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# Deterministic and Probabilistic Inventory Model. What is Operations Research?

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Introduction

DETERMINISTIC AND PROBABILISTIC INVENTORY MODEL. What is Operations Research? Before: Application of mathematics and the scientific method to military operations Today: Scientific approach to decision making. Seeks to determine best way to design and operate system, usually requiring allocation of scarce resources OR Methodology Inventory Model Inventory refers to idle goods or materials that are held by an organization for use sometime in future. Example. � Raw materials � Purchased parts � Components � Subassemblies � Work-in-process � Finished goods Key issues: 1. What are possible costs? * Order cost. There is a fixed cost when an order is placed. * Holding cost. It represents the costs of carrying inventory in stock (e.g. interest on invested capital, storage, handling, depreciation, and maintenance) * Purchasing cost. A discount or price break when the order quantity reaches certain level. * Shortage cost. It is a penalty when stock is run out (e.g. lost of customers goodwill and potential loss of income). 2. How much should be ordered when the inventory is replenished? * This is about order quantity. The idea is to minimize the total cost. 3. When should the inventory be replenished? There are two ways. * Periodic review. To order at a fixed time interval. ...read more.

Middle

In this case a reorder point occurs when the inventory level drops to LD units. Now effective lead time becomes Le = L- nt0 Where n is the largest integer no exceeding L/ t0.The following figure provides the pictorial representation of the static EOQ method. The realistic inventory model. Q = order batch size, B = order point, S = the buffer inventory, L = order lead-time. Dynamic EOQ Model This model differs from the static model in 2 aspects 1. The inventory level is reviewed periodically over finite number of equal periods 2. The demand per period is dynamic in the sense that it may vary from one period to the next. The situation in which dynamic deterministic demand occurs is material requirement planning MRP Probabilistic Inventory Model The technique presented in the above method assumes that data area known with certainty which is not true for all situations and there are several cases in which it is not correct to represent the demand by a single deterministic value. In such cases we observe the historical data to represent the demand as a probabilistic distribution Probability is associated with performing an experiment whose outcome occurs randomly. The conjunction of all the outcomes of an experiment is referred to as the sample space and subset of the sample space is known as an event. ...read more.

Conclusion

To develop the total cost function per unit time let f(x) = pdf of demand x during lead time D = Expected demand per unit time h = Holding cost per inventory per unit time p = Shortage Cost per inventory unit K = Setup Cost Based on these definitions the elements of the cost function are now determined 1. Setup cost = KD/Y 2. Holding Cost = hI where I is the average inventory which is given by I = (Y+E{R-x})+E{R-x}/2 => Y/2 + R - E{x} 3. Expected Shortage Cost When x > R and S = ??R (x-R)f(x)dx Single- and Multi- Period Models The classification applies to the probabilistic demand case. In a single-period model, the items unsold at the end of the period are not carried over to the next period. The unsold items, however, may have some salvage values. In a multi-period model; all the items unsold at the end of one period are available in the next period. In the single-period model and in some of the multi-period models, there remains only one question to answer: how much to order. Trade-offs in Single-Period Models * Loss resulting from the items unsold * ML= Purchase price - Salvage value * Profit resulting from the items sold * MP= Selling price - Purchase price * Given costs of overestimating / underestimating demand and the probabilities of various demand sizes. ...read more.

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