In solving Equation (1) of the paper, which described the state of the inventory level at any time, t, in the inventory cycle, a mistake was made. We now solve the equation again so as to correct the mistake.
The differential equation is given as
To determine the correct solution of the equation, we proceed as follows:
. This is a linear differential equation where the Integrating factor is
so that the solution is
In trying to determine the integral on the right hand side of Equation (2), it is noted that the exponent is not linear so the exponential cannot be integrated directly. One needs to use the series solution of the exponential and then integrate term-by-term (see Wataru, 2014  ).
The Integral then becomes
where Q is a constant.
Applying the initial boundary condition i.e at t = 0, I(t) = I0
Substituting Equation (5) into Equation (4), we get
Applying the boundary condition at t = T, I(t) = 0 into Equation (6) gives
Substituting Equation (8) into Equation (6) gives
The total demand during the cycle period
is given as follows:
The number of deteriorated units is given as initial order quantity minus the total demand in the cycleperiod
. Thus the number of deteriorated units is
Deterioration cost (DC) for the cycle [O, T] = Ac × (the number of deteriorated units)
To find the total inventory holding cost (IHC) for the cycle [O, T] we note that
Therefore the total inventory holding cost (IHC) is given by
Substituting Equation (15) into Equation (16) gives
However, the total variable cost = Ordering cost (OC) + Deterioration cost + Inventory Holding cost (IHC) and the total variable Cost per unit time TC(T) is
The main objective is to find the cycle length T that will give the minimum variable cost per unit time. The necessary and sufficient conditions to minimize TC(T) are respectively,
Therefore to satisfy the necessary condition we have to differentiate Equation (19) with respect to T, as follows
We now equate Equation (20) to zero and simplify by multiplying with
on both sides in order to determine the T which minimizes the variable cost per unit time as follows:
The value of T obtained, gives the minimum cost provided it satisfies the following condition
Equation (21) is highly nonlinear and therefore difficult to solve by any analytic method. Likewise the same problem will exist in trying to check the inequality in (22) above. However, in all our examples below, we use direct search method with the help of EXCEL to obtain the root of the equation and also confirm that the sufficient Condition (21) is satisfied.
4. Numerical Example
To illustrate the model developed an example is considered based on the following values of parameters: N0 = ₦5000 per order, K = 250, C = ₦2.50 per unit, β= 1, a= 0.8, b= 2, i = 0.1 per Naira per unit time, Ac= ₦3.00, and h = 0.5, taking some parameter values from Dash et al. (2014) . Substituting and simplifying the above parameters into Equation (21), gives
(490 days). On substitution of this optimal value
in Equations (19) and (8), we obtain the minimum total cost per unit time
= ₦4578.88 and economic order quantity
units. Note that the
5. Sensitivity Analysis
A sensitivity analysis was carried out to study the effect of changes in the values of the system parameters N0, K, β, a, b, C, i, Acand h on the optimal length of the cycle (
), the economic order quantity (
) and the minimum total cost per unit time (
). The sensitivity analysis was performed for each of the parameters by changing its value by 50%, 25%, 5%, 2%, −2%, −5%. −25%, −50%, while keeping the remaining parameters at their original values. The analysis showed the following:
1) With increase in the value of the parameter a, the values of
increase. This is because when a increases, deterioration increases and so to avoid much deterioration the model forces
increases probably to compensate for the deteriorated units.
increases due to the cost of deterioration. The increase/decrease in the values is moderate hence the decision variables are moderately sensitive to changes in a.
2) With increase in the value of parameter b, the values of
decrease while the value of
increases. This is expected because when b increases, deterioration with respect to time also increases and so the model forces
increases due to the cost of deterioration. The decreases/increase in the values are moderate hence the decision variables are moderately sensitive to changes in b.
3) With increase in the value of parameter Ac, the values of
increases. This is also expected since when the cost of a deteriorated unit increases then the model will avoid much quantity in a supply and so both
increases because of the cost of deteriorated items. The increase/decrease in the values is moderate hence the decision variables are moderately sensitive to changes in Ac
4) With increase in the value of the parameter β,the values of
is not stable. This is not expected because when β increases, it is expected that
should decrease. The fact that
is not stable indicates that the model tries to adjust
to the minimum value, at the expense of increasing
. The increases/decrease in the values are low hence the variables are lowly sensitive to changes in β.
5) With increase in the value of the parameter N0, the values of
increase. This is also expected since when ordering cost increases then the model will avoid more orders and so both
will however increase due to increase in stockholding cost. The increases in the values are high hence the decision variables
are highly sensitive to changes in N0.
6) With increase in the value of parameter K, the values of
decreases. This is because when K increases, there will be more demand and so the economic order quantity (
) will increase. This will result in increase in the optimal total cost (
). The cycle period (
) decreases probably as a result of the model trying to avoid much deterioration. The increases/decrease in the values are moderate hence the decision variables are moderately sensitive to changes in K.
7) With increase in the value of parameter h, the values of
decreases. This is because as the demand increases the economic order quantity also increases, hence the total variable cost,
also increases. On the other hand however, the cycle period decreases which is probably due to higher demand. The increase in the values is moderate hence the decision variables are moderately sensitive to changes in h.
8) With increase in the value of parameteri, the values of
increases. This is expected because when the inventory carrying charge, i is increased there will be more stockholding cost so the model will avoid that by reducing quantity in an order which results in decreasing
increases due to the increase in the carrying charge (i). The increase/decrease in the values is moderate hence the decision variables are moderately sensitive to changes ini.
With increase in the value of parameter C, the values of
increases. This is also expected because when the unit cost of an item, C, is increased there will be more stockholding cost so the model will avoid that by reducing quantity in an order, which results in decreasing
increases due to the increase in the unit cost. The increase/decrease in the values is moderate hence the decision variables are moderately sensitive to changes in C.
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