# A Production Model for Deteriorating Inventory Items with Production Disruptions

Authors: Yong He; Ju He
Journal: Discrete Dynamics in Nature and Society
 (2010)

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## Abstract

Disruption management has recently become an active area of research. In this study, an extension is made to consider the fact that some products may deteriorate during storage. A production-inventory model for deteriorating items with production disruptions is developed. Then the optimal production and inventory plans are provided, so that the manufacturer can reduce the loss caused by disruptions. Finally, a numerical example is used to illustrate the model.


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## Body

## 1. Introduction

In real life, the effect of decay and deterioration is very important in many inventory systems. In general, deterioration is defined as decay, damage, spoilage, evaporation, obsolescence, pilferage, loss of utility, or loss of marginal value of a commodity that results in decreasing usefulness [1]. Most of the physical goods undergo decay or deterioration over time, the examples being medicine, volatile liquids, blood banks, and others. Consequently, the production and inventory problem of deteriorating items has been extensively studied by researchers. Ghare and Schrader [2] were the first to consider ongoing deterioration of inventory with constant demand. As time progressed, several researchers developed inventory models by assuming either instantaneous or finite production with different assumptions on the patterns of deterioration. In this connection, researchers may refer to [3–7]. Interested readers may refer to review [8, 9]. Recently, several related papers were presented, dealing with such inventory problems [10–17].At the beginning of each cycle, the manufacturer should decide the optimal production time, so that the production quantity should satisfy the following two requirements: one, it should meet demand and deterioration; second, all products should be sold out in each cycle, that is, at the end of each cycle, the inventory level should decrease to zero. Some researchers have studied such production model for deteriorating items under different condition. For example, Yang and Wee [18] derived the optimal production time for a single-vendor, multiple-buyers system. Liao [19] derived a production model for the lot-size inventory system with finite production rate, taking into consideration the effect of decay and the condition of permissible delay in payments. Lee and Hsu [20] developed a two-warehouse inventory model with time-dependent demand. He et al. [21] provided a solution procedure to find the optimal production time under the premise that the manufacturer sells his products in multiple markets. The above papers all assume that production rate is known and keeps constant during each cycle. They do not consider how to adjust the production plan once the production rate is changed during production time.However, after the plan was implemented, the production run is often disrupted by some emergent events, such as supply disruptions, machine breakdowns, earthquake, H1N1 epidemic, financial crisis, political event and policy change. For example, the Swedish mining company Boliden AB suffered the production disruptions at its Tara zinc mine in Ireland due to an electric motor breakdown at one of the grinding mills. As a result of the breakdown, the production of zinc and lead concentrates is expected to fall by some 40% over the next six weeks [22]. These production disruptions will lead to a hard decision in production and inventory plans. Recently, there is a growing literature on production disruptions. For example, some researchers studied the production rescheduling problems with the machine disruptions [23–26]. Some researchers analyzed the optimal inventory policy with supply disruptions [27–30].In most of the existing literature, products are assumed to be no deterioration when the production disruptions are considered. But, in real situation the deterioration is popular in many kinds of products. Hence, if the deterioration rate is not small enough, the deterioration factor cannot be ignored when the production system is disrupted.Therefore, in this paper, we develop a production-inventory model for deteriorating items with production disruptions. Once the production rate is disrupted, the following questions are considered in this paper.(i)
Whether to replenish from spot markets or not?(ii)
How to adjust the production plan if the new production system can still satisfy the demand?(iii)
How to replenish from spot markets if the new production system no longer satisfies the demand?The paper is organized as follows. Section2 is concerned with the mathematical development and the method for finding the optimal solutions. In Section 3, we present a numerical example to illustrate the model. In Section 4, conclusions and topics for further research are presented.


## 2. Mathematical Modeling and Analysis

Suppose a manufacturer produces a certain product and sells it in a market. All items are produced and sold in each cycle. The following assumptions are used to formulate the problem.(a)
A single product and a single manufacturer are assumed.(b)
Demand rate is deterministic and constant.(c)
Normal Production rate is greater than demand rate.(d)
Lead time is assumed to be negligible.(e)
Deterioration rate is deterministic and constant.(f)
Shortages are not allowed.(g)
Time horizon is finite.(h)
There is only one chance to order the products from spot markets during the planning horizon.Let the basic parameters be as follows:p:
normal production rate,d:
demand rate,θ:
constant deterioration rate of finished products,H:
planning horizon,Tp:
the normal production period without disruptions,Td:
the production disruptions time,Tpd:
the new production period with disruptions,Tr:
the replenishment time from spot markets once shortage appears,Qr:
the order quantity from spot markets once shortage appears,Ii(t):
inventory level in theith interval (i=1,2,…n), n can be different in different scenario.


### 2.1. The Basic Model without Disruptions

At first, the manufacturer makes decisions about the optimal production timeTp under the normal production rate. The inventory model for deteriorating items with normal production rate can be depicted as in Figure 1.Figure 1
Inventory system without disruptions.The instantaneous inventory level at any timet∈[0,H] is governed by the following differential equations:(2.1)dI1(t)dt+θI1(t)=p-d,0≤t≤Tp,dI2(t)dt+θI2(t)=-d,Tp≤t≤H.
Using the boundary condition I1(0)=0 and I2(H)=0, the solutions of above differential equations are(2.2)I1(t)=p-dθ(1-e-θt),0≤t≤Tp,I2(t)=dθ[eθ(H-t)-1],Tp≤t≤H.
The condition I1(Tp)=I2(Tp) yields(2.3)p-dθ(1-e-θTp)=dθ[eθ(H-Tp)-1].
From (2.3), the production time Tp satisfies the following equation:(2.4)Tp=1θlnp-d+deθHp.
In order to facilitate analysis, we do an asymptotic analysis for Ii(t). Expanding the exponential functions and neglecting second and higher power of θ for small value of θ, (2.2) becomes(2.5)I1(t)≈(p-d)(t-12θt2),0≤t≤Tp,I2(t)≈d[(H-t)+12θ(H-t)2],Tp≤t≤H,
and Tp approximately satisfies the equation(2.6)(p-d)(Tp-12θTp2)=d[(H-Tp)+12θ(H-Tp)2].
From Misra [31], we have(2.7)Tp≈dp-d(H-Tp)[1+12θ(H-Tp)].
Since(2.8)dTpdθ=12d(H-Tp)2p+θd(H-Tp)>0,
we can get the following corollary.Corollary 2.1.
Assuming thatθ≪1, then Tp is increasing in θ.Corollary2.1 implies that the manufacturer has to produce more products when deterioration rate increases. Hence, decreasing deterioration rate is an effective way to reduce the product cost of manufacture.


### 2.2. The Production-Inventory Model under Production Disruptions

In the above model, the production rate is assumed to be deterministic and known. In practice, the production system is often disrupted by various unplanned and unanticipated events. Here, we assume the production disruptions time isTd. Without loss of generality, we assume that the new disrupted production rate is p+Δp, where Δp<0 if production rate decreases suddenly, or Δp>0 if production rate increases.Proposition 2.2.
IfΔp≥-(p-d)(1-e-θH)/(1-e-θ(H-Td)), then the manufacturer can still satisfy the demand after production disruptions. Otherwise, that is, -p≤Δp<-(p-d)(1-e-θH)/(1-e-θ(H-Td)), there will exist shortages due to the production disruptions.Proof.
Without considering the stop time of production or replenishment, the inventory system with production disruptions can be depicted as Figure2.
From Section2.1, we know(2.9)I1(t)=p-dθ(1-e-θt),0≤t≤Td.
The inventory system after disruptions can be represented by the following differential equation:
(2.10)dI2(t)dt+θI2(t)=p+Δp-d,Td≤t≤H.
Using I1(Td)=I2(Td)=((p-d)/θ)(1-e-θTd), we have
(2.11)I2(t)=1θ[p+Δp-d-Δpe-θ(t-Td)-(p-d)e-θt],Td≤t≤H.
Hence, we know that
(2.12)I2(H)=1θ[p+Δp-d-Δpe-θ(H-Td)-(p-d)e-θH].
Hence, if I2(H)≥0, that is, Δp≥-(p-d)(1-e-θH)/(1-e-θ(H-Td)), this means that the manufacturer can still satisfy the demand after production disruptions.
But ifI2(H)<0, that is, -p≤Δp<-(p-d)(1-e-θH)/(1-e-θ(H-Td)), we know that the manufacturer will face shortage since the production rate decreases deeply. The proposition is proved.Figure 2
Inventory system with production disruptions.From Proposition2.2, we know that if Δp≥-(p-d)(1-e-θH)/(1-e-θ(H-Td)), the production-inventory problem is to find the new optimal production period Tpd. If -p≤Δp<-(p-d)(1-e-θH)/(1-e-θ(H-Td)), the production-inventory problem is to find the optimal replenishment time Tr and replenishment quantity Qr.Proposition 2.3.
IfΔp≥-(p-d)(1-e-θH)/(1-e-θ(H-Td)), then the manufacturer’s production time with production disruptions is
(2.13)Tpd=1θlnp+d(eθH-1)+ΔpeθTdp+Δp.Proof.
From Proposition2.2, we know that the new production time Tpd∈[Td,H] if Δp≥-(p-d)(1-e-θH)/(1-e-θ(H-Td)). The inventory model can be depicted as in Figure 3.
So ifΔp≥-(p-d)(1-e-θH)/(1-e-θ(H-Td)), the inventory system after disruptions can be represented by the following differential equations:(2.14)dI2(t)dt+θI2(t)=p+Δp-d,Td≤t≤Tpd,dI3(t)dt+θI3(t)=-d,Tpd≤t≤H.
Using the boundary conditions I1(Td)=I2(Td)=((p-d)/θ)(1-e-θTd) and I3(H)=0, we know
(2.15)I2(t)=1θ[p+Δp-d-Δpe-θ(t-Td)-(p-d)e-θt],Td≤t≤Tpd,I3(t)=dθ[eθ(H-t)-1],Tpd≤t≤H.
Using the boundary condition I2(Tpd)=I3(Tpd), we have
(2.16)Tpd=1θlnp+d(eθH-1)+ΔpeθTdp+Δp.
The proposition is proved.Figure 3
Inventory system (Tpd∈[Td,H]).SincedTpd/dTd=(p+Δp)ΔpeθTd/(p+d(eθH-1)+ΔpeθTd), we can easily get Corollary 2.4.Corollary 2.4.
If-(p-d)(1-e-θH)/(1-e-θ(H-Td))≤Δp<0, then Tpd is decreasing in Td.
IfΔp>0, then Tpd is increasing in Td.Expanding the exponential functions and neglecting second and higher power ofθ for small value of θ, (2.15) becomes(2.17)I2(t)≈Δp(t-Td)[1-12θ(t-Td)]+(p-d)t(1-12θt),Td≤t≤Tpd,I3(t)≈d(H-t)[1+12θ(H-t)],Tpd≤t≤H,
and Tpd approximately satisfies the equation(2.18)Δp(Tpd-Td)[1-12θ(Tpd-Td)]+(p-d)Tpd[1-12θTpd]=d(H-Tpd)[1+12θ(H-Tpd)].
From Misra [31], we have(2.19)Tpd≈ΔpTd+d(H-Tpd)[1+(1/2)θ(H-Tpd)]p+Δp-d.
According to (2.19), we know that(2.20)dTpddθ=12d(H-Tpd)2p+Δp+dθ(H-Tpd)>0.
Hence, we can get the following corollary.Corollary 2.5.
Assuming thatθ≪1, then Tpd is increasing in θ.If-p≤Δp<-(p-d)(1-e-θH)/(1-e-θ(H-Td)), there will exist shortage. The manufacturer will have to produce products during the whole planning horizon, that is, Tpd=H. In order to avoid shortage, the manufacturer needs to order products from spot markets to satisfy the demand. The inventory model can be depicted as in Figure 4.Figure 4
Inventory system (Tpd=H).Proposition 2.6.
If-p≤Δp<-(p-d)(1-e-θH)/(1-e-θ(H-Td)), then the replenishment time and quantity are
(2.21)Tr=1θlnp-d+ΔpeθTdp+Δp-d,Qr=p+Δp-dθ[1-eθ(H-Tr)].Proof.
First, we need to determine the order time pointTr. Let I2(t)=(1/θ)[p+Δp-d-Δpe-θ(t-Td)-(p-d)e-θt]=0, we have
(2.22)Tr=1θlnp-d+ΔpeθTdp+Δp-d.
So,
(2.23)dI3(t)dt+θI3(t)=p+Δp-d,Tr≤t≤H.
Using the boundary condition I3(H)=0, we have
(2.24)I3(t)=p+Δp-dθ[1-eθ(H-t)],Tr≤t≤H.
Hence, the order quantity is
(2.25)Qr=I3(Tr)=p+Δp-dθ[1-eθ(H-Tr)].
The proposition is proved.If-p≤Δp<-(p-d)(1-e-θH)/(1-e-θ(H-Td)), according to (2.22), we have(2.26)dTrdTd=p+Δp-dp-d+ΔpeθTdΔpeθTd<0,dQrdTd=(p+Δp-d)eθ(H-Tr)dTrdTd<0.
Hence, we can obtain the following corollary.Corollary 2.7.
If-p≤Δp<-(p-d)(1-e-θH)/(1-e-θ(H-Td)), then Tr and Qr are decreasing in Td.


## 2.1. The Basic Model without Disruptions

At first, the manufacturer makes decisions about the optimal production timeTp under the normal production rate. The inventory model for deteriorating items with normal production rate can be depicted as in Figure 1.Figure 1
Inventory system without disruptions.The instantaneous inventory level at any timet∈[0,H] is governed by the following differential equations:(2.1)dI1(t)dt+θI1(t)=p-d,0≤t≤Tp,dI2(t)dt+θI2(t)=-d,Tp≤t≤H.
Using the boundary condition I1(0)=0 and I2(H)=0, the solutions of above differential equations are(2.2)I1(t)=p-dθ(1-e-θt),0≤t≤Tp,I2(t)=dθ[eθ(H-t)-1],Tp≤t≤H.
The condition I1(Tp)=I2(Tp) yields(2.3)p-dθ(1-e-θTp)=dθ[eθ(H-Tp)-1].
From (2.3), the production time Tp satisfies the following equation:(2.4)Tp=1θlnp-d+deθHp.
In order to facilitate analysis, we do an asymptotic analysis for Ii(t). Expanding the exponential functions and neglecting second and higher power of θ for small value of θ, (2.2) becomes(2.5)I1(t)≈(p-d)(t-12θt2),0≤t≤Tp,I2(t)≈d[(H-t)+12θ(H-t)2],Tp≤t≤H,
and Tp approximately satisfies the equation(2.6)(p-d)(Tp-12θTp2)=d[(H-Tp)+12θ(H-Tp)2].
From Misra [31], we have(2.7)Tp≈dp-d(H-Tp)[1+12θ(H-Tp)].
Since(2.8)dTpdθ=12d(H-Tp)2p+θd(H-Tp)>0,
we can get the following corollary.Corollary 2.1.
Assuming thatθ≪1, then Tp is increasing in θ.Corollary2.1 implies that the manufacturer has to produce more products when deterioration rate increases. Hence, decreasing deterioration rate is an effective way to reduce the product cost of manufacture.


## 2.2. The Production-Inventory Model under Production Disruptions

In the above model, the production rate is assumed to be deterministic and known. In practice, the production system is often disrupted by various unplanned and unanticipated events. Here, we assume the production disruptions time isTd. Without loss of generality, we assume that the new disrupted production rate is p+Δp, where Δp<0 if production rate decreases suddenly, or Δp>0 if production rate increases.Proposition 2.2.
IfΔp≥-(p-d)(1-e-θH)/(1-e-θ(H-Td)), then the manufacturer can still satisfy the demand after production disruptions. Otherwise, that is, -p≤Δp<-(p-d)(1-e-θH)/(1-e-θ(H-Td)), there will exist shortages due to the production disruptions.Proof.
Without considering the stop time of production or replenishment, the inventory system with production disruptions can be depicted as Figure2.
From Section2.1, we know(2.9)I1(t)=p-dθ(1-e-θt),0≤t≤Td.
The inventory system after disruptions can be represented by the following differential equation:
(2.10)dI2(t)dt+θI2(t)=p+Δp-d,Td≤t≤H.
Using I1(Td)=I2(Td)=((p-d)/θ)(1-e-θTd), we have
(2.11)I2(t)=1θ[p+Δp-d-Δpe-θ(t-Td)-(p-d)e-θt],Td≤t≤H.
Hence, we know that
(2.12)I2(H)=1θ[p+Δp-d-Δpe-θ(H-Td)-(p-d)e-θH].
Hence, if I2(H)≥0, that is, Δp≥-(p-d)(1-e-θH)/(1-e-θ(H-Td)), this means that the manufacturer can still satisfy the demand after production disruptions.
But ifI2(H)<0, that is, -p≤Δp<-(p-d)(1-e-θH)/(1-e-θ(H-Td)), we know that the manufacturer will face shortage since the production rate decreases deeply. The proposition is proved.Figure 2
Inventory system with production disruptions.From Proposition2.2, we know that if Δp≥-(p-d)(1-e-θH)/(1-e-θ(H-Td)), the production-inventory problem is to find the new optimal production period Tpd. If -p≤Δp<-(p-d)(1-e-θH)/(1-e-θ(H-Td)), the production-inventory problem is to find the optimal replenishment time Tr and replenishment quantity Qr.Proposition 2.3.
IfΔp≥-(p-d)(1-e-θH)/(1-e-θ(H-Td)), then the manufacturer’s production time with production disruptions is
(2.13)Tpd=1θlnp+d(eθH-1)+ΔpeθTdp+Δp.Proof.
From Proposition2.2, we know that the new production time Tpd∈[Td,H] if Δp≥-(p-d)(1-e-θH)/(1-e-θ(H-Td)). The inventory model can be depicted as in Figure 3.
So ifΔp≥-(p-d)(1-e-θH)/(1-e-θ(H-Td)), the inventory system after disruptions can be represented by the following differential equations:(2.14)dI2(t)dt+θI2(t)=p+Δp-d,Td≤t≤Tpd,dI3(t)dt+θI3(t)=-d,Tpd≤t≤H.
Using the boundary conditions I1(Td)=I2(Td)=((p-d)/θ)(1-e-θTd) and I3(H)=0, we know
(2.15)I2(t)=1θ[p+Δp-d-Δpe-θ(t-Td)-(p-d)e-θt],Td≤t≤Tpd,I3(t)=dθ[eθ(H-t)-1],Tpd≤t≤H.
Using the boundary condition I2(Tpd)=I3(Tpd), we have
(2.16)Tpd=1θlnp+d(eθH-1)+ΔpeθTdp+Δp.
The proposition is proved.Figure 3
Inventory system (Tpd∈[Td,H]).SincedTpd/dTd=(p+Δp)ΔpeθTd/(p+d(eθH-1)+ΔpeθTd), we can easily get Corollary 2.4.Corollary 2.4.
If-(p-d)(1-e-θH)/(1-e-θ(H-Td))≤Δp<0, then Tpd is decreasing in Td.
IfΔp>0, then Tpd is increasing in Td.Expanding the exponential functions and neglecting second and higher power ofθ for small value of θ, (2.15) becomes(2.17)I2(t)≈Δp(t-Td)[1-12θ(t-Td)]+(p-d)t(1-12θt),Td≤t≤Tpd,I3(t)≈d(H-t)[1+12θ(H-t)],Tpd≤t≤H,
and Tpd approximately satisfies the equation(2.18)Δp(Tpd-Td)[1-12θ(Tpd-Td)]+(p-d)Tpd[1-12θTpd]=d(H-Tpd)[1+12θ(H-Tpd)].
From Misra [31], we have(2.19)Tpd≈ΔpTd+d(H-Tpd)[1+(1/2)θ(H-Tpd)]p+Δp-d.
According to (2.19), we know that(2.20)dTpddθ=12d(H-Tpd)2p+Δp+dθ(H-Tpd)>0.
Hence, we can get the following corollary.Corollary 2.5.
Assuming thatθ≪1, then Tpd is increasing in θ.If-p≤Δp<-(p-d)(1-e-θH)/(1-e-θ(H-Td)), there will exist shortage. The manufacturer will have to produce products during the whole planning horizon, that is, Tpd=H. In order to avoid shortage, the manufacturer needs to order products from spot markets to satisfy the demand. The inventory model can be depicted as in Figure 4.Figure 4
Inventory system (Tpd=H).Proposition 2.6.
If-p≤Δp<-(p-d)(1-e-θH)/(1-e-θ(H-Td)), then the replenishment time and quantity are
(2.21)Tr=1θlnp-d+ΔpeθTdp+Δp-d,Qr=p+Δp-dθ[1-eθ(H-Tr)].Proof.
First, we need to determine the order time pointTr. Let I2(t)=(1/θ)[p+Δp-d-Δpe-θ(t-Td)-(p-d)e-θt]=0, we have
(2.22)Tr=1θlnp-d+ΔpeθTdp+Δp-d.
So,
(2.23)dI3(t)dt+θI3(t)=p+Δp-d,Tr≤t≤H.
Using the boundary condition I3(H)=0, we have
(2.24)I3(t)=p+Δp-dθ[1-eθ(H-t)],Tr≤t≤H.
Hence, the order quantity is
(2.25)Qr=I3(Tr)=p+Δp-dθ[1-eθ(H-Tr)].
The proposition is proved.If-p≤Δp<-(p-d)(1-e-θH)/(1-e-θ(H-Td)), according to (2.22), we have(2.26)dTrdTd=p+Δp-dp-d+ΔpeθTdΔpeθTd<0,dQrdTd=(p+Δp-d)eθ(H-Tr)dTrdTd<0.
Hence, we can obtain the following corollary.Corollary 2.7.
If-p≤Δp<-(p-d)(1-e-θH)/(1-e-θ(H-Td)), then Tr and Qr are decreasing in Td.


## 3. A Numerical Example

Our objective in this section is to gain further insights based on a numerical example. We use the following numbers as the base values of the parameters:p=350, d=200, θ=0.03, H=20, Td=8, and Δp=-200. Using (2.4), we obtain Tp=12.8. Next, we observe how Tpd, Tr, and Qr would change as θ and Td. Figures 5 and 6 depict Tpd with respect to θ and Td, respectively. Figures 7 and 9 depict Tr with respect to θ and Td, respectively. Figures 8 and 10 depict Qr with respect to θ and Td, respectively.Figure 5
Tpd with respect to θ.Figure 6
Tpd with respect to Td.Figure 7
Tr with respect to θ.Figure 8
Qr with respect to θ.Figure 9
Tr with respect to Td.Figure 10
Qr with respect to Td.From Figure5, we can find that Tpd is increasing in θ when θ≤0.068. When θ>0.068, since the deterioration rate is so high that the manufacturer cannot satisfy the demand by self-producing, he has to buy products from spot markets in order to avoid shortage. From Figures 7 and 8, we can see that Tr is decreasing in θ, and Qr is increasing in θ when θ>0.068.From Figure6, we can find that Tpd is decreasing in Td when 6.2≤Td≤12.8. If 0≤Td<6.2, the manufacturer will have to replenish inventory from spot markets. From Figures 9 and 10, we can see that Tr is increasing in Td, and Qr is decreasing in θ when 0≤Td<6.2.


## 4. Conclusions

In this paper, we propose a production-inventory model for a deteriorating item with production disruptions. Here, we analyze this inventory system under different situations. We have showed that our method helps the manufacturer reduce the loss caused by production disruptions.In this study, the proposed model considers the deterioration rate as constant. In real life, we may consider the deterioration rate as a function of time, stock, and so on. This will be done in our future research.



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