Simulation of Economic Production Quantity Model for Deteriorating Item

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Figure 2. Production System after disruption

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,boundary condition

(1)

,boundary condition

(2)

On solving Equation (1) and Equation (2) with boundary conditions we get

(3)

(4)

As per Figure 1 inventory level I₁(t) and I₂(t) are equal at time T_p.

i.e. I₁(T_p) = I₂(T_p) yields

(5)

If θ <<1, then

(6)

Proposition 1. If θ <<1 then T_p is in increasing in θ.

By Equation (6) one can write

(7)

This proved the result_*

As θ increases optimal production time T_p increases that is more product required to producing. One can conclude that to keep low deterioration is effective way to keep lower cost of production of items.

4. Production System with Disruption

As previous section production rate remains unchanged but in practice production system is always disruption due to unplanned and thus we consider the production system little changed by and disruption time is T_d . If, then production rate decreases and, if then production rate increases.

Proposition 2.

then manufacturer still satisfies the demand even production system has been disrupted, otherwise If then production system unable to satisfy demand that is there will be shortages due to production disruption.

Proof: Suppose the production system disrupted at time T_d as (see Figure 2) and there after the production rate will be thus presentations of two differential equations for intervals [0, T_d] and [T_d, H] are

(8)

(9)

with boundary condition

On solving Equation (9) with boundary condition we get

(10)

If this means production system satisfy the demand of items.

That isthen still satisfy the demand.

If this means production system does not satisfy the demand of items.

That is then there will be shortages in the system.

This proved the lemma_*

Again if then we find optimal production time (with disruption) such that at time H entire stock will be sold-out and inventory level will be zero.

If there will be shortages in the system and in this situation we will find the optimum time T_r of placing the order and respective order quantity Q_r.

Proposition 3. If then production time with disruption is obtained by.

Proof: If

or that is on hand inventory is .

Therefore we will find out the optimal time(see Figure 2) when we stopped the production after disruption in such a manner that stock remains zero at time H. the presentations of two differential equations for intervals [T_{d ,}T_p^d] and [T_p^d, H] are

(11)

Boundary condition

, boundary condition

(12)

On solving (11) with boundary condition we get

(13)

(14)

Using condition

(15)

By Equation (10),

(16)

and thus by Equation (14), order quantity will be

(17)

This proved the result_*

4.1. An Application with Simulation

For application we assumed a particular case when p = 350, µ = 300, Δp = -100, θ = 0.01, H = 15 and T_d=1, on applying then we get I₂(H)>0 and thus by Equation (6) and Equation (15) gives T_p=0.14, = 13.71 days. We simulate the application on same data in which other parameters are invariant.

Figure 3. (Effect of Td on

)

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Figure 4. (Effect of H on

)

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Figure 5. (Effect of θ on

)

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Figure 6. (Comparison between

and T_p with and without disruption w.r.t. θ)

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Table 1. (Variations in parameters)

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