Fixed lifetime inventory system with double order under useful lifetime based model

The problem of outdating in the fixed lifetime inventory system has been the focus of researchers in the last decade. Results show that while outdating is being minimized, shortages have become a problem. To eliminate or reduce shortages in the fixed lifetime inventory system, we propose a model where, there are two orders, one period apart in arriving into the inventory system. The second order is designed to satisfy any demand that cannot be satisfied by the first order.


INTRODUCTION
One of the biggest problems of the fixed lifetime inventory system is outdating of products. Products are said to have expired (or outdated) if they have not been used to meet demand at the end of their useful lifetime in inventory. Many companies in Nigeria and Africa suffer huge financial losses annually due to outdating of fixed lifetime products. Besides the financial losses, there is the danger of citizens consuming outdated products which can lead to outbreak of diseases. The research focus [Chiu (1995), Liu and Lian (1999), Olsson and Tydesjo (2010), Nahmias (1982), Nahmias (2011), Silver et al. (2012, Mahmoodi et al (2015), Sheng-Chih et al. (2016), Izevbizua and Omosigho (2017), Izevbizua and Apanapudor (2019), Izevbizua and Emuefe (2019)] has been on how to reduce the quantity of items outdating in the fixed lifetime inventory system. While the outdate quantity was been minimized, shortages was observed to be on the increase. Many inventory managers have struggled with the problem of stocking because of outdating. Some have over stock and others under stock. Over stocking is having too many items on hand, leading to an increase in outdated products, while under stocking is having too little on hand leading to shortages. The desired inventory policy is one that order in such a way that there will always be items on hand to meet demand, while reducing the outdate quantity. This is the main focus of this work namely; to develop inventory policies that will make goods available at all times and also reduce the amount of items outdating from the fixed lifetime inventory system. To do this, a double order fixed lifetime inventory model was introduced. The model consist of two orders which may or may not be equal in size and arrives into inventory one period apart (that is, the first order arrive in period i and the second order arrive in period 1  i ). The second order is the backup order designed to take care of excess demand that cannot be satisfied by the first order. Items from the first order will outdate before items from the second order. The model allows us to trace items from the point of entry to the point of leaving the system either by demand or

Model assumptions
1) Order is placed for new products when the useful lifetime remaining on the items on hand is one period.
2) There are two orders 4) The issuing policy is FIFO. 5) The age of items arriving into inventory is zero. Items not used to meet demand at the end of their useful periods outdate and are discarded. 6) Excess demand that cannot be satisfy from the back up order are lost.

Model description
In Table 1 Observe that when the useful lifetime remaining on items from the first order is one, a new order 2 , 1 y (first order of set 2) arrives and when the useful lifetime remaining on the second order is one, another order 2 , 2 y (which is the second order of set 2) arrives. The new arrivals will undergo the same periodic movement until they expires, if not used to meet demand. In a fixed lifetime inventory system, it is necessary to track items of a particular order from the time they arrive in inventory to their last useful period in inventory. Table 2 shows the quantity of items on hand and their age distribution from Table 1.
Again, observe that at the end of period 1  m , no item(s) from the first set is left in the system. Either they have been used to meet demand or have expired. Next, we look at the amount of items on hand and their age categories. This enables us keep track of items from a particular order.
Shortage cost: One of the advantage of the double order model is that shortages are highly minimized. This is because, the second order of each set act as a backup for the first order, by satisfying any demand that cannot be satisfied from the first order. However, if the demand in a period cannot be completely satisfied by items from both orders, the excess demand is lost. Such excess demand is referred to as shortages. Since total demand in a period is t , shortage occur if 2 1 y y t   . So that our shortage quantity is given as  Outdates from Hence, the total outdate quantity from the set is given as; And the outdate cost for the model is Holding cost: There is a fixed cost 0  h for each item(s) held in inventory, so that our holding cost is; Therefore, our total cost function is the sum of all the cost components.
A computer programme in MATHEMATICA 8 was used to solve the total cost function in Equation 8. Table 3 gives the quantity of items ordered in a departmental shop in Benin City, Nigeria, the demand for each order, the shortage and outdates associated with each order. At the end of 30 days of applying the ordering policy of the model, the number of shortages was zero while the 2 Table 3. Orders, demand, shortage and outdates for a product with 4useful-lifetime and zero lead time.    outdates was 9. The problem of shortages have been addressed by the model as the second order mops up excess demand.

Day
From Table 3, Table 4 is obtained showing the cost associated with each set of orders. Constant parameters are k = 150, v = 10, h = 20,  =0.005.

Conclusion
The double order model reduces the shortages associated with the fixed lifetime inventory system. The second order act as backup order, meeting demands that cannot be satisfied by the first order. This will restore customers' confidence in the inventory manager. The double order alongside the ordering policy based on the remaining useful lifetime have the advantage of making goods available and minimizing outdating as shown in the numerical example.