Journal of Naval Architecture and Marine Engineering December, 2016 http://dx.doi.org/10.3329/jname.v13i2.23905 http://www.banglajol.info
ESTIMATION OF INVENTORY FOR DETERIORATING ITEMS WITH NON-LINEAR DEMAND CONSIDERING DISCOUNT AND WITHOUT DISCOUNT POLICY M. Mondal1*, M. F. Uddin2, M. S.Hossain3 1
Assistant Professor, Basic Science Division, World University of Bangladesh, Dhaka-1205, Bangladesh, *Email:
[email protected] Professor, Department of Mathematics,Bangladesh University of Engineering & Technology, Dhaka-1000, Bangladesh, E-mail:
[email protected] 3 Lecturer, Department of Mathematics, Hamdard University Bangladesh, E-mail:
[email protected] 2
Abstract: This paper develops an inventory model for deteriorating items consisting of the ordering cost, unit cost, opportunity cost, deterioration cost and shortage cost. In this inventory model instead of linear demand function nonlinear exponential function of time for deteriorating items with deterioration rate has been considered. The effects of inflation and cash flow are also taken into account under a trade-credit policy of discount and without discount with time. In order to validate the model, numerical examples have been solved by bisection method deploying MATLAB. Further, in order to estimate the cash flow the sensitivity of different parameters is considered. Keywords: Inventory, deterioration, non-linear demand, inflation, trade-credit policy
NOMENCLATURE D(t) T I C(t) A(t) h r
: : : : : : : : :
is the demand rate at any time t Cycle length Order quantity in the cycle Inventory Carrying charge Unit cost of the item at any time t Ordering cost of the item at any time t Inflation rate per unit time Opportunity cost per unit time Rate of deterioration
1. Introduction Inventory or stock refers to the goods and materials that a business holds for the ultimate purpose of resale (or repair). Inventory management is a science primarily about specifying the shape and percentage of stock goods. It is required at different locations within a facility or within many locations of a supply network to precede the regular and planned course of production and stock of materials. The scope of inventory management concerns the fine lines between replenishment lead time, carrying costs of inventory, asset management, inventory forecasting, inventory valuation, inventory visibility, future inventory price forecasting, physical inventory, available physical space for inventory, quality management, replenishment, returns and defective goods, and demand forecasting. Balancing these competing requirements leads to optimal inventory levels, which is an ongoing process as the business needs shift and react to the wider environment. Inventory management involves a retailer seeking to acquire and maintain a proper merchandise assortment while ordering, shipping, handling, and related costs are kept in check. It also involves systems and processes that identify inventory requirements, set targets, provide replenishment techniques, report actual and projected inventory status and handle all functions related to the tracking and management of material. This would include the monitoring of material moved into and out of stockroom locations and the reconciling of the inventory balances. Supply Chain Management (SCM) is an essential element for operational efficiency. SCM can be applied to customer satisfaction and company success, as well as within societal settings, including medical missions; disaster relief operations and other kinds of emergencies; cultural evolution; and it can help improve quality of 1813-8235 (Print), 2070-8998 (Online) © 2016 ANAME Publication. All rights reserved.
Received on: June, 2015
M. Mondal, M. F. Uddin, M. S. Hossain /Journal of Naval Architecture and Marine Engineering, 13(2016) 151-164
life. Because of the vital role SCM plays within organizations, employers seek employees with an abundance of SCM skills and knowledge. Deterioration of physical goods is one of the important factors in any inventory and production system. The deteriorating items with shortages have received much attention of several researches in the recent year because most of the physical goods undergo decay or deterioration over time. Commodities such as fruits, vegetables and food stuffs are kept in store by direct decay trick from reduction. The demand function relates price and quantity. It tells how many units of a goods will be purchased at different prices. In general, at higher prices, less will be purchased. Thus, the graphical representation of the demand function has a negative slope. The market demand function is calculated by adding up all of the individual consumers' demand functions. Economic order quantity (EOQ) model is the method that provides the company with an order quantity. By using this model, the companies can minimize the costs associated with the ordering and inventory holding. In 1913, Ford W. Harris developed this formula whereas R. H. Wilson is given credit for the application and indepth analysis on this model. The model is used to calculate the optimal quantity which can be purchased or produced to minimize the cost of both the carrying inventory and the processing of purchase orders or production set-ups. Basu et al. (2007) proposed a general inventory model with due consideration to the factors of time dependent partial backlogging and time dependent deterioration. Sharma, et al. (2013) established an inventory model for deteriorating items, the rate of deterioration following the Weibull distribution with two parameters. The demand rate is time dependent. Ghare and Schrader (1963) industrialized a model for an exponentially decaying inventory. Inventory models with a time dependent rate of deterioration were considered by Covert and Philip (1973), Mishra (1975) and Deb and Chaudhuri (1986). In these models shortages are completely backlogged. In developing mathematical inventory model, it is assumed that payments will be made to the supplier for the goods immediately after receiving the consignment. In day-to-day dealings, it is observed that the suppliers offer different trade credit policies to the buyers. One such trade credit policy is " net T" which means that a discount on sale price is granted if payments are made within days and the full sale price is due within days from the date of invoice if the discount is not taken. Ben-Horim and Levy (1982), Chung (1989), Aggarwal and Jaggi (1994) discussed trade-credit policy of type " net T" in their models. Aggarwal et al. (1997) investigated an inventory model taking into consideration inflation, time- value of money and a trade credit policy " net T" with a constant demand rate. Lal and Staelin (1994) examined an optimal discounting pricing policy. In this paper, an attempt has been made to extend the model of Aggarwal et al. (1997) to an inventory of deteriorating items at a constant rate . Also the demand rate is taken to be non-linear instead of a constant demand rate. Sensitivity of the optimal solution is examined to see how far the output of the model is affected by changes in the values of its input parameters. So far I know in literature inventory with non-linear demand for deteriorating items has not considered. That is why in our model we have considered deteriorating with shortage inventory as a non-linear demand function. The objectives of this study are to (i) develop an inventory model for deteriorating items with deterioration rate when the demand is considered as a nonlinear function of time and, (ii) determine an optimal solution of the inventory model with cash flow. Further, it is considered that a percent discount of sale price is granted if payments are made within days and the full sale price is due within M days from the date of invoice if the discount is not taken.
1.1
Outline of the paper
In this paper we have introduced several elementary properties of the stationary inventory process and literature review in Section 1. We have also constructed the mathematical formulation and solved it in Section 2. Section 3 describes the numerical results and graphically representation of the formulated model. Finally, we have summarized the results and proposed directions for future research in Section 4.
Estimation of inventory for deteriorating items with non-linear demand considering discount and without discount policy
152
M. Mondal, M. F. Uddin, M. S. Hossain /Journal of Naval Architecture and Marine Engineering, 13(2016) 151-164
2.
Mathematical Formulation
The exponential function has the form: where is a constant called the base of the exponential function, and tis the independent variable. Thus exponential function has a constant base raised to a variable exponent. In economics exponential functions are important for looking at growth or decay. Examples are the value of an investment that increases by a constant percentage each period, sales of a company that increase at a constant percentage each period, models of economic growth or models of the spread of an epidemic. Mathematically
and
Following assumptions are made: 1. 2. 3. 4. Let
Lead time is zero. Shortage is not allowed. Replenishment is instantaneous. The time horizon of the inventory system is infinite. and
at any time interval
be the ordering cost and unit cost of the item at any time t. Then and , assuming continuous compounding of inflation. Let be the instantaneous inventory level in the th cycle. The differential equation governing the instantaneous states of in the is (2.1)
, Where
, i=0,1,2….
and
(2.2)
Now from equation (2.1) (2.3)
When [
When [
Let
2.1
]
(2.4)
]
be the replenishment points and
(2.5)
so that
With discount
Here purchases made at time the th cycle is
are paid after
The present worth of cash-flows for the
days and purchase price in real terms for
units at time
in
th cycle is
Estimation of inventory for deteriorating items with non-linear demand considering discount and without discount policy
153
M. Mondal, M. F. Uddin, M. S. Hossain /Journal of Naval Architecture and Marine Engineering, 13(2016) 151-164
∫
Where
(assuming R=r-h and using
∫
Now,
(
∫ (
)
(
)
(
(2.6)
)
)
Therefore,
[ (
)
(2.7) ]
taking P=2r-h The present worth of all future cash flows is
∑
(
The solution of the equation
Now
)
(2.8)
gives the optimum value of T provided it satisfies the condition (2.9)
yields the equation
Estimation of inventory for deteriorating items with non-linear demand considering discount and without discount policy
154
M. Mondal, M. F. Uddin, M. S. Hossain /Journal of Naval Architecture and Marine Engineering, 13(2016) 151-164
[ ]
(
)
(
)
(2.10)
This equation being highly nonlinear cannot be solved analytically. It can be solved numerically for given parameter values. Its solution gives the optimum quantities from and , the optimum present value of all future cash flows from (2.8).
2.2
Without discount
Here purchases made at time in the
th cycle is
The present worth of cash flows for the
th cycle is
[
(2.11) ]
[ (taking
(
)]
,
) i=0,1,2,…….
The present worth of all future cash flows is ∑
(2.12) The solution of the equation
gives the optimum value of T provided it satisfies the condition (2.13)
Now
yields the equation
Estimation of inventory for deteriorating items with non-linear demand considering discount and without discount policy
155
M. Mondal, M. F. Uddin, M. S. Hossain /Journal of Naval Architecture and Marine Engineering, 13(2016) 151-164
[
]
(2.14) (
)
(
)
This equation also needs to be solved numerically.
3.
Result Discussion and Computation Analysis
In this section, a numerical example is given to illustrate this maintenance model. We consider parameter numerical example to valid our model for discount. Let in appropriate units. Solving the highly non-linear equation (2.10) for with discount by Bisection method, we get the optimum value of T, Substituting T in equation (2.8), we get the optimum value of as . We consider parameter numerical example to valid our model for without discount. Let appropriate units. Solving the equation (2.13) for without discount by the same method as in with discount case, we get the optimum value of T, . Substituting T in equation (2.8), we get the optimum value of as . Based on the numerical examples considered above, a sensitivity analysis of T, , is performed by changing (increasing or decreasing) the parameters by 10% and 50% and taking one parameter at a time, keeping the remaining parameters at their original values. The percentage error is calculated by the formula: (measured value- actual value) *100/actual value.
4.
Sensitivity Analysis
Sensitivity analysis of the model is also performed with respect to unit price of purchase, replenishment cost, inventory carrying cost, deteriorating cost and inflation rate. The model also judges the sensitive analysis of the parameters on ordering quantity and total system cost. Those results provided further insight in the nature of the problem studied and summarizes in Table 2.1 and 2.2.
Estimation of inventory for deteriorating items with non-linear demand considering discount and without discount policy
156
M. Mondal, M. F. Uddin, M. S. Hossain /Journal of Naval Architecture and Marine Engineering, 13(2016) 151-164
Table 2.1: Sensitivity of different parameters with discount Changing Parameters
A(0)
b
C(0)
r
I
(%) Change
T*
-50 -40 -30 -20 -10 +10 +20 +30 +40 +50 -50 -40 -30 -20 -10 +10 +20 +30 +40 +50 -50 -40 -30 -20 -10 +10 +20 +30 +40 +50 -50 -40 -30 -20 -10 +10 +20 +30 +40 -50 -40 -30 -20 -10 +10 +20 +30 +40 +50
149.108 154.808 159.659 163.882 167.622 174.023 176.809 179.378 181.761 183.983 193.281 187.381 182.411 178.119 174.345 167.941 165.176 162.639 160.296 158.121 193.281 187.381 182.411 178.119 174.345 167.941 165.176 162.639 160.296 158.121 291.113 242.767 211.944 189.029 156.294 144.082 133.749 124.884 117.189 175.909 174.858 173.842 172.857 171.903 170.079 169.205 168.355 167.528 166.723
(%) change in T* 1128.639 1338.683 1547.681 1755.867 1963.404 2376.962 2583.135 2788.977 2994.528 3199.824 2111.643 2124.891 2137.246 2148.875 2159.899 2180.467 2190.134 2199.450 2208.453 2217.173 2111.643 2124.891 2137.246 2148.875 2159.899 2180.467 2190.134 2199.450 2208.453 2217.173 3397.790 2610.291 2355.637 2236.697 2129.201 2101.584 2082.018 2067.554 2056.495 2150.433 2154.509 2158.544 2162.537 2166.491 2174.286 2178.130 2181.940 2185.718 2189.464
-12.791 -9.457 -6.620 -4.150 -1.963 +1.781 +3.411 +4.913 +6.307 +7.606 +13.044 9.594 6.686 4.177 1.969 -1.776 -3.393 -4.876 -6.247 -7.519 +13.044 9.594 6.686 4.177 1.969 -1.776 -3.393 -4.876 -6.247 -7.519 70.263 41.987 23.960 10.557 -8.587 -15.730 -21.773 -26.958 -31.459 2.884 2.269 1.675 1.099 0.541 -0.525 -1.036 -1.533 -2.017 -2.489
(%) change in -47.999 -38.321 -28.691 -19.099 -9.538 +9.516 +19.016 +28.500 +37.971 +47.429 -2.707 -2.097 -1.527 -0.992 -0.484 0.463 0.909 1.338 1.752 +2.155 -2.707 -2.097 -1.527 -0.992 -0.484 0.463 0.909 1.338 1.752 +2.155 56.550 20.267 8.534 3.054 -1.898 -3.170 -4.072 -4.738 -5.248 -0.920 -0.732 -0.546 -0.362 -0.180 0.178 0.355 0.531 0.705 0.878
Estimation of inventory for deteriorating items with non-linear demand considering discount and without discount policy
157
M. Mondal, M. F. Uddin, M. S. Hossain /Journal of Naval Architecture and Marine Engineering, 13(2016) 151-164
Changing Parameters
M
h
α
θ
ρ
(%) Change
T*
-50 -40 -30 -20 -10 +10 +20 +30 +40 +50 -50 -40 -30 -20 -10 +10 +20 +30 +40 +50 -50 -40 -30 -20 -10 +10 +20 +30 +40 +50 -50 -40 -30 -20 -10 +10 +20 +30 +40 +50 -50 -40 -30 -20 -10 +10 +20 +30 +40 +50
180.588 178.660 176.735 174.813 172.894 169.065 167.156 165.250 163.347 161.449 130.186 136.895 144.226 152.270 161.142 181.950 194.277 208.238 224.212 242.734 169.254 169.591 169.932 170.277 170.625 171.334 171.695 172.060 172.429 172.803 230.092 214.712 201.544 190.080 179.974 162.904 155.611 148.985 142.936 137.388 216.109 205.507 195.993 187.219 178.942 163.171 155.374 147.433 139.160 130.301
2203.977 2196.805 2189.869 2183.162 2176.677 2164.346 2158.488 2152.827 2147.357 2142.073 2116.476 2123.462 2131.847 2142.042 2154.619 2190.633 2217.195 2253.154 2303.732 2378.502 2176.048 2174.929 2173.807 2172.679 2171.545 2169.262 2168.112 2166.957 2165.795 2164.629 2065.298 2084.253 2104.347 2125.471 2147.524 2194.032 2218.319 2243.194 2268.592 2294.453 2063.221 2080.166 2098.918 2119.852 2143.461 2201.606 2238.375 2282.688 2337.666 2408.604
(%) change in T* -5.573 -4.462 -3.350 -2.235 -1.118 1.120 2.243 3.367 4.492 5.620 -23.858 -19.933 -15.646 -10.941 -5.752 6.417 13.626 21.792 31.135 41.968 -1.008 -0.810 -0.611 -0.409 -0.206 0.208 0.419 0.633 0.848 1.068 34.574 25.578 17.877 11.172 5.262 -4.721 -8.987 -12.862 -16.401 -19.646 26.396 20.195 14.630 9.499 4.658 -4.565 -9.125 -13.770 -18.609 -23.791
(%) change in 1.547 1.216 0.896 0.587 0.288 -0.279 -0.549 -0.809 -1.061 -1.305 -2.485 -2.162 -1.776 -1.306 -0.727 0.931 2.155 3.812 6.142 9.588 0.259 0.208 0.156 0.104 0.052 -0.052 -0.105 -0.158 -0.212 -0.266 -4.842 -3.969 -3.043 -2.070 -1.054 1.088 2.207 3.353 4.523 5.715 -4.939 -4.157 -3.293 -2.329 -1.241 3.131 1.437 5.173 7.706 10.975
Estimation of inventory for deteriorating items with non-linear demand considering discount and without discount policy
158
M. Mondal, M. F. Uddin, M. S. Hossain /Journal of Naval Architecture and Marine Engineering, 13(2016) 151-164
Table 2.2: Sensitivity of different parameters without discount Changing Parameters
A(0)
b
C(0)
r
I
(%) Change
T*
-50 -40 -30 -20 -10 +10 +20 +30 +40 +50 -50 -40 -30 -20 -10 +10 +20 +30 +40 +50 -50 -40 -30 -20 -10 +10 +20 +30 +40 +50 -50 -40 -30 -20 -10 +10 +20 +30 +40 -50 -40 -30 -20 -10 +10 +20 +30 +40 +50
148.941 154.640 159.489 163.711 167.451 173.851 176.637 179.206 181.588 183.809 193.107 187.208 182.238 177.946 174.173 167.770 165.006 162.469 160.127 157.952 193.107 187.208 182.238 177.946 174.173 167.770 165.006 162.469 160.127 157.951 290.809 242.507 211.721 188.835 156.142 143.944 133.624 124.769 117.084 175.737 174.686 173.670 172.686 171.732 169.908 169.034 168.185 167.358 166.553
1129.043 1339.123 1548.151 1756.367 1963.930 2377.536 2583.731 2789.593 2995.165 3200.479 2112.013 2125.303 2137.696 2149.361 2160.418 2181.049 2190.744 2200.089 2209.119 2217.864 2112.013 2125.303 2137.696 2149.361 2160.418 2181.049 2190.744 2200.089 2209.119 2217.864 3400.421 2611.769 2356.630 2237.420 2129.636 2101.936 2082.308 2067.795 2056.698 2150.912 2155.004 2159.053 2163.060 2167.028 2174.851 2178.709 2182.534 2186.325 2190.085
(%) change in T* -12.802 -9.465 -6.625 -4.153 -1.964 1.782 3.413 4.917 6.312 7.612 13.056 9.602 6.692 4.180 1.970 -1.777 -3.396 -4.881 -6.252 -7.526 13.056 9.602 6.692 4.180 1.970 -1.777 -3.396 -4.881 -6.252 -7.526 70.256 41.977 23.953 10.554 -8.585 -15.726 -21.768 -26.952 -31.453 2.886 2.271 1.676 1.100 0.541 -0.526 -1.037 -1.534 -2.019 -2.491
(%) change in -47.993 -38.316 -28.688 -19.097 -9.536 9.515 19.013 28.495 37.965 47.422 -2.715 -2.103 -1.532 -0.994 -0.485 0.464 0.911 1.341 1.757 2.161 -2.715 -2.103 -1.532 -0.994 -0.485 0.464 0.911 1.341 1.757 2.161 56.632 20.304 8.552 3.061 -3.179 -1.903 -4.083 -4.751 -5.263 -0.923 -0.734 -0.548 -0.363 -0.181 0.179 0.357 0.533 0.707 0.881
Estimation of inventory for deteriorating items with non-linear demand considering discount and without discount policy
159
M. Mondal, M. F. Uddin, M. S. Hossain /Journal of Naval Architecture and Marine Engineering, 13(2016) 151-164
Changing Parameters
M
h
θ
ρ
(%) Change
T*
-50 -40 -30 -20 -10 +10 +20 +30 +40 +50 -50 -40 -30 -20 -10 +10 +20 +30 +40 +50 -50 -40 -30 -20 -10 +10 +20 +30 +40 +50 -50 -40 -30 -20 -10 +10 +20 +30 +40 +50
159.700 161.911 164.128 166.349 168.576 173.042 175.281 177.525 179.772 182.024 128.841 135.738 143.276 151.551 160.680 182.108 194.809 209.201 225.672 244.771 229.881 214.510 201.351 189.895 179.796 162.740 155.453 148.833 142.788 137.245 215.934 205.332 195.819 187.046 178.770 163.001 155.206 147.266 138.995 130.139
2210.808 2202.205 2193.929 2185.970 2178.316 2163.886 2157.089 2150.558 2144.284 2138.258 2120.718 2127.104 2134.837 2144.313 2156.087 2190.114 2215.392 2249.761 2298.300 2370.336 2065.539 2084.557 2104.713 2125.900 2148.014 2194.644 2218.991 2243.925 2269.380 2295.299 2063.416 2080.417 2099.232 2120.236 2143.923 2202.261 2239.152 2283.612 2338.771 2409.942
(%) change in T* -6.502 -5.207 -3.910 -2.609 -1.305 1.308 2.619 3.933 5.248 6.567 -24.568 -20.531 -16.118 -11.273 -5.928 6.616 14.052 22.478 32.121 43.302 34.585 25.586 17.882 11.175 5.263 -4.722 -8.988 -12.864 -16.403 -19.648 26.420 20.213 14.643 9.507 4.662 -4.569 -9.133 -13.781 -18.624 -23.809
(%) change in 1.836 1.439 1.058 0.691 0.338 -0.325 -0.638 -0.939 -1.228 -1.506 -2.314 -2.020 -1.663 -1.227 -0.684 0.882 2.046 3.629 5.865 9.183 -4.856 -3.979 -3.051 -2.075 -1.056 1.091 2.212 3.361 4.533 5.727 -4.954 -4.170 -3.303 -2.336 -1.2453 1.441 3.141 5.189 7.729 11.008
Figures 2.1 (a), 2.2 (a), 2.3(a), 2.4(a), 2.5(a) and 2.6(a) show the estimated significant values of the parameters ordering cost, unit cost, opportunity cost, inventory carrying charge, inflation rate and deterioration rate with discount. On the other hand, Figures 2.1 (b), 2.2 (b), 2.3(b), 2.4(b), 2.5(b), and 2.6(b) show the estimated significant values of the parameters ordering cost, unit cost, opportunity cost, inventory carrying charge inflation rate and deterioration rate without discount.
Estimation of inventory for deteriorating items with non-linear demand considering discount and without discount policy
160
M. Mondal, M. F. Uddin, M. S. Hossain /Journal of Naval Architecture and Marine Engineering, 13(2016) 151-164
a.
b. Ordering cost vs cash flow
4000
3500 3000 2500 2000 1500 1000 500 0
3000 Cash flow
Cash flow
Ordering cost vs cash flow
2000 1000 0
2
3
4 5 6 7 8 9 10 Ordering cost of different case
1
11
2
3
4
5
6
7
8
9
10
9
10
Ordering cost of different case
Figure 2.1: Effect of ordering cost on cash flow; (a) Discount Case, (b) Without discount Case
a.
b. Unit cost vs cash flow
2250
2250
2200
2200 Cash flow
Cash flow
Unit cost vs cash flow
2150
2150 2100
2100 2050
2050 1
2
3
4 5 6 7 8 Unit cost of different case
9
10
1
2
3 4 5 6 7 8 Unit cost of different case
Figure 2.2: Effect of unit cost on cash flow: (a) Discount Case; (b) Without discount case
a.
b. Opportunity cost vs cash flow 4000 3500 3000 Cash flow
Cash flow
Opportunity cost vs cash flow 4000 3500 3000 2500 2000 1500 1000 500 0
2500 2000 1500 1000 500
1
2
3 4 5 6 7 8 Opportunity cost of different case
9
0 1
2 3 4 5 6 7 8 Opportunity cost of different case
Figure 2.3: Effect of Opportunity cost on cash flow: (a) Discount Case; (b) Without discount Case
Estimation of inventory for deteriorating items with non-linear demand considering discount and without discount policy
161
9
M. Mondal, M. F. Uddin, M. S. Hossain /Journal of Naval Architecture and Marine Engineering, 13(2016) 151-164
a.
b. Inventory carrying charge vs cash flow 2200
2190
2190
2180
2180 Cash flow
Cash flow
2200
Inventory carrying charge vs cash flow
2170 2160 2150
2170 2160 2150 2140
2140
2130
2130 1
1
2 3 4 5 6 7 8 9 10 Inventory carrying charge of different case
2 3 4 5 6 7 8 9 10 Inventory carrying charge of different case
Figure2.4: Effect of Inventory carrying charge on cash flow: (a) Discount Case; (b) Without discount case b. Inflation rate vs cash flow
2400 2350 2300 2250 2200 2150 2100 2050 2000 1950 1
2
3
4
5
6
7
8
9
Inflation rate vs cash flow
2400 2350 2300 2250 2200 2150 2100 2050 2000 1950
Cash flow
Cash flow
a.
10
1
2
Inflation rate of different case
3 4 5 6 7 8 Inflation rate of different case
9
10
Figure2.5: Effect of Inflation rate on cash flow: (a) Discount case; (b) Without discount case a.
b. 2350
2300
2300
2250
2250
2200
2200
Cash flow
Cash flow
Deterioration rate vs cash flow 2350
2150 2100
Deterioration rate vs cash flow
2150 2100
2050
2050
2000
2000 1950
1950 1
2 3 4 5 6 7 Deterioration rate of different case
8
9
10
1
2 3 4 5 6 7 8 9 Deterioration rate of different case
Figure 2.6: Effect of deterioration rate on cash flow: (a) Discount Case; (b) Without discount case It is observed that if the value of parameter ordering cost, unit cost, inflation rate, inventory carrying charge and deterioration rate increase then the cash flows of ordering cost, unit cost, inflation rate, inventory carrying Estimation of inventory for deteriorating items with non-linear demand considering discount and without discount policy
162
10
M. Mondal, M. F. Uddin, M. S. Hossain /Journal of Naval Architecture and Marine Engineering, 13(2016) 151-164
charge and deterioration rate increase in both case discount and without discount (see Figures 2.1, 2.2, 2.3, 2.5 and 2.6). On the other hand, the value of opportunity cost increase then the cash flows of opportunity cost decrease in both case discount and without discount (see Figure 2.3). Further, discount and without discount case economic order quantity and incurred cost decrease as ordering cost increases (see Tables 2.1 and 2.2), but this increment is nonlinear.
5. Conclusion The model has developed here deals with the optimum replenishment policy of a deteriorating item in the presence of inflation and a trade credit policy. It is observed that there is no significant change in the optimum value of the present worth of all future cash flows in the discount case and without discount case. It is observed that if the ordering cost and unit cost decrease, then the cash flow also decrease both cases. On the other hand, in both cases cash flow increases if opportunity cost decreases. Hence the discount case is considered to be better economically.
Acknowledgements The authors express their sincerest thanks to Bangladesh University of Engineering and Technology (BUET) Dhaka, Bangladesh for its infrastructural and financial support to carry out this work.
References Aggarwal, S.P. and Jaggi, C.K. (1994): Credit financing in economic ordering policies of deteriorating items, International Journal of Production Economics, 34, 151-155. https://doi.org/10.1016/0925-5273(94)90031-0 Aggarwal, K.K., Aggarwal, S.P. and Jaggi, C.K. (1997): Impact of inflation and credit policies on economic ordering, Bull. Pure Appl. Sci., 16(1) 93-100. Basu, M. and Sinha, S. (2007): An inflationary inventory model with time dependent demand with Weibull deterioration and partial backlogging under permissible delay in payments, Control and Cybernetics, vol. 36. Ben-Horim, M. and Levy, H. (1982): Inflation and the trade credit period, Management Science, 28(6) 646-651. https://doi.org/10.1287/mnsc.28.6.646 Chung, K.H. (1989): Inventory control and trade-credit revisited, Journal of Operational Research Society, 4(5) 495-498. https://doi.org/10.1057/jors.1989.77 Covert, R.P. and Philip, G. C. (1973): An EOQ model for items with Weibull distribution deterioration, AIIE Transactions, 5, 323-326. https://doi.org/10.1080/05695557308974918 Deb, M. and Chaudhuri, K. S. (1986): An EOQ model for items with finite rate of production and variable rate of deterioration, Operation research, 23, 175-181. Ghare, P.M. and Schrader, G.F. (1993): An inventory model for exponentially deteriorating items, Journal of Industrial Engineering, 14, 238-243. Lal, R., and Staelin, R. (1997): An approach for developing an optimal discount pricing policy, Management Science, 30 1524-1539. https://doi.org/10.1287/mnsc.30.12.1524 Mishra, R.B. (1975): Optimum production lot-size model for a system with deteriorating inventory, International Journal of Production Research, 1975; 3, 495-505. https://doi.org/10.1080/00207547508943019 Sharma, V. and Chaudhary, R.R. (2013): An inventory Model for deteriorating items with Weibull Deterioration with Time Dependent Demand and Shortages, Research Journal of Management Sciences, Vol. 2(3), 28-30.
Estimation of inventory for deteriorating items with non-linear demand considering discount and without discount policy
163
M. Mondal, M. F. Uddin, M. S. Hossain /Journal of Naval Architecture and Marine Engineering, 13(2016) 151-164
Appendix A Assuming
, we have ∑
∑
∑
∑
∑
∑
Estimation of inventory for deteriorating items with non-linear demand considering discount and without discount policy
164
