Deterministic Inventory Models for Delayed Deteriorating Items With Inventory Level Dependent Demand Rate
Chapter One
Aim and Objectives of the Study
The main aim of this research is to develop Deterministic Inventory Models for Delayed Deteriorating Items with Inventory Level Dependent Demand Rate. This aim will be achieved through the following objectives.
- To develop an EOQ model for delayed deteriorating items with inventory level dependent demand rate and constant deterioration
- To develop an EOQ model for delayed deteriorating items with inventory level- dependent demand rate and shortages.
- To develop an EPQ model for delayed deteriorating items with stock-dependent and linear time dependent holding
- To develop an EPQ model for delayed deteriorating items with stock- dependent demand rate and time-dependent deterioration rate.
- To develop an EOQ model for delayed deteriorating items with inventory level dependent demand rate and linear time-dependent holding
CHAPTER TWO
LITERATURE REVIEW
Introduction
In many inventory models a general assumption is that products have indefinitely long lives. However, in general, almost all items deteriorate over time. Often the rate of deterioration is so low that there is little need to consider the deterioration in determining the inventory models. Nevertheless, there are many items in the real life situations that are subject to a significant rate of deterioration. Hence, in such situation, the impact of item deterioration should not be neglected in the inventory decision – making. More specifically, those inventory models where deterioration is a function of the on-hand level of inventory are considered.
There is no clear consensus on the definition of deterioration of items. Raafat (1991) defines deterioration as any process that prevents an item from being used for its intended original use such as: (i) spoilage, as in perishable foodstuffs, fruits, vegetables, periodicals (such as newspapers and magazines); (ii) physical depletion, as in pilferage or highly volatile liquids such as gasoline, alcohol, turpentine, and so on, which undergo physical depletion over time through the process of evaporation. (iii) decay, as in radioactive substances, degradation, as in electronic components, or loss of potential or utility with the passage of time as in photographic films and pharmaceutical drugs. Goyal and Giri (2001) in their review of deteriorating inventory models classified all such models based on obsolescence, deterioration or neither. They reported “items are subject to obsolescence if they lose their value over time because of rapid changes of technology or the introduction of a new product by a competitor or because they go out of fashion” (Goyal and Giri, 2001, p. 1). Examples of items that are subject to obsolescence are computer chips, mobile phones, fashion and seasonal goods, and so on. Others are style goods such as spare parts for military which become obsolete when a replacement is introduced.
In many real-life situations, for certain types of consumer goods (e.g., fruits, vegetables, meat, bread, beans, cassava, yams and others), the consumption rate is sometimes influenced by stock-level. It is usually observed that a large pile of goods displayed on shelves in a shop will attract customers to buy more and which in turn generate higher sales. The consumption rate may go up or down depending on the on-hand stock level (Gupta and Vrat, 1986). The inventory level as a function of time will decrease rapidly initially, since the quantity demanded will be greater at a high level of inventory. As the inventory is depleted, the quantity demanded will decrease, resulting in the inventory level decreasing more slowly (Baker and Urban, 1988). This type of demand rate which depends on the level of on-hand stock is termed, „„inventory level- dependent demand rate‟‟ (Gupta and Vrat, 1986, p. 19; Baker and Urban, 1988, p. 824). This implies demand rate of this pattern is a function of instantaneous level of inventory.
In the literature of inventory systems, inventory models for deteriorating items assume that deterioration starts as soon as the retailer receives the commodities. However, in real life, many products would have a span of maintaining quality or the original condition for some period. That is during that period there is no deterioration occurring until later, and that phenomenon is termed as “non – instantaneous deterioration” in Wu et al. (2006, p. 369) or “delayed deterioration” in Musa and Sani (2012, p. 75).
The aim of this chapter is to review the published literature on mathematical modelling of deteriorating items with inventory level dependent demand pattern which are related to this research. The models have been classified and grouped under specific sections and organized as follows:
- Classical square root EOQ Model
- Inventory Models with constant / variable demand
- Delayed(Non-instantaneous) deteriorating inventory models
- Variable holding deteriorating inventory models
- Production Inventory Models for Deteriorating Items
- Deteriorating inventory models with shortages
CHAPTER THREE
AN EOQ MODEL FOR DELAYED DETERIORATING ITEM WITH INVENTORY LEVEL – DEPENDENT DEMAND RATE AND CONSTANT DETERIORATION RATE
Advertisements
Introduction
In many real-life situations, for certain types of consumer goods such as yams, fruits, vegetables, doughnuts, and so on, the demand rate is sometimes influenced by the stock level. It is usually observed that a large pile of goods displayed on shelves in a shop will lead the customer to buy more and then generate higher demand. The consumption rate may go up or down with the on-hand stock level.
Depletion of inventory items may occur for some reasons other than demand, such as by direct spoilage or physical decay or deterioration. Some products such as fruits, vegetables, pharmaceuticals, volatile liquids, and others of this nature deteriorate continuously due to evaporation, obsolescence, spoilage, and so on. Ghare and Schrader (1963) first derived an economic order quantity (EOQ) model by assuming exponential decay.
In reality, not all kinds of inventory items deteriorate as soon as they are received in stock. In the fresh product time, when the product has no deterioration the inventory items retain their original quality. Wu et al. (2006) named this phenomenon as “non- instantaneous deterioration”, and they established an inventory model for non- instantaneous deteriorating items with permissible delay in payments.
CHAPTER FOUR
AN EOQ MODEL FOR DELAYED DETERIORATING ITEMS WITH INVENTORY LEVEL DEPENDENT DEMAND RATE AND SHORTAGES
Introduction
In practice, when shortages occur, some customers are willing to wait for backorder while others are impatient to wait and therefore would turn to buy from other competitors or buy alternative item. For inventory models with stock dependent consumption rate, some authors assumed shortages to be completely backlogged while others assume the shortages to be partially backlogged.
CHAPTER FIVE
AN EPQ MODEL FOR DELAYED DETERIORATING ITEMS WITH STOCK- DEPENDENT DEMAND RATE AND LINEAR TIME – DEPENDENT HOLDING COST WITH TIME- DEPENDENT DETERIORATION RATE
Introduction
In this chapter, two economic production quantity (EPQ) models are presented. In both cases, there is a delay in deterioration and the production rate is constant, demand rate is inventory level dependent in a linear functional form, both before and after production. In the first model, the holding cost is a linear function of time and the deterioration rate is a constant while in the second model, the holding cost is a constant but the deterioration rate is time dependent.
CHAPTER SIX
AN EOQ MODEL FOR DELAYED DETERIORATING ITEMS WITH INVENTORY LEVEL DEPENDENT DEMAND RATE AND LINEAR TIME- DEPENDENT HOLDING COST
Introduction
To keep sales high, inventory level needs to remain high. Of course, this would also result in high holding or procurement costs (Baker and Urban, 1988). Larson and DeMarais (1990) suggested that only those items with high direct product profitability (DPP) and high sales volume need to be considered for positive ending- inventory. An inventory system which possesses inventory -level-dependent demand rate, and in which the inventory level falls to zero at the end of the order cycle, may not provide the optimal solution (Urban, 1992 and chang, 2004). However, Urban (1992) stated that having inventory remaining at the end of the order cycle is not a general requirement but a possibility. Maximising profit rather than minimising cost is justified in this model (see Urban, 1992 and chang, 2004 for instance) since the cost of holding an item in stock increases with each passing time and the aim in this environment is to hold more stock so as to generate more profit.
CHAPTER SEVEN
SUMMARY, CONCLUSION AND RECOMMENDATIONS
Summary
The main aim of this study is to develop deterministic inventory models for delayed deteriorating items with inventory level dependent demand rate. The aim has been achieved through five main objectives. These objectives (as already stated in section 1.10) were accomplished by way of developing five models of Economic Order Quantity (EOQ) and Economic Production Quantity (EPQ) for delayed deteriorating items with inventory -level- dependent demand rate.
The first EPQ model presented is that of an inventory system for delayed deteriorating items with stock-dependent demand rate which is in polynomial functional form. In the first stage, inventory depletes down to a certain level of the inventory due to market demand only which is inventory level dependent. In the second stage, the inventory level gets depleted due to the effect of both market demand and deterioration but still dependent on stock until the inventory level falls to zero at the end of the cycle. The aim is to find the optimal ordering quantity and optimal cycle time for the inventory system, in order to minimize the total average system cost per unit time. Newton-Raphson method has been used to find the optimal order quantity and the optimal cycle length. Furthermore, some numerical examples have been presented to illustrate the application of the model developed and use the examples to study the effect of various changes in some possible combinations of model parameters on the decision variables.
The second model is an EOQ model for delayed deteriorating items with stock- dependent demand, allowing shortages and with fixed backlogging rate. The model is an amendment of the first model by allowing shortages. We assume a fixed fraction of demand rate to be backlogged during the shortage period as in Wee (1995).
The next two models presented are economic production quantity (EPQ) models. In both models, there is a delay in deterioration and the production rate is constant, demand rate is inventory level dependent in a linear functional form, both before and after production. In the first model, the holding cost is a linear function of time and the deterioration rate is a constant while in the second model, the holding cost is a constant but the deterioration rate is time dependent. Deterioration starts immediately production stops and so the delay in deterioration is that period when production is in progress. The main objective in both cases is to determine the optimal replenishment cycle time such that the total variable cost is minimised. Some numerical examples are presented to illustrate the application of the models developed and the examples are used to study the effect of various changes in some possible combinations of model parameters on the decision variables of the system.
The last model is for an optimal replenishment policy for delayed deteriorating items with power-form inventory level dependent demand rate. The holding cost is assumed to be linear while the deterioration rate is a constant. The difference between this model and the first model is that the holding cost in this model is linear time dependent and not a constant as in the first model. Moreover, the aim in this model is to maximize profit while cost is minimized in the first model. We consider two cases in this model: (i) when replenishment occurs after deterioration sets in (td < T) and (ii) when replenishment occurs before deterioration sets in (td ³ T or 0 < T £ td ).
The inventory
sstem in each case is zero-ending and the objective is to determine the optimal replenishment policy to maximize average net profit per unit time. If the holding cost per unit were to be a constant, then (i) our model of section 6.3.2 would reduce to the model of Baker and Urban (1988), (ii) if
td = 0 , our model of section 6.3.1 would reduce to Pal et al. (1993), provided its terminal inventory iT = 0 . The application of the model is illustrated with the help of numerical examples and a sensitivity analysis is carried out on the decision variables to see the effect of changes in the model parameters.
Conclusion
Different model parameters and assumptions govern the different inventory systems investigated in this thesis. Specific conclusions were given at the end of each of the inventory models investigated. However, general conclusions are drawn across the entire research work which is summarized as follows:
A model has been presented of an inventory system for delayed deterioratingitems with stock-dependent demand rate which is in polynomial functional form. As expected by the result shown in Table 3.1, the deterioration rate q and the stock- dependent parameter b have impact on the optimal solutions and hence, should not be
neglected in modelling deteriorating inventory models with inventory dependent demand. This model helps to determine the optimum ordering quantity and optimum ordering cycle for situations where shortages are not permitted and the replenishment rate is infinite. For that reason, any inventory system with the kind of characteristics ofB the first model assumptions in Section 1.10 can use the model to determine its inventory policy.
AnEOQ model is presented for delayed deteriorating items with stock- dependent demand, allowing shortages and fixed backlogging rate. The impact of stock dependent demand rate, constant rate of deterioration and partial backlogging parameters on order quantity, maximum inventory level and total system cost per unit time were reported. We find from the results (Tables 4.1 – 4.4) that the effects of stock dependent demand rate, holding cost parameter, deterioration and backlogging rate on the optimal replenishment policy are significant, and hence should not be ignored in developing such inventory models. The proposed model can be used in controlling the inventory of certain delayed deteriorating items such as food items, electronic components, fashionable commodities, vegetables, fruits, yams, potatoes, and so
The first EPQ model presented under Section 5.2 in chapter five is for singleproduct with delayed deterioration in which the production rate is constant, demand rate is inventory level dependent and in a linear functional form before and after production. The holding cost is a linear function of time. The results as tabulated in Table 5.1 reveal that for various values of inventory level dependent demand rate parameter
(b = 0.70, 0.60, 0.50, 0.04, 0.10) and deterioration rate ( q = 0.10, 0.20, 0.05, 0.08, 0.02),
the optimum values of production quantity and average total cost function per unit time were determined using the expressions for (4.2.4) and (4.2.10) respectively. This structure of linear time dependent holding cost is representative of many real-life situations and this is particularly true in the storage of some deteriorating and perishable items such as food products.
The second EPQ model under Section 5.3 in chapter five, is for a single productwith delayed deterioration in which the production rate is constant, demand rate is inventory level dependent and in a linear functional form before and after production. The deteriorating rate is a linear function of We find from the results (Tables 5.2 –
5.5) that the effects of changing the model parameters
b , w, Cs. and l on the optimal replenishment policy reveal the following:
- Whenthe stock-dependent consumption rate b is increasing, the optimal cost is
- Whenthe deterioration rate w is increasing, the optimal cost is
- Whenthe set up cost Cs is increasing, the optimal cost is
- Whenthe production rate l is increasing, the optimal cost is
The model helps in determining the optimum ordering quantity for stock dependent demand rate items under linear time dependent holding cost and delayed deterioration.
Inchapter six, an EOQ model for delayed deteriorating items with inventory level dependent demand rate is The model considers linear time dependent holding cost but taking a constant deterioration rate. This structure of linear time dependent holding cost is representative of many real-life situations and this is particularly true in the storage of some deteriorating and perishable items such as food products. Our results indicate that the effect of stock dependent demand parameter b and holding cost parameter R on the profit is significant.
The results of sensitivity analysis of the decision variables with regard to changes in the model parameters indicate that Q* and Z* are sensitive to over estimation and under estimation of the parameters Co, a , b and R while they are more or less insensitive to the parameters td , c and q . This is equivalent to saying that the optimum ordering quantity and net profit per unit time are sensitive to over estimation and under estimation of ordering cost, stock dependent demand parameter and instantaneous rate of holding cost while they are more or less insensitive to the parameters of fresh product time, unit purchase cost and rate of deterioration.
Our findings show that if the holding cost per unit were to be a constant, then (i)our model of section 3.2 would reduce to the model of Baker and Urban (1988) and
(ii) if td = 0 , our model of section 6.3.1 would reduce to zero – ending inventory of Pal et al. (1993).
The contributions of this research include the following:
A model has been developed of an inventory system for delayed deterioratingitems with stock-dependent demand rate by extending the work of Musa and Sani (2009) where the demand rate is a constant by considering a situation where the demand rate is a power-form function as in Urban and Baker (1988). As expected by the result of the model developed it has been shown that, the deterioration rate q and the stock- dependent parameter b have impact on the optimal solutions and hence, should not be left out in modelling delayed deteriorating items which are inventory level dependent demand rate.
A model has been presented of an inventory system for delayed deterioratingitems with inventory level dependent demand rate as a linear function of inventory level and assumed a fixed fraction of demand rate to be the backlogging rate during the shortage period as in Wee (1995). This is an extension of the first contribution where shortages are not allowed and the demand rate function is a power-form function.
An attempt has been made to extend the model of Sugapriya and Jeyarama(2008a) who established an inventory model for non-instantaneous deteriorating item with constant demand rate for both in- production run and out-production run, while we have developed two EPQ models for delayed deteriorating item in which the production rate is constant, demand rate is inventory level dependent and (1) the holding cost is a linear function of time to reflect the fact that holding cost increases linearly with time,
(2) the deteriorating rate is a linear function of time to also reflect the fact that deterioration increases with each passing time.
(4) Some authors including Urban (1992) and Chang (2004) have suggested that an inventory system that possesses an inventory -level-dependent demand rate, in which the inventory level falls to zero at the end of the order cycle may not provide the optimal solution. However, we find from the results of the models developed that the terminal condition of zero-ending inventory provide optimal solution in which inventory item is the stock dependent and the holding cost is linear time-dependent. In addition, our findings show that if the holding cost per unit were to be a constant, then (i) our model of section 6.3.2 would reduce to the model of Baker and Urban (1988) and (ii) if td = 0 , our model of section 6.3.1 would reduce to zero – ending inventory of Pal et al. (1993).
Recommendations
The models presented in this research provide basis for several possible extensions including incorporating inflation, time – value of money, discount rates, salvage cost, quantity discounts, trade credits; other variable forms of deterioration rate, variable partial backlogging rate and variable holding cost functions, and so on.
REFERENCES
- Abad, P. L. (1996). Optimal pricing and lot sizing under conditions of perishability and partial backordering. Management Science, 42(8), 1093-1104.
- Abad, P.L. (2001). Optimal price and order size for a reseller under partial backordering. Computers and Operations Research, 28, 53–65.
- Abad, P. L. (2003). Optimal pricing and lot-sizing under conditions of perishability, finite production and partial back-ordering and lost sale. European Journal of Operational Research, 144, 677-685
- Alfares, H.K. (2007). Inventory model with stock-level dependent demand rate and variable holding cost. International Journal of Production Economics, 108, 259
- Baker, R. C. and Urban, T. L. (1988). A deterministic inventory system with an inventory level – dependent demand rate. Journal of Operational Research Society, 39, 823–831.
- Balkhi, Z. T. (2001). On a finite horizon production lot size inventory model for deteriorating items: An optimal solution. European Journal of Operational Research, 132, 210-223.
- Baraya, Y. M., and Sani, B. (2011a). An Inventory Model for Delayed Deteriorating Items with Inventory – Level – Depended Demand Rate. ABACUS: Journal of the Mathematical Association of Nigeria, 38(2), 146-160.
- Baraya, Y. M., and Sani, B. (2011b). An economic production quantity (EPQ) model for delayed deteriorating items with stock-dependent demand rate and linear time dependent holding cost. Journal of the Nigerian Association of Mathematical
