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INVENTORY THEORY

Inventory theory is concerned with the design of inventory systems to minimize costs. Observation of almost any company balance sheet reveals that a very significant part of its assets comprise inventories of raw materials, products within the production process, or finished products.

How do companies use operations research to improve their inventory policy for when and how much to replenish their inventory? They use scientific inventory management comprising of the following steps:

1. Formulate a mathematical model describing the behavior of the inventory system.
2. Seek an optimal inventory policy with respect to this model.
3. Use a computerized information processing system to maintain a record of the current inventory levels.
4. Using this record of current inventory levels, apply the optimal inventory policy to signal when and how much to replenish inventory.

The mathematical inventory models used with this approach can be divided into two broad categories—deterministic models and scholastic models—according to the predictability of demand  involved. The demand for a product in inventory is the number of units that will need to be withdrawn from inventory for some use (e.g., sales) during a specific period. If the demand in future periods can be forecast with considerable precision, it is reasonable to use an inventory policy that assumes that all forecasts will always be completely accurate. This is the case of known demand where a deterministic inventory model would be used. However, when demand cannot be predicted very well, it becomes necessary to use a scholastic  inventory model where the demand in any period is a random variable rather than a known constant.



Cost that is affected by a firm’s decision to maintain a particular level of inventory is called Relevant Cost. Certain important costs and terms are given below:

  • Ordering Cost (c(z)): This is the cost of placing an order to an outside supplier or releasing a production order to a manufacturing shop. The amount ordered is z . c(z) is often a nonlinear function. The dimension of ordering cost is ($).
  • Setup Cost (K): A common assumption is that the ordering cost consists of a fixed cost, that is independent of the amount ordered, and a variable cost that depends on the amount ordered. The fixed cost is called the setup cost. ($).
  • Product Cost (c): This is the unit cost of purchasing the product as part of an order. If the cost is independent of the amount ordered, the total cost is cz, where c is the unit cost and z is the amount ordered. Alternatively, the product cost may be a decreasing function of the amount ordered. ($/unit).
  •  Holding Cost (h): This is the cost of holding an item in inventory for some given unit of time. It usually includes the lost investment income caused by having the asset tied up in inventory. This is not a real cash flow, but it is an important component of the cost of inventory. If c is the unit cost of the product, this component of the cost is c , where is the discount or interest rate. The holding cost may also include the cost of storage, insurance, and other factors that are proportional to the amount stored in inventory. ($/unit-time).
  • Shortage Cost (p): When a customer seeks the product and finds the inventory empty, the demand can either go unfulfilled or be satisfied later when the product becomes available. The former case is called a lost sale, and the latter is called a backorder. Although lost sales are often important in inventory analysis, they are not considered in this section, so no notation is assigned to it. The total backorder cost is assumed to be proportional to the number of units backordered and the time the customer must wait. The constant of proportionality is p, the per unit backorder cost per unit of time. ($/unit-time).
  • Demand Rate (a): This is the constant rate at which the product is withdrawn from inventory. (units / time).
  • Lot Size (Q): This is the fixed quantity received at each inventory replenishment. (units).
  • Order Level (S): The maximum level reached by the inventory is the order level. When backorders are not allowed, this quantity is the same as Q. When backorders are allowed, it is less than Q. (units).
  • Cycle Time: The time between consecutive inventory replenishments is the cycle time. For the models of this section = Q/a. (time). 
  • Cost per Time (T): This is the total of all costs related to the inventory system that are affected by the decision under consideration. ($/time).
  • Optimum Quantities (Q*, S*, T*): The quantities defined above that maximize profit or minimize cost for a given model are the optimum solution.

 The two models will be further explained  in the forthcoming posts. 
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