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Imperfection with inspection policy and variable demand under trade-credit: A deteriorating inventory model

The author, Magfura Pervin is very thankful to University Grants Commission (UGC) of India for providing financial support to continue this research work under [MANF(UGC)] scheme: Sanctioned letter number [F1-17.1/2012-13/MANF-2012-13-MUS-WES-19170 /(SA-Ⅲ/Website)] dated 28/02/2013

The research of Gerhard-Wilhelm Weber (Institute of Applied Mathematics, METU, 06800, Ankara, Turkey) is partially supported by the Portuguese Foundation for Science and Technology ("FCT-Fundação para a Ciência e a Tecnologia"), through the CIDMA - Center for Research and Development in Mathematics and Applications, within project UID/MAT/ 04106/2013.
Abstract / Introduction Full Text(HTML) Figure(4) / Table(4) Related Papers Cited by
  • A deteriorating inventory model with imperfect product and variable demand is formulated in this paper. A time-dependent deterioration factor is considered because the rate of deterioration is highly hinging on time. We introduce imperfect quality of production which leads to imperfect items in our proposed model. The retailer adopts inspection policy to pick over the perfect items from imperfect. Type Ⅰ and Type Ⅱ, both type of errors are included and the retailer invest some capital to improve the production process quality of the supplier. There is also a penalty cost for the retailer if they deliver some defective items by mistake. Sometime, there is a high amount of demand and, consequently, we assume shortages and partial backorder in our formulated model. The retailer adopts the trade-credit policy for his customers in order to promote market competition. The main objective of the paper is to show that the total cost is globally minimized and we have aimed at reducing the total cycle length, defectiveness of the system and the optimal order size by maximizing the total profit of the system. Then, we present three theorems and prove them to find an easy solution procedure to reduce the total cost of a system. The results are discussed with the help of numerical examples to approve the proposed model. A sensitivity analysis of the optimal solutions for the parameters is also provided. The paper ends with the conclusions and an outlook to possible future studies.

    Mathematics Subject Classification: Primary: 90B05, 91A10; Secondary: 90C26.

    Citation:

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  • Figure 1.  Graphical representation of our proposed inventory control model

    Figure 2.  Graphical description to show the convexity of the total cost. Included are $ T $, $ \alpha $ and the total cost $ R_2(T) $, along the x-axis, the y-axis and the z-axis, respectively

    Figure 3.  Graphical presentation to show the convexity of the total cost. The Portrayed are $ T $, $ q $ and the total cost $ R_1(T) $, along the x-axis, the y-axis and the z-axis, respectively

    Figure 4.  Graphical representation to show the convexity of the total cost. Represented are $ \alpha $, $ q $ and the total cost $ R_3(T) $, along the x-axis, the y-axis and the z-axis, respectively

    Table 1.  Contributions of many authors related to inventory model

    Author(s) Imperfec tness Variable demand Deterio rations Trade-credit policy Short ages
    Sana et al. (2004) $ \surd $ $ \surd $ $ \surd $
    Khan et al. (2014) $ \surd $
    Lin and Hou (2015) $ \surd $
    Teng et al. (2002) $ \surd $ $ \surd $ $ \surd $
    Ghosh and Chaudhuri (2004) $ \surd $ $ \surd $ $ \surd $
    Kumar et al. (2012) $ \surd $ $ \surd $
    Pervin et al. (2016, a) $ \surd $ $ \surd $ $ \surd $
    Mahata (2012) $ \surd $ $ \surd $
    Goswami and Chaudhuri (1991) $ \surd $ $ \surd $ $ \surd $
    Aggarwal and Jaggi (1995) $ \surd $ $ \surd $
    Ting (2015) $ \surd $ $ \surd $
    Ouyang et al. (2005) $ \surd $ $ \surd $ $ \surd $
    Tripathi (2015) $ \surd $ $ \surd $ $ \surd $
    Annadurai and Uthayakumar (2015) $ \surd $ $ \surd $ $ \surd $
    Balkhi (2004) $ \surd $ $ \surd $
    Pervin et al. (2016, b) $ \surd $ $ \surd $ $ \surd $
    Our paper $ \surd $ $ \surd $ $ \surd $ $ \surd $ $ \surd $
     | Show Table
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    Table 2.  Computational result. Bold style represents the optimal solution

    $ q $ $ \alpha^* $ $ T_1^* $ $ T_2^* $ $ T_3^* $ $ R_1(T,q,\alpha) $ $ R_2(T,q,\alpha) $ $ R_3(T,q,\alpha) $
    $ 1 $ 0.1077 1.235 1.302 1.187 3478.12 3599.45 3517.30
    $ 2 $ 0.1125 1.301 1.213 1.171 3560.82 3011.67 3069.57
    $ 3 $ 0.1092 1.420 1.387 1.432 3673.19 3095.07 3112.70
    $ {\bf{4}} $ 0.1089 1.324 1.112 1.108 3488.22 2973.16 2844.93
    $ 5 $ 0.1107 1.403 1.274 1.392 3579.08 3077.45 3205.71
    $ 6 $ 0.1073 1.449 1.171 1.321 3678.99 3360.21 3119.54
    $ 7 $ 0.128 1.610 1.534 1.558 3879.04 3232.70 3374.56
    $ 8 $ 0.116 1.791 1.645 1.325 3927.34 3306.18 3317.84
    $ 9 $ 0.125 1.570 1.349 1.102 4012.57 3812.05 3518.38
    $ 10 $ 0.144 1.397 1.747 1.560 3939.55 3670.10 3575.68
     | Show Table
    DownLoad: CSV

    Table 3.  Sensitivity analysis for various parameters involved in Example 1

    Para meter $ \% $ change value $ T_1^* $ $ T_2^* $ $ T_3^* $ $ R_1(T) $ $ R_2(T) $ $ R_3(T) $ Total Cost Profit
    $ A $ +50 1350 1.832 1.572 1.624 3122.05 2705.91 2812.83 $ \downarrow $ $ \uparrow $
    +30 1170 1.510 1.480 1.613 2988.71 2633.61 2694.74 $ \downarrow $ $ \uparrow $
    +10 990 1.257 1.437 1.589 2734.00 2482.09 2503.15 $ \downarrow $ $ \uparrow $
    -10 810 0.941 1.428 1.541 2613.47 2290.41 2387.40 $ \downarrow $ $ \uparrow $
    -30 630 0.703 1.410 1.516 2497.38 2100.16 2206.85 $ \downarrow $ $ \uparrow $
    -50 450 0.627 1.377 1.487 2206.16 2000.00 2183.57 $ \downarrow $ $ \uparrow $
    $ s $ +50 600 0.904 0.867 0.880 3385.10 2834.61 2791.38 $ \downarrow $ $ \uparrow $
    +30 520 0.760 0.843 0.861 3174.51 2690.42 2614.50 $ \downarrow $ $ \uparrow $
    +10 440 0.601 0.829 0.847 2893.67 2517.83 2576.31 $ \downarrow $ $ \uparrow $
    -10 360 0.472 0.820 0.831 2635.85 2385.76 2344.10 $ \downarrow $ $ \uparrow $
    -30 280 0.275 0.816 0.822 2483.77 2189.02 2208.47 $ \downarrow $ $ \uparrow $
    -50 200 0.064 0.803 0.815 2305.38 1985.47 2083.17 $ \downarrow $ $ \uparrow $
    $ h_1 $ +50 150 1.966 1.973 1.989 3192.70 2911.56 2810.55 $ \downarrow $ $ \uparrow $
    +30 130 1.728 1.854 1.910 2948.16 2738.64 2655.18 $ \downarrow $ $ \uparrow $
    +10 110 1.601 1.779 1.878 2749.03 2500.53 2451.38 $ \downarrow $ $ \uparrow $
    -10 90 1.486 1.611 1.763 2605.74 2374.95 2180.76 $ \downarrow $ $ \uparrow $
    -30 70 1.310 1.572 1.683 2477.19 2285.31 2090.59 $ \downarrow $ $ \uparrow $
    -50 50 1.173 1.489 1.557 2204.60 2069.73 1877.92 $ \downarrow $ $ \uparrow $
    $ h_2 $ +50 300 2.765 2.581 2.593 3592.00 2746.31 2522.20 $ \downarrow $ $ \uparrow $
    +30 260 2.303 2.505 2.512 3386.17 2506.57 2257.46 $ \downarrow $ $ \uparrow $
    +10 220 2.007 2.488 2.502 3152.04 2344.10 2037.18 $ \downarrow $ $ \uparrow $
    -10 180 1.841 2.394 2.581 2819.11 2183.51 1734.62 $ \downarrow $ $ \uparrow $
    -30 140 1.573 2.353 2.473 2540.83 1982.86 1504.27 $ \downarrow $ $ \uparrow $
    -50 100 1.250 2.279 2.371 2274.62 1710.27 1374.02 $ \downarrow $ $ \uparrow $
    $ a $ +50 22.5 2.791 2.880 2.898 2916.47 3721.40 2883.45 $ \downarrow $ $ \uparrow $
    +30 19.5 2.675 2.741 2.805 2803.15 3569.00 2761.74 $ \downarrow $ $ \uparrow $
    +10 16.5 2.533 2.689 2.775 2785.32 3477.54 2653.14 $ \downarrow $ $ \uparrow $
    -10 13.5 2.412 2.537 2.643 2511.63 3307.11 2578.39 $ \downarrow $ $ \uparrow $
    -30 10.5 2.370 2.481 2.550 2479.59 3258.46 2386.55 $ \downarrow $ $ \uparrow $
    -50 7.5 2.264 2.372 2.445 2333.16 3117.35 2290.28 $ \downarrow $ $ \uparrow $
    $ b $ +50 30 2.672 2.503 2.475 3599.89 2800.00 2795.64 $ \downarrow $ $ \uparrow $
    +30 26 2.538 2.484 2.449 3512.04 2715.35 2683.05 $ \downarrow $ $ \uparrow $
    +10 22 2.460 2.336 2.429 3475.68 2610.97 2500.34 $ \downarrow $ $ \uparrow $
    -10 18 2.397 2.260 2.405 3381.62 2573.61 2435.65 $ \downarrow $ $ \uparrow $
    -30 14 2.200 2.175 2.391 3260.91 2483.29 2362.41 $ \downarrow $ $ \uparrow $
    -50 10 2.822 2.124 2.364 3129.56 2321.79 2155.20 $ \downarrow $ $ \uparrow $
    $ \gamma $ +50 0.60 1.754 2.134 2.431 3529.87 3138.56 2490.18 $ \downarrow $ $ \uparrow $
    +30 0.52 1.694 2.089 2.379 3348.72 2763.79 2317.93 $ \downarrow $ $ \uparrow $
    +10 0.44 1.504 1.905 2.248 3095.23 2480.58 2188.37 $ \downarrow $ $ \uparrow $
    -10 0.36 1.438 1.881 2.179 2860.93 2201.49 2020.99 $ \downarrow $ $ \uparrow $
    -30 0.28 1.215 1.753 2.098 2547.65 1973.10 1845.73 $ \downarrow $ $ \uparrow $
    -50 0.20 1.093 1.668 1.979 2196.37 1748.53 1549.27 $ \downarrow $ $ \uparrow $
    $ r_1 $ +50 75 2.402 1.402 1.657 3891.57 3124.51 3054.36 $ \downarrow $ $ \uparrow $
    +30 65 2.763 1.435 1.661 3522.56 2910.28 2839.17 $ \downarrow $ $ \uparrow $
    +10 55 2.911 1.492 1.704 3205.84 2764.33 2780.10 $ \downarrow $ $ \uparrow $
    -10 45 3.250 1.555 1.783 2935.27 2459.37 2638.61 $ \downarrow $ $ \uparrow $
    -30 35 3.619 1.784 2.044 2789.42 2218.06 2529.50 $ \downarrow $ $ \uparrow $
    -50 25 3.873 1.976 2.423 2642.17 2049.62 2485.73 $ \downarrow $ $ \uparrow $
     | Show Table
    DownLoad: CSV

    Table 4.  Sensitivity analysis for various parameters involved in Example 1

    Para meter $ \% $ change value $ T_1^* $ $ T_2^* $ $ T_3^* $ $ R_1(T) $ $ R_2(T) $ $ R_3(T) $ Total Cost Profit
    $ r_2 $ +50 60 2.402 2.265 1.875 3891.57 3124.51 3054.36 $ \downarrow $ $ \uparrow $
    +30 52 2.257 2.203 1.713 3522.56 2910.28 2839.17 $ \downarrow $ $ \uparrow $
    +10 44 2.105 2.165 1.681 3205.84 2764.33 2780.10 $ \downarrow $ $ \uparrow $
    -10 36 1.943 2.081 1.578 2935.27 2459.37 2638.61 $ \downarrow $ $ \uparrow $
    -30 28 1.870 1.863 1.475 2789.42 2218.06 2529.50 $ \downarrow $ $ \uparrow $
    -50 20 1.655 1.756 1.409 2642.17 2049.62 2485.73 $ \downarrow $ $ \uparrow $
    $ r_3 $ +50 15 2.795 2.531 2.320 3891.57 3124.51 3054.36 $ \downarrow $ $ \uparrow $
    +30 13 2.661 2.457 2.259 3522.56 2910.28 2839.17 $ \downarrow $ $ \uparrow $
    +10 11 2.573 2.347 2.188 3205.84 2764.33 2780.10 $ \downarrow $ $ \uparrow $
    -10 9 2.412 2.201 2.096 2935.27 2459.37 2638.61 $ \downarrow $ $ \uparrow $
    -30 7 2.345 2.179 2.016 2789.42 2218.06 2529.50 $ \downarrow $ $ \uparrow $
    -50 5 2.235 2.153 1.875 2642.17 2049.62 2485.73 $ \downarrow $ $ \uparrow $
    $ r_4 $ +50 60 1.987 1.853 1.760 3891.57 3124.51 3054.36 $ \downarrow $ $ \uparrow $
    +30 52 1.664 1.571 1.483 3522.56 2910.28 2839.17 $ \downarrow $ $ \uparrow $
    +10 44 1.370 1.356 1.338 3205.84 2764.33 2780.10 $ \downarrow $ $ \uparrow $
    -10 36 1.297 1.283 1.275 2935.27 2459.37 2638.61 $ \downarrow $ $ \uparrow $
    -30 28 1.264 1.257 1.241 2789.42 2218.06 2529.50 $ \downarrow $ $ \uparrow $
    -50 20 1.239 1.221 1.195 2642.17 2049.62 2485.73 $ \downarrow $ $ \uparrow $
    $ r_5 $ +50 45 2.354 2.279 2.165 3567.10 3049.31 3098.21 $ \downarrow $ $ \uparrow $
    +30 39 2.170 2.086 1.952 3417.05 2994.22 2947.18 $ \downarrow $ $ \uparrow $
    +10 33 1.941 1.876 1.740 3307.53 2885.31 2856.37 $ \downarrow $ $ \uparrow $
    -10 27 1.769 1.635 1.509 3268.37 2765.42 2664.00 $ \downarrow $ $ \uparrow $
    -30 21 1.526 1.470 1.358 3191.30 2670.51 2509.11 $ \downarrow $ $ \uparrow $
    -50 15 1.391 1.258 1.174 3034.67 2500.00 2481.03 $ \downarrow $ $ \uparrow $
    $ \beta $ +50 0.45 1.310 1.465 1.589 2473.11 1730.22 2017.27 $ \uparrow $ $ \downarrow $
    +30 0.39 1.574 1.640 1.736 2705.43 1918.47 2384.53 $ \uparrow $ $ \downarrow $
    +10 0.33 1.812 1.922 1.989 3058.24 2164.38 2673.40 $ \uparrow $ $ \downarrow $
    -10 0.27 2.123 2.070 2.145 3347.80 2351.06 2945.82 $ \uparrow $ $ \downarrow $
    -30 0.21 2.492 2.195 2.287 3692.47 2537.64 3255.70 $ \uparrow $ $ \downarrow $
    -50 0.15 2.760 2.358 2.460 3903.12 2799.00 3509.35 $ \uparrow $ $ \downarrow $
    $ \delta $ +50 30 2.385 2.072 2.110 3392.46 2918.36 2840.79 $ \downarrow $ $ \uparrow $
    +30 26 2.200 1.915 2.048 3175.13 2875.42 2711.54 $ \downarrow $ $ \uparrow $
    +10 22 2.071 1.854 1.991 2917.00 2704.60 2657.18 $ \downarrow $ $ \uparrow $
    -10 18 1.848 1.774 1.867 2739.26 2538.11 2302.63 $ \downarrow $ $ \uparrow $
    -30 14 1.639 1.640 1.790 2543.35 2326.83 2199.29 $ \downarrow $ $ \uparrow $
    -50 10 1.325 1.573 1.615 2319.67 2100.00 1978.63 $ \downarrow $ $ \uparrow $
    $ M $ +50 1.8 1.367 1.470 1.585 3141.47 2758.10 2890.05 $ \uparrow $ $ \downarrow $
    +30 1.56 1.532 1.609 1.779 3250.26 2893.55 2918.62 $ \uparrow $ $ \downarrow $
    +10 1.32 1.727 1.826 1.963 3319.51 2964.28 3022.37 $ \uparrow $ $ \downarrow $
    -10 1.08 1.904 1.995 2.057 3471.43 3027.83 3126.29 $ \uparrow $ $ \downarrow $
    -30 0.84 2.157 2.257 2.376 3520.79 3119.44 3257.81 $ \uparrow $ $ \downarrow $
    -50 0.6 2.370 2.450 2.538 3752.00 3348.63 3486.90 $ \uparrow $ $ \downarrow $
    $ \theta $ +50 0.6 2.752 2.568 2.330 4180.57 3042.78 3506.79 $ \downarrow $ $ \uparrow $
    +30 0.52 2.695 2.418 2.257 3912.36 2965.03 3481.20 $ \downarrow $ $ \uparrow $
    +10 0.44 2.516 2.343 2.194 3764.25 2860.24 3358.18 $ \downarrow $ $ \uparrow $
    -10 0.36 2.404 2.267 2.210 3580.19 2725.57 3293.49 $ \downarrow $ $ \uparrow $
    -30 0.28 2.279 2.001 1.935 3318.38 2530.29 3167.16 $ \downarrow $ $ \uparrow $
    -50 0.20 2.011 1.883 1.520 3127.93 2350.03 2948.57 $ \downarrow $ $ \uparrow $
     | Show Table
    DownLoad: CSV
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