.7274 2.1605 15 22.6078 2.9691 0.5264 3.7246 1.9840 2.4427 20 20.1058 3.0618 0.4074 3.7811 2.1880 2.6544 25 17.8250 3.1382 0.3159 3.8343 2.3590 2.8223 30 15.7056 3.2037 0.2439 3.8858 2.5067 2.9598 $a$ 0.08 16.1844 2.8147 0.1615 0 2.1550 2.6532 0.19 18.3422 3.0251 0.3453 3.8236 2.1963 2.6798 0.3 20.1058 3.0618 0.4074 3.7811 2.1880 2.6544 0.41 21.2412 3.0765 0.4378 3.4653 2.1819 2.6387 0.52 22.0291 3.0843 0.4559 3.1586 2.1777 2.6284 $\lambda$ 0.8 13.8535 2.8817 0.3308 3.6707 2.6957 2.5509 0.9 16.9434 2.9760 0.3708 3.7274 2.4150 2.6052 1 20.1058 3.0618 0.4074 3.7811 2.1880 2.6544 1.1 23.3336 3.1408 0.4413 3.8320 2.0008 2.6995 1.2 26.6212 3.2141 0.4729 3.8804 1.8437 2.7412 $h$ 0.2 22.0652 3.3883 0.5140 4.1190 2.2519 2.8743 0.25 20.9776 3.2045 0.4531 3.9328 2.2166 2.7514 0.3 20.1058 3.0618 0.4074 3.7811 2.1880 2.6544 0.35 19.3801 2.9463 0.3716 3.6531 2.1640 2.5747 0.4 18.7597 2.8498 0.3425 3.5425 2.1432 2.5073 $s$ 45 9.0587 2.8849 0.1786 3.2113 2.4050 2.7063 50 14.4533 2.9806 0.2948 3.5260 2.2877 2.6858 55 20.1058 3.0618 0.4074 3.7811 2.1880 2.6544 60 25.9663 3.1330 0.5129 3.9930 2.1035 2.6201 65 31.9997 3.1966 0.6107 4.1732 2.0311 2.5859 $p$ 10 33.3378 3.2234 0.6074 2.9528 2.0160 2.6160 15 26.5681 3.1444 0.5121 3.4419 2.0954 2.6323 20 20.1058 3.0618 0.4074 3.7811 2.1880 2.6544 25 13.9445 2.9707 0.2945 3.9925 2.2981 2.6762 30 8.1104 2.8650 0.1771 4.0781 2.4301 2.6879 $\gamma$ 3 26.2330 4.5559 0.8063 4.9638 2.5784 3.7496 3.5 22.6169 3.6210 0.5504 4.2757 2.3429 3.0706 4 20.1058 3.0618 0.4074 3.7811 2.1880 2.6544 4.5 18.2543 2.6932 0.3182 3.4071 2.0788 2.3750 5 16.8301 2.4333 0.2582 3.1140 1.9977 2.1751 $\beta$ 0.32 19.2226 2.9215 0.2756 3.5565 2.2976 2.6459 0.36 19.6213 2.9920 0.3412 3.6710 2.2467 2.6508 0.4 20.1058 3.0618 0.4074 3.7811 2.1880 2.6544 0.44 20.6777 3.1314 0.4738 3.8865 2.1236 2.6576 0.48 21.3397 3.2011 0.5399 3.9873 2.0550 2.6612
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