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Hello! Welcome to our session on periodic review system related to inventory modeling! In this session, we will be covering concepts related to Periodic Review System and at the end we will be discussing comparison between P System and Q System, Q-System we have discussed in the last module and in this particular module will be discussing about P-System and then finally will draw this comparison. Periodic Review System is an alternative inventory control system where in it is a system in which an items inventory position is reviewed periodically rather than continuously. Inventory position in the last module we had discussed is basically, the On-handed inventory plus the schedule receipts minus the back order. Periodic review system, which is basically the P-system, is also termed as fixed interval reorder system or periodic reorder system or P-system; here, the review period is fixed but the order size can vary. In this periodic review-based inventory control system, inventory position is reviewed at predetermined fixed points in time therefore, these systems are known as fixed order interval systems. If the first status review happens at time P, the second review would be carried out at a point 2P, and so on. At the time of review, an order is placed; an order size is such that it will replenish the inventory position to a predetermined level which we denote by T; this parameter T is also known as order-up-to level and this order-up-to level basically dictates the order quantity which is variable order quantity; this T is also known as target inventory level and at times referred to as the reorder level in a periodic inventory system. So, the decision variables in this system are that target stock level or target inventory level T, the variable order size Q and the optimal time between the review points that is P that means the review period. So, we have to determine 3 things T, Q and P. Let us, understand this system again as before we denote the stock level may be on an inventory or the inventory position along the y-axis and x-axis denotes the time. The target inventory level T is being denoted by these red line. The stock level is denoted by these blue lines one-hand inventory. When these on-hand inventory depletes and falls down to this particular position say and that was the beginning of a or that was a point when that stock was reviewed and order size equivalent to that difference between the target stock level and the inventory position IP1 that is target stock level minus the inventory position IP1 this amount was placed as an order on to the supply which is Q1 and we planned it in such a way that the inventory position will be up to the target stock level then depletion takes place and the order finally arise at this point. So, this is order received, again there is depletion of stock consumption takes place from this point in time when the first or inventory position was reviewed and then order was placed a time period of P has elapsed when we review the position of inventory and we find that it is IP2 at that point in time we place another order which is the difference between again the target stock level which is fixed and IP2. So, target stock level T minus IP2 is the amount Q2 which is placed as an order on to the suppler at this point in time. So, order placed, but we do not immediately get the replenishment, the replenishment comes at a point in time which is in here, which is after a lead time period L the order is received again the stock fluctuates this is on-hand inventory and again at the review period we placed an order amounting to Q3 which is nothing but the difference between the target stock level and the inventory position IP3 and this cycle continue. So, one thing to note here you see that this Q1, Q2 and Q3 are not same that means the order quantities they vary at each review so, variable order size but, the review period is fixed. So, the first decision variable is to determine the target stock level. To find the target stock level, we will assume that the lead time for an item is constant and demand is normally distributed, that assumption will hold good, one thing you must notated that even though, whenever we place an order we plan in such a way that the actual stock level should reach the target stock level but, in reality there is a lead time during which the stock falls because, consumption takes places during that lead time period before the delivery arrives. So, if you refer to that figure here you will see that even after receipt of an order the actual stock level never reaches the target stock level. The actual stock never actually reaches the target level, the size of order Q1 is determined by the stock level IP1 but, when this Q1 actually arrives, the actual stock level by that time has declined. Now, this order Q1 has to satisfy all demand until the next order arrives. So, all demand over the period P plus L need to be satisfied by Q1, you see this order size Q1 you are receiving this order here but, the next order is received at this point in time. So, this order size Q1 must satisfy all the demand over the period P plus L which is basically called the Protection interval because, still that point in time you are not getting any fresh order received. So, the demand over the period P plus L, P means the review period plus L which is the lead time has been assume to be normally distributed with a mean of P plus L star d bar and standard deviation of sigma into root of R of P plus L, sigma d basically into root of R of P plus L. So, the target stock level is a mean demand over P plus L plus the safety stock and as discussed before the safety stock is nothing but Z multiplied by the standard deviation of demand over P plus L, where Z is a standardize normal random variant. So, finally we get the expression for the target stock level T which is nothing but, P plus L star d bar plus Z star sigma star root of R of P plus L, where sigma denotes the standard deviation of demand. So, we first discuss this case when target inventory level has to be determined under a situation when demand is variable and lead time is constant as before target stock level T is d bar into P plus L where d bar is the average demand plus safety stock over the protection interval, T is a target inventory level, P is the review period, L is the lead time when an order is placed but it does not arrive instantaneously. So, safety stock under such a circumstance becomes Z times the distribution of demand over a period P plus L where, this sigma P plus L like we discussed before is sigma d star or times root over of R of P plus L. Another important thing to be notated is that if, the review period P is not specified than an approximation to the optimal value of P can be made by the use of Economic Order Quantity formula. Since, P is the time between orders, it is related to the economic order quantity by an expression which is P is equal to economic order quantity divided by the annual demand D which is nothing but Q by D. Which is nothing but 1 by D multiplied by the formula for economic order quantity which is root of R up to 2DS by i into C where, D is annual demand, S is the cost of placing an order, i is the interest rate, C is the unit cost of the item. This particular expression then gives you an approximately optimal review interval P you will use this particular value of P in a problem where the review period is not given and you have to determine that. Thank you! These are the references for this particular session. Thank you all! Hello! Welcome to our session on periodic review system related to inventory modelling! Let us take one numerical example. Selecting the target inventory level when demand is variable and lead time is constant. This problem has been taken from this book, Operations management by Krajewski. Problem is, the demand for a bird feeder is normally distributed with a mean 18 units per week and the standard deviation of 5 units per week. The lead time is constant and is given as 2 weeks. The business operates over 52 weeks per year. If you try to design a Q-system for this similar problem, for this particular problem, you will find that the EOQ works out to be 75 units and the safety stock of 9 units for a cycle service level of 90 percent; you can easily calculate this and verify. Now, what is the equivalent P-system that is the problem? So, if we have to work out that first we have to find out what is the annual demand. So, D is given as 18 units per week multiplied by 52 weeks per year gives us 936 units that is the annual demand D. Now, here the review period is not given because we are trying to design an equivalent P system. So, the review period P is equal to EOQ by D, D is given as 18 units per week, so 75 by 936 into 52 weeks per year. If we work out, this becomes 4.2 that is equal to 4 weeks, we have rounded it. We now find out the standard deviation of demand over the protection interval. So, the protection interval is nothing but the review period plus the lead time which is equal to 4 plus 2, which is 6 weeks. Therefore, standard deviation of demand over the period P plus L that is standard deviation of demand over the protection interval is nothing but sigma D multiplied by root over P plus L which is nothing but 5 into root over of 6, which is 12.25 units. The business operates 52 weeks per year and the lead time is given. For a 90 percent cycle service level, Z which is a standardized normal random variant can be found out to be 1.28. Hence, the safety stock works out to be 1.28 multiplied by 12.25 which is 15.68 say 16 units, the target inventory level will work out to be D bar into P plus L plus safety stock which is 18 units per week into 6 weeks plus 16 units, which is 124 units, the safety stock was worked out to be 16 units. Now, let us discuss the advantages of using periodic review system. The main benefit of a periodic review system is that it is simple to operate and convenient to administer. There is a routine where stock is checked at regular intervals, orders are placed, delivery is arranged, goods arriving are checked and so on, very simple to administer, and this inventory system is particularly beneficial for cheap items with a high demand. This routine also means that the stock level is only checked at specific intervals and does not have to be monitored continuously unlike fixed order quantity methods in the Q-system where the stock has to be checked continuously to mark the point when the inventory position falls to the reorder level. Another advantage of periodic review system is the ease at which we can combine orders for several items into a single order, and this will result into larger orders that will encourage suppliers to offer price discounts. On the other hand, a major advantage of fixed order quantity method that is a Q-system is that orders of constant size are easier to administer in comparison to the variable ones. Suppliers know how much to send and the administration and transport can be tailored according to the specific needs, perhaps supplying a truckload at a time. They also mean that orders can be tailored according to the needs of each item and unlike periodic review method that commonly uses the same period for many diverse items, and items with no demands are ordered as frequently as those with high demand that is the advantage of Q-system. So, the major advantage of fixed order quantity method that is the Q-system is that it operates with lower stocks. The safety stock has to cover uncertainty in the lead time L for a continuous review system, while the safety stock in a periodic review system has to cover uncertainty in the cycle length plus the lead time. So, the protection interval P plus L is greater than L. So, naturally you need to carry more safety stock in a periodic review system in comparison to a Q system. So, basically this allows smaller safety stock in a Q-system and thereby lower overall stocks in the continuous review system. At times, it is possible to get the benefits of both the approaches, continuous review system and periodic review system by using a hybrid method. And two common methods of hybrid approaches are the periodic review with reorder level that means at any point in time when you are reviewing the stock. You will try to compare the inventory position with the reorder point and if your inventory position is above the reorder point we will not issue any fresh order. Another is the hybrid approach combining reorder level and target stock, both these things are widely used in practice. So, here is the table which gives you the comparison between P system and Q system. In a periodic review system the order interval is fixed. But in a continuous review system, order interval is varying, because basically it depends when the stock or inventory position will reach the reorder point, variable order sizes in a periodic review system. Whereas, a continuous review system is a fixed order size, periodic review system convenient to administer whereas, a continuous review system it allows individual review frequencies. In a periodic review system orders may be combined, continuous review system is possible to provide quantity discounts. Periodic review system - more safety stock; in a continuous review system - lower safety stock. Periodic review systems are more responsive to changes in demand; in a Q-system, it is less responsive to changes in demand. Ease of implementation in case of periodic review system; in a continuous review system in practice, it is implemented using 2-bin system. Periodic review systems are suited for high value items whereas continuous review system is suited for medium and low value items. Thank you! These are the references for this particular session. Thank you!
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