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March  2020, 16(2): 553-578. doi: 10.3934/jimo.2018167

## A two-warehouse probabilistic model with price discount on backorders under two levels of trade-credit policy

 1 Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore-721102, West Bengal, India 2 Faculty of Engineering Management, Chair of Marketing and Economic Engineering, Poznan University of Technology, ul. Strzelecka 11, 60-965 Poznan, Poland

* Corresponding author: sankroy2006@gmail.com

The research of Sankar Kumar Roy 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, University of Aveiro, Portugal, within project UID/MAT/04106/2013.
The author, Magfura Pervin is very much 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, Middle East Technical University, 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.

Received  January 2018 Revised  July 2018 Published  March 2020 Early access  October 2018

It is impossible in this competitive era to assess the demand for items in advance. So, it is essential to refer to a stochastic demand function. In this paper, a probabilistic inventory model for deteriorating items is unfolded. Here, the supplier as well as the retailer adopt the trade-credit policy for their customers with the aim of promoting the market competition. Shortages are included into the model, and when stock on hand is zero, the retailer offers a price discount to those customers who are willing to back-order their demands. We consider two different warehouses in which the first one is an Own Warehouse (OW) where the deterioration is constant over time and the other is a Rented Warehouse (RW), and where the deterioration rate follows a Weibull distribution. An algorithm is provided for finding the solutions of the formulated model.Global convexity of the cost function is established which shows that our proposed model is very helpful for any supplier or retailer to finalize the optimal ordering policy. Beside of this, we target to increase the total profit for retailer by reducing the corresponding total inventory cost. The theoretical concept is justified with the help of some numerical examples. A sensitivity analysis of the optimal solution with respect to the major parameters is also provided in order to stabilize our model. We finalize the paper through a conclusion and a preview onto possible future studies.

Citation: Sankar Kumar Roy, Magfura Pervin, Gerhard Wilhelm Weber. A two-warehouse probabilistic model with price discount on backorders under two levels of trade-credit policy. Journal of Industrial & Management Optimization, 2020, 16 (2) : 553-578. doi: 10.3934/jimo.2018167
##### References:

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##### References:
Characteristic path of the OW
Characteristic path of the RW
Convexity of the function of total cost
Convexity nature of total cost in case of joint effect
Convexity nature of total cost in case of joint effect
Variation of total cost $TC$ with respect to demand factor $x$
Variation of total cost $TC$ with respect to shape parameter $\alpha$
Variation of total cost $TC$ with respect to ordering cost $A$
Variation of total cost $TC$ with respect to OW capacity $W$
Previous works of different authors in this field including our work
 Author(s) Two warehouse Probabilistic demand Trade credit Deterio-rations Shortage Price discount Datta and Pal (1988) $\surd$ $\surd$ Bhunia and Maiti (1994) $\surd$ Shah and Shah (1998) $\surd$ $\surd$ Shah (1997) $\surd$ $\surd$ Palanivel et al. (2016) $\surd$ $\surd$ Jaggi et al. (2017) $\surd$ $\surd$ Benkherouf (1997) $\surd$ $\surd$ $\surd$ Singh et al. (2010) $\surd$ $\surd$ Kaliraman et al. (2017) $\surd$ $\surd$ $\surd$ Bhunia et al. (2014) $\surd$ $\surd$ $\surd$ Chung and Liao (2004) $\surd$ Yang (2006) $\surd$ $\surd$ $\surd$ De and Goswami (2009) $\surd$ $\surd$ $\surd$ Chung and Huang (2007) $\surd$ $\surd$ $\surd$ Kurdhi et al. (2015) $\surd$ $\surd$ Pervin et al. (2018) $\surd$ $\surd$ Pervin et al. (2017) $\surd$ $\surd$ $\surd$ $\surd$ Goyal (1985) $\surd$ $\surd$ $\surd$ Yang and Chang (2013) $\surd$ $\surd$ $\surd$ Sarkar et al. (2015) $\surd$ $\surd$ Ray and Chaudhuri (1997) $\surd$ $\surd$ Hariga (1995) $\surd$ $\surd$ Our paper $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
 Author(s) Two warehouse Probabilistic demand Trade credit Deterio-rations Shortage Price discount Datta and Pal (1988) $\surd$ $\surd$ Bhunia and Maiti (1994) $\surd$ Shah and Shah (1998) $\surd$ $\surd$ Shah (1997) $\surd$ $\surd$ Palanivel et al. (2016) $\surd$ $\surd$ Jaggi et al. (2017) $\surd$ $\surd$ Benkherouf (1997) $\surd$ $\surd$ $\surd$ Singh et al. (2010) $\surd$ $\surd$ Kaliraman et al. (2017) $\surd$ $\surd$ $\surd$ Bhunia et al. (2014) $\surd$ $\surd$ $\surd$ Chung and Liao (2004) $\surd$ Yang (2006) $\surd$ $\surd$ $\surd$ De and Goswami (2009) $\surd$ $\surd$ $\surd$ Chung and Huang (2007) $\surd$ $\surd$ $\surd$ Kurdhi et al. (2015) $\surd$ $\surd$ Pervin et al. (2018) $\surd$ $\surd$ Pervin et al. (2017) $\surd$ $\surd$ $\surd$ $\surd$ Goyal (1985) $\surd$ $\surd$ $\surd$ Yang and Chang (2013) $\surd$ $\surd$ $\surd$ Sarkar et al. (2015) $\surd$ $\surd$ Ray and Chaudhuri (1997) $\surd$ $\surd$ Hariga (1995) $\surd$ $\surd$ Our paper $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Effect of change in capacity of OW and trade-credit period
 $W$ $M$ $N$ $T^*$ $b(T^*)$ $TC$ Case 0.5 0.4 0.088 521 1923.41 $M\le T$ 800 0.9 0.8 0.097 560 1979.05 $N\le T $WMNT^*b(T^*)TC$Case 0.5 0.4 0.088 521 1923.41$M\le T$800 0.9 0.8 0.097 560 1979.05$N\le T
Sensitivity analysis for different parameters involved in Example 1
 Parameter $\%$ change value $T_1$ $T_2$ $T_3$ $T_4$ $T_5$ $T^*$ $b(T^*)$ $TC$ $\%$ change of TC +50 600 1 1.5 ... ... ... 0.274 594 2284.24 +36.64 $A$ +20 480 1 1.5 2 ... ... 0.240 573 2170.33 +28.41 -20 320 1 1.5 2 2.5 ... 0.208 557 2001.16 +9.71 -50 200 1 1.5 2 2.3 3 0.173 529 1981.07 -0.85 +50 90 1 1.5 ... ... ... 0.198 468 2478.21 +27.53 $h_o$ +20 72 1 1.5 2 ... ... 0.183 450 2356.34 +21.23 -20 48 1 1.5 2 2.5 ... 0.175 438 2213.08 +18.73 -50 30 1 1.5 2 2.5 3 0.166 426 2087.60 +9.39 +50 105 1 1.5 ... ... ... 0.098 537 3798.26 +47.07 $h_r$ +20 84 1 1.5 2 ... ... 0.082 522 3523.65 +31.67 -20 56 1 1.5 2 2.5 ... 0.076 517 3247.43 +23.81 -50 35 1 1.5 2 2.5 3 0.680 510 3068.11 -1.27 +50 1.2 1 1.5 ... ... ... 0.176 526 2109.87 +28.45 $\theta$ +20 0.96 1 1.5 2 ... ... 0.248 538 2084.21 +17.39 -20 0.64 1 1.5 2 2.5 ... 0.273 559 1985.06 -1.87 -50 0.4 1 1.5 2 2.5 3 0.296 570 1867.30 -2.64 +50 0.075 1 1.5 ... ... ... 0.211 523 2075.32 +12.37 $\alpha$ +20 0.06 1 1.5 2 ... ... 0.250 538 1924.00 +10.22 -20 0.04 1 1.5 2 2.5 ... 0.310 550 1775.71 -0.79 -50 0.025 1 1.5 2 2.5 3 0.352 567 1528.66 -2.85 +50 4.5 1 1.5 ... ... ... 0.560 413 1968.20 +25.75 $\beta$ +20 3.6 1 1.5 2 ... ... 0.581 399 1876.11 +10.29 -20 2.4 1 1.5 2 2.5 ... 0.615 307 1718.53 -2.20 -50 1.5 1 1.5 2 2.5 3 0.672 279 1528.04 -5.43 +50 0.12 1 1.5 ... ... ... 0.736 578 1727.34 +24.74 $\delta$ +20 0.096 1 1.5 2 ... ... 0.703 530 1783.05 +18.42 -20 0.064 1 1.5 2 2.5 ... 0.682 492 1816.11 +10.53 -50 0.04 1 1.5 2 2.5 3 0.644 454 1874.20 -8.64 +50 0.725 1 1.5 ... ... ... 0.675 649 2665.93 +43.25 $M$ +20 0.6 1 1.5 2 ... ... 0.510 687 2682.75 +21.36 -20 0.4 1 1.5 2 2.5 ... 0.509 760 2789.77 +13.85 -50 0.25 1 1.5 2 2.5 3 0.588 781 2895.34 +7.04 +50 0.6 1 1.5 ... ... ... 0.322 300 2541.11 +40.52 $N$ +20 0.48 1 1.5 2 ... ... 0.379 349 2562.47 +37.06 -20 0.32 1 1.5 2 2.5 ... 0.401 373 2580.63 -7.43 -50 0.2 1 1.5 2 2.5 3 0.419 388 2558.47 -8.65 +50 15 1 1.5 ... ... ... 0.411 644 1563.72 +15.27 $R$ +20 12 1 1.5 2 ... ... 0.458 541 1571.08 +9.04 -20 8 1 1.5 2 2.5 ... 0.392 520 1584.60 -10.11 -50 5 1 1.5 2 2.5 3 0.450 501 1599.01 -5.14 +50 75 1 1.5 ... ... ... 0.749 385 1932.84 +29.27 $c$ +20 60 1 1.5 2 ... ... 0.755 337 1920.03 +21.43 -20 40 1 1.5 2 2.5 ... 0.759 249 1907.32 +37.19 -50 25 1 1.5 2 2.5 3 0.780 277 1871.92 +22.48 +50 15 1 1.5 ... ... ... 0.753 495 1884.67 +23.42 $s$ +20 12 1 1.5 2 ... ... 0.734 327 1940.59 +17.99 -20 8 1 1.5 2 2.5 ... 0.690 224 1982.47 -2.33 -50 5 1 1.5 2 2.5 3 0.638 200 2027.59 -7.21 +50 1200 1 1.5 ... ... ... 0.922 540 1825.49 +29.36 $W$ +20 960 1 1.5 2 ... ... 0.870 511 1871.52 +22.15 -20 640 1 1.5 2 2.5 ... 0.761 487 1905.14 -5.22 -50 400 1 1.5 2 2.5 3 0.739 475 1917.26 -9.46 +50 75 1 1.5 ... ... ... 0.875 610 1932.34 +31.06 $x$ +20 60 1 1.5 2 ... ... 0.852 587 1956.07 +26.17 -20 40 1 1.5 2 2.5 ... 0.830 551 1988.23 +13.50 -50 25 1 1.5 2 2.5 3 0.781 513 2130.54 +4.21
 Parameter $\%$ change value $T_1$ $T_2$ $T_3$ $T_4$ $T_5$ $T^*$ $b(T^*)$ $TC$ $\%$ change of TC +50 600 1 1.5 ... ... ... 0.274 594 2284.24 +36.64 $A$ +20 480 1 1.5 2 ... ... 0.240 573 2170.33 +28.41 -20 320 1 1.5 2 2.5 ... 0.208 557 2001.16 +9.71 -50 200 1 1.5 2 2.3 3 0.173 529 1981.07 -0.85 +50 90 1 1.5 ... ... ... 0.198 468 2478.21 +27.53 $h_o$ +20 72 1 1.5 2 ... ... 0.183 450 2356.34 +21.23 -20 48 1 1.5 2 2.5 ... 0.175 438 2213.08 +18.73 -50 30 1 1.5 2 2.5 3 0.166 426 2087.60 +9.39 +50 105 1 1.5 ... ... ... 0.098 537 3798.26 +47.07 $h_r$ +20 84 1 1.5 2 ... ... 0.082 522 3523.65 +31.67 -20 56 1 1.5 2 2.5 ... 0.076 517 3247.43 +23.81 -50 35 1 1.5 2 2.5 3 0.680 510 3068.11 -1.27 +50 1.2 1 1.5 ... ... ... 0.176 526 2109.87 +28.45 $\theta$ +20 0.96 1 1.5 2 ... ... 0.248 538 2084.21 +17.39 -20 0.64 1 1.5 2 2.5 ... 0.273 559 1985.06 -1.87 -50 0.4 1 1.5 2 2.5 3 0.296 570 1867.30 -2.64 +50 0.075 1 1.5 ... ... ... 0.211 523 2075.32 +12.37 $\alpha$ +20 0.06 1 1.5 2 ... ... 0.250 538 1924.00 +10.22 -20 0.04 1 1.5 2 2.5 ... 0.310 550 1775.71 -0.79 -50 0.025 1 1.5 2 2.5 3 0.352 567 1528.66 -2.85 +50 4.5 1 1.5 ... ... ... 0.560 413 1968.20 +25.75 $\beta$ +20 3.6 1 1.5 2 ... ... 0.581 399 1876.11 +10.29 -20 2.4 1 1.5 2 2.5 ... 0.615 307 1718.53 -2.20 -50 1.5 1 1.5 2 2.5 3 0.672 279 1528.04 -5.43 +50 0.12 1 1.5 ... ... ... 0.736 578 1727.34 +24.74 $\delta$ +20 0.096 1 1.5 2 ... ... 0.703 530 1783.05 +18.42 -20 0.064 1 1.5 2 2.5 ... 0.682 492 1816.11 +10.53 -50 0.04 1 1.5 2 2.5 3 0.644 454 1874.20 -8.64 +50 0.725 1 1.5 ... ... ... 0.675 649 2665.93 +43.25 $M$ +20 0.6 1 1.5 2 ... ... 0.510 687 2682.75 +21.36 -20 0.4 1 1.5 2 2.5 ... 0.509 760 2789.77 +13.85 -50 0.25 1 1.5 2 2.5 3 0.588 781 2895.34 +7.04 +50 0.6 1 1.5 ... ... ... 0.322 300 2541.11 +40.52 $N$ +20 0.48 1 1.5 2 ... ... 0.379 349 2562.47 +37.06 -20 0.32 1 1.5 2 2.5 ... 0.401 373 2580.63 -7.43 -50 0.2 1 1.5 2 2.5 3 0.419 388 2558.47 -8.65 +50 15 1 1.5 ... ... ... 0.411 644 1563.72 +15.27 $R$ +20 12 1 1.5 2 ... ... 0.458 541 1571.08 +9.04 -20 8 1 1.5 2 2.5 ... 0.392 520 1584.60 -10.11 -50 5 1 1.5 2 2.5 3 0.450 501 1599.01 -5.14 +50 75 1 1.5 ... ... ... 0.749 385 1932.84 +29.27 $c$ +20 60 1 1.5 2 ... ... 0.755 337 1920.03 +21.43 -20 40 1 1.5 2 2.5 ... 0.759 249 1907.32 +37.19 -50 25 1 1.5 2 2.5 3 0.780 277 1871.92 +22.48 +50 15 1 1.5 ... ... ... 0.753 495 1884.67 +23.42 $s$ +20 12 1 1.5 2 ... ... 0.734 327 1940.59 +17.99 -20 8 1 1.5 2 2.5 ... 0.690 224 1982.47 -2.33 -50 5 1 1.5 2 2.5 3 0.638 200 2027.59 -7.21 +50 1200 1 1.5 ... ... ... 0.922 540 1825.49 +29.36 $W$ +20 960 1 1.5 2 ... ... 0.870 511 1871.52 +22.15 -20 640 1 1.5 2 2.5 ... 0.761 487 1905.14 -5.22 -50 400 1 1.5 2 2.5 3 0.739 475 1917.26 -9.46 +50 75 1 1.5 ... ... ... 0.875 610 1932.34 +31.06 $x$ +20 60 1 1.5 2 ... ... 0.852 587 1956.07 +26.17 -20 40 1 1.5 2 2.5 ... 0.830 551 1988.23 +13.50 -50 25 1 1.5 2 2.5 3 0.781 513 2130.54 +4.21
Sensitivity analysis for the combined effect of total cost involved in Example 1
 $\%$ change $x$ $M$ $N$ $\delta$ $T^*$ $b(T^*)$ $TC$ $\%$ change of TC +50 75 0.725 0.6 0.12 0.165 610 2685.00 +25.16 +40 70 0.70 0.56 0.112 0.137 589 2576.27 +32.00 +30 65 0.65 0.52 0.104 0.114 576 2450.08 +23.76 +20 60 0.6 0.48 0.096 0.105 558 2329.18 +21.34 +10 55 0.55 0.44 0.088 0.098 543 2249.71 +19.47 0 50 0.5 0.4 0.08 0.089 522 1923.46 ... -10 45 0.45 0.36 0.072 0.068 516 1879.34 +13.32 -20 40 0.4 0.32 0.064 0.062 511 1794.11 -11.06 -30 35 0.35 0.28 0.056 0.058 504 1720.57 +5.48 -40 30 0.30 0.24 0.048 0.051 497 1685.00 -2.21 -50 25 0.25 0.2 0.04 0.048 483 1649.27 -0.23
 $\%$ change $x$ $M$ $N$ $\delta$ $T^*$ $b(T^*)$ $TC$ $\%$ change of TC +50 75 0.725 0.6 0.12 0.165 610 2685.00 +25.16 +40 70 0.70 0.56 0.112 0.137 589 2576.27 +32.00 +30 65 0.65 0.52 0.104 0.114 576 2450.08 +23.76 +20 60 0.6 0.48 0.096 0.105 558 2329.18 +21.34 +10 55 0.55 0.44 0.088 0.098 543 2249.71 +19.47 0 50 0.5 0.4 0.08 0.089 522 1923.46 ... -10 45 0.45 0.36 0.072 0.068 516 1879.34 +13.32 -20 40 0.4 0.32 0.064 0.062 511 1794.11 -11.06 -30 35 0.35 0.28 0.056 0.058 504 1720.57 +5.48 -40 30 0.30 0.24 0.048 0.051 497 1685.00 -2.21 -50 25 0.25 0.2 0.04 0.048 483 1649.27 -0.23

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