A Production Model for Deteriorating Inventory Items with Production Disruptions

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.

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. Section 2 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,𝑇_𝑝: the normal production period without disruptions,𝑇_𝑑: the production disruptions time,𝑇^𝑑_𝑝: the new production period with disruptions,𝑇_𝑟: the replenishment time from spot markets once shortage appears,𝑄_𝑟: the order quantity from spot markets once shortage appears,𝐼_𝑖(𝑡): inventory level in the 𝑖th 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 time 𝑇_𝑝 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 time 𝑡∈[0,𝐻] is governed by the following differential equations:𝑑𝐼₁(𝑡)𝑑𝑡+𝜃𝐼₁(𝑡)=𝑝−𝑑,0≤𝑡≤𝑇_𝑝,𝑑𝐼₂(𝑡)𝑑𝑡+𝜃𝐼₂(𝑡)=−𝑑,𝑇_𝑝≤𝑡≤𝐻. (2.1) Using the boundary condition 𝐼₁(0)=0 and 𝐼₂(𝐻)=0, the solutions of above differential equations are𝐼₁(𝑡)=𝑝−𝑑𝜃(1−𝑒^−𝜃𝑡),0≤𝑡≤𝑇_𝑝,𝐼₂𝑑(𝑡)=𝜃[𝑒^{𝜃(𝐻−𝑡)}−1],𝑇_𝑝≤𝑡≤𝐻. (2.2) The condition 𝐼₁(𝑇_𝑝)=𝐼₂(𝑇_𝑝) yields𝑝−𝑑𝜃(1−𝑒^{−𝜃𝑇𝑝}𝑑)=𝜃[𝑒^{𝜃(𝐻−𝑇𝑝)}−1]. (2.3) From (2.3), the production time 𝑇_𝑝 satisfies the following equation:𝑇_𝑝=1𝜃ln𝑝−𝑑+𝑑𝑒^𝜃𝐻𝑝. (2.4) In order to facilitate analysis, we do an asymptotic analysis for 𝐼_𝑖(𝑡). Expanding the exponential functions and neglecting second and higher power of 𝜃 for small value of 𝜃, (2.2) becomes𝐼₁1(𝑡)≈(𝑝−𝑑)(𝑡−2𝜃𝑡²),0≤𝑡≤𝑇_𝑝,𝐼₂1(𝑡)≈𝑑[(𝐻−𝑡)+2𝜃(𝐻−𝑡)²],𝑇_𝑝≤𝑡≤𝐻, (2.5) and 𝑇_𝑝 approximately satisfies the equation(𝑝−𝑑)(𝑇_𝑝−12𝜃𝑇²_𝑝)=𝑑[(𝐻−𝑇_𝑝1)+2𝜃(𝐻−𝑇_𝑝)²]. (2.6) From Misra [31], we have𝑇_𝑝≈𝑑𝑝−𝑑(𝐻−𝑇_𝑝1)[1+2𝜃(𝐻−𝑇_𝑝)]. (2.7) Sinced𝑇_𝑝=1d𝜃2𝑑(𝐻−𝑇_𝑝)²𝑝+𝜃𝑑(𝐻−𝑇_𝑝)>0, (2.8) we can get the following corollary.

Corollary 2.1. Assuming that 𝜃≪1, then 𝑇_𝑝 is increasing in 𝜃.

Corollary 2.1 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 is 𝑇_𝑑. 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. If Δ𝑝≥−(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 Figure 2.
From Section 2.1, we know𝐼₁(𝑡)=𝑝−𝑑𝜃(1−𝑒^−𝜃𝑡),0≤𝑡≤𝑇_𝑑. (2.9) The inventory system after disruptions can be represented by the following differential equation: 𝑑𝐼₂(𝑡)𝑑𝑡+𝜃𝐼₂(𝑡)=𝑝+Δ𝑝−𝑑,𝑇_𝑑≤𝑡≤𝐻. (2.10) Using 𝐼₁(𝑇_𝑑)=𝐼₂(𝑇_𝑑)=((𝑝−𝑑)/𝜃)(1−𝑒^{−𝜃𝑇𝑑}), we have 𝐼₂1(𝑡)=𝜃[𝑝+Δ𝑝−𝑑−Δ𝑝𝑒^{−𝜃(𝑡−𝑇𝑑)}−(𝑝−𝑑)𝑒^−𝜃𝑡],𝑇_𝑑≤𝑡≤𝐻. (2.11) Hence, we know that 𝐼₂1(𝐻)=𝜃[𝑝+Δ𝑝−𝑑−Δ𝑝𝑒^{−𝜃(𝐻−𝑇𝑑)}−(𝑝−𝑑)𝑒^−𝜃𝐻]. (2.12) 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 if 𝐼₂(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 Proposition 2.2, 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 𝑇^𝑑_𝑝. If −𝑝≤Δ𝑝<−(𝑝−𝑑)(1−𝑒^−𝜃𝐻)/(1−𝑒^{−𝜃(𝐻−𝑇𝑑)}), the production-inventory problem is to find the optimal replenishment time 𝑇_𝑟 and replenishment quantity 𝑄_𝑟.

Proposition 2.3. If Δ𝑝≥−(𝑝−𝑑)(1−𝑒^−𝜃𝐻)/(1−𝑒^{−𝜃(𝐻−𝑇𝑑)}), then the manufacturer’s production time with production disruptions is 𝑇^𝑑_𝑝=1𝜃ln𝑝+𝑑(𝑒^𝜃𝐻−1)+Δ𝑝𝑒^𝜃𝑇𝑑𝑝+Δ𝑝. (2.13)

Proof. From Proposition 2.2, we know that the new production time 𝑇^𝑑_𝑝∈[𝑇_𝑑,𝐻] if Δ𝑝≥−(𝑝−𝑑)(1−𝑒^−𝜃𝐻)/(1−𝑒^{−𝜃(𝐻−𝑇𝑑)}). The inventory model can be depicted as in Figure 3.
So if Δ𝑝≥−(𝑝−𝑑)(1−𝑒^−𝜃𝐻)/(1−𝑒^{−𝜃(𝐻−𝑇𝑑)}), the inventory system after disruptions can be represented by the following differential equations:𝑑𝐼₂(𝑡)𝑑𝑡+𝜃𝐼₂(𝑡)=𝑝+Δ𝑝−𝑑,𝑇_𝑑≤𝑡≤𝑇^𝑑_𝑝,𝑑𝐼₃(𝑡)𝑑𝑡+𝜃𝐼₃(𝑡)=−𝑑,𝑇^𝑑_𝑝≤𝑡≤𝐻. (2.14) Using the boundary conditions 𝐼₁(𝑇_𝑑)=𝐼₂(𝑇_𝑑)=((𝑝−𝑑)/𝜃)(1−𝑒^{−𝜃𝑇𝑑}) and 𝐼₃(𝐻)=0, we know 𝐼₂1(𝑡)=𝜃[𝑝+Δ𝑝−𝑑−Δ𝑝𝑒^{−𝜃(𝑡−𝑇𝑑)}−(𝑝−𝑑)𝑒^−𝜃𝑡],𝑇_𝑑≤𝑡≤𝑇^𝑑_𝑝,𝐼₃𝑑(𝑡)=𝜃[𝑒^{𝜃(𝐻−𝑡)}−1],𝑇^𝑑_𝑝≤𝑡≤𝐻. (2.15) Using the boundary condition 𝐼₂(𝑇^𝑑_𝑝)=𝐼₃(𝑇^𝑑_𝑝), we have 𝑇^𝑑_𝑝=1𝜃ln𝑝+𝑑(𝑒^𝜃𝐻−1)+Δ𝑝𝑒^𝜃𝑇𝑑𝑝+Δ𝑝. (2.16)
The proposition is proved.

Figure 3 

Inventory system (𝑇𝑑𝑝∈[𝑇𝑑,𝐻]).

Since d𝑇^𝑑_𝑝/d𝑇_𝑑=(𝑝+Δ𝑝)Δ𝑝𝑒^𝜃𝑇𝑑/(𝑝+𝑑(𝑒^𝜃𝐻−1)+Δ𝑝𝑒^𝜃𝑇𝑑), we can easily get Corollary 2.4.

Corollary 2.4. If −(𝑝−𝑑)(1−𝑒^−𝜃𝐻)/(1−𝑒^{−𝜃(𝐻−𝑇𝑑)})≤Δ𝑝<0, then 𝑇^𝑑_𝑝 is decreasing in 𝑇_𝑑.
If Δ𝑝>0, then 𝑇^𝑑_𝑝 is increasing in 𝑇_𝑑.

Expanding the exponential functions and neglecting second and higher power of 𝜃 for small value of 𝜃, (2.15) becomes𝐼₂(𝑡)≈Δ𝑝(𝑡−𝑇_𝑑1)[1−2𝜃(𝑡−𝑇_𝑑1)]+(𝑝−𝑑)𝑡(1−2𝜃𝑡),𝑇_𝑑≤𝑡≤𝑇^𝑑_𝑝,𝐼₃1(𝑡)≈𝑑(𝐻−𝑡)[1+2𝜃(𝐻−𝑡)],𝑇^𝑑_𝑝≤𝑡≤𝐻, (2.17) and 𝑇^𝑑_𝑝 approximately satisfies the equationΔ𝑝(𝑇^𝑑_𝑝−𝑇_𝑑1)[1−2𝜃(𝑇^𝑑_𝑝−𝑇_𝑑)]+(𝑝−𝑑)𝑇^𝑑_𝑝1[1−2𝜃𝑇^𝑑_𝑝]=𝑑(𝐻−𝑇^𝑑_𝑝1)[1+2𝜃(𝐻−𝑇^𝑑_𝑝)]. (2.18) From Misra [31], we have𝑇^𝑑_𝑝≈Δ𝑝𝑇_𝑑+𝑑(𝐻−𝑇^𝑑_𝑝)[1+(1/2)𝜃(𝐻−𝑇^𝑑_𝑝)]𝑝+Δ𝑝−𝑑. (2.19) According to (2.19), we know thatd𝑇^𝑑_𝑝=1d𝜃2𝑑(𝐻−𝑇^𝑑_𝑝)²𝑝+Δ𝑝+𝑑𝜃(𝐻−𝑇^𝑑_𝑝)>0. (2.20) Hence, we can get the following corollary.

Corollary 2.5. Assuming that 𝜃≪1, then 𝑇^𝑑_𝑝 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, 𝑇^𝑑_𝑝=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 (𝑇𝑑𝑝=𝐻).

Proposition 2.6. If −𝑝≤Δ𝑝<−(𝑝−𝑑)(1−𝑒^−𝜃𝐻)/(1−𝑒^{−𝜃(𝐻−𝑇𝑑)}), then the replenishment time and quantity are 𝑇_𝑟=1𝜃ln𝑝−𝑑+Δ𝑝𝑒^𝜃𝑇𝑑,𝑄𝑝+Δ𝑝−𝑑_𝑟=𝑝+Δ𝑝−𝑑𝜃[1−𝑒^{𝜃(𝐻−𝑇𝑟)}]. (2.21)

Proof. First, we need to determine the order time point 𝑇_𝑟. Let 𝐼₂(𝑡)=(1/𝜃)[𝑝+Δ𝑝−𝑑−Δ𝑝𝑒^{−𝜃(𝑡−𝑇𝑑)}−(𝑝−𝑑)𝑒^−𝜃𝑡]=0, we have 𝑇_𝑟=1𝜃ln𝑝−𝑑+Δ𝑝𝑒^𝜃𝑇𝑑𝑝+Δ𝑝−𝑑. (2.22) So, 𝑑𝐼₃(𝑡)𝑑𝑡+𝜃𝐼₃(𝑡)=𝑝+Δ𝑝−𝑑,𝑇_𝑟≤𝑡≤𝐻. (2.23) Using the boundary condition 𝐼₃(𝐻)=0, we have 𝐼₃(𝑡)=𝑝+Δ𝑝−𝑑𝜃[1−𝑒^{𝜃(𝐻−𝑡)}],𝑇_𝑟≤𝑡≤𝐻. (2.24) Hence, the order quantity is 𝑄_𝑟=𝐼₃(𝑇_𝑟)=𝑝+Δ𝑝−𝑑𝜃[1−𝑒^{𝜃(𝐻−𝑇𝑟)}]. (2.25)
The proposition is proved.

If −𝑝≤Δ𝑝<−(𝑝−𝑑)(1−𝑒^−𝜃𝐻)/(1−𝑒^{−𝜃(𝐻−𝑇𝑑)}), according to (2.22), we haved𝑇_𝑟d𝑇_𝑑=𝑝+Δ𝑝−𝑑𝑝−𝑑+Δ𝑝𝑒^𝜃𝑇𝑑Δ𝑝𝑒^𝜃𝑇𝑑<0,d𝑄_𝑟d𝑇_𝑑=(𝑝+Δ𝑝−𝑑)𝑒^{𝜃(𝐻−𝑇𝑟)}d𝑇_𝑟d𝑇_𝑑<0. (2.26) Hence, we can obtain the following corollary.

Corollary 2.7. If −𝑝≤Δ𝑝<−(𝑝−𝑑)(1−𝑒^−𝜃𝐻)/(1−𝑒^{−𝜃(𝐻−𝑇𝑑)}), then 𝑇_𝑟 and 𝑄_𝑟 are decreasing in 𝑇_𝑑.

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, 𝑇_𝑑=8, and Δ𝑝=−200. Using (2.4), we obtain 𝑇_𝑝=12.8. Next, we observe how 𝑇^𝑑_𝑝, 𝑇_𝑟, and 𝑄_𝑟 would change as 𝜃 and 𝑇_𝑑. Figures 5 and 6 depict 𝑇^𝑑_𝑝 with respect to 𝜃 and 𝑇_𝑑, respectively. Figures 7 and 9 depict 𝑇_𝑟 with respect to 𝜃 and 𝑇_𝑑, respectively. Figures 8 and 10 depict 𝑄_𝑟 with respect to 𝜃 and 𝑇_𝑑, respectively.

Figure 5 

𝑇𝑑𝑝 with respect to 𝜃.

Figure 6 

𝑇𝑑𝑝 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 Figure 5, we can find that 𝑇^𝑑_𝑝 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 𝑇_𝑟 is decreasing in 𝜃, and 𝑄_𝑟 is increasing in 𝜃 when 𝜃>0.068.

From Figure 6, we can find that 𝑇^𝑑_𝑝 is decreasing in 𝑇_𝑑 when 6.2≤𝑇_𝑑≤12.8. If 0≤𝑇_𝑑<6.2, the manufacturer will have to replenish inventory from spot markets. From Figures 9 and 10, we can see that 𝑇_𝑟 is increasing in 𝑇_𝑑, and 𝑄_𝑟 is decreasing in 𝜃 when 0≤𝑇_𝑑<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.

Acknowledgments

The authors thank the valuable comments of the referees for an earlier version of this paper. Their comments have significantly improved the paper. This research is supported by a grant from the Ph.D. Programs Foundation of Ministry of Education of China (no. 200802861030). Also, this research is partly supported by the Project Sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, the National Natural Science Foundation of China, the China Postdoctoral Science Foundation (no. 20070411043), and the Postdoctoral Foundation of Jiangsu Province of China (no. 0701045C).