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The **economic production quantity** model (also known as the **EPQ model**) determines the quantity a company or retailer should order to minimize the total inventory costs by balancing the inventory holding cost and average fixed ordering cost. The EPQ model was developed by E.W. Taft in 1918. This method is an extension of the economic order quantity model (also known as the EOQ model). The difference between these two methods is that the EPQ model assumes the company will produce its own quantity or the parts are going to be shipped to the company while they are being produced, therefore the orders are available or received in an incremental manner while the products are being produced. While the EOQ model assumes the order quantity arrives complete and immediately after ordering, meaning that the parts are produced by another company and are ready to be shipped when the order is placed.

In inventory management, **economic order quantity** (**EOQ**) is the order quantity that minimizes the total holding costs and ordering costs. It is one of the oldest classical production scheduling models. The model was developed by Ford W. Harris in 1913, but R. H. Wilson, a consultant who applied it extensively, and K. Andler are given credit for their in-depth analysis.

- Overview
- Assumptions
- Variables
- Total cost function and derivation of EPQ formula
- Relevant formulas
- See also
- References

In some literature, "economic manufacturing quantity" model (EMQ) is used for "economic production quantity" model (EPQ). Similar to the EOQ model, EPQ is a single product lot scheduling method. A multiproduct extension to these models is called *product cycling problem*.

EPQ only applies where the demand for a product is constant over the year and that each new order is delivered/produced incrementally when the inventory reaches zero. There is a fixed cost charged for each order placed, regardless of the number of units ordered. There is also a holding or storage cost for each unit held in storage (sometimes expressed as a percentage of the purchase cost of the item).

We want to determine the optimal number of units of the product to order so that we minimize the total cost associated with the purchase, delivery and storage of the product

The required parameters to the solution are the total demand for the year, the purchase cost for each item, the fixed cost to place the order and the storage cost for each item per year. Note that the number of times an order is placed will also affect the total cost, however, this number can be determined from the other parameters

- Demand for items from inventory is continuous and at a constant rate
- Production runs to replenish inventory are made at regular intervals
- During a production run, the production of items is continuous and at a constant rate
- Production set-up/ordering cost is fixed (independent of quantity produced)
- The lead time is fixed
- The purchase price of the item is constant, i.e. no discount is available
- The replenishment is made incrementally

- K = ordering/setup cost per production run
- D = yearly demand rate
- h = yearly holding cost per product
- T = cycle length
- P = yearly production rate
- Q = order quantity

- Holding Cost per Year =

Where is the average inventory level, and is the average holding cost. Therefore, multiplying these two results in the holding cost per year.

- Ordering Cost per Year =

Where are the orders placed in a year, multiplied by K results in the ordering cost per year.

We can notice from the equations above that the total ordering cost decreases as the production quantity increases. Inversely, the total holding cost increases as the production quantity increases. Therefore, in order to get the optimal production quantity we need to set holding cost per year equal to ordering cost per year and solve for quantity (Q), which is the EPQ formula mentioned below. Ordering this quantity will result in the lowest total inventory cost per year.

- Average holding cost per unit time:

- Average ordering and holding cost as a function of time:

- Infinite fill rate for the part being produced: Economic order quantity
- Demand is random: classical Newsvendor model
- Demand varies over time: Dynamic lot size model
- Several products produced on the same machine: Economic lot scheduling problem
- Reorder point

The **newsvendor****model** is a mathematical model in operations management and applied economics used to determine optimal inventory levels. It is (typically) characterized by fixed prices and uncertain demand for a perishable product. If the inventory level is , each unit of demand above is lost in potential sales. This model is also known as the *newsvendor problem* or *newsboy problem* by analogy with the situation faced by a newspaper vendor who must decide how many copies of the day's paper to stock in the face of uncertain demand and knowing that unsold copies will be worthless at the end of the day.

The **economic lot scheduling problem** (**ELSP**) is a problem in operations management and inventory theory that has been studied by a large number of researchers for more than 50 years. The term was first used in 1958 by professor Jack D. Rogers of Berkeley, who extended the economic order quantity model to the case where there are several products to be produced on the same machine, so that one must decide both the lot size for each product and when each lot should be produced. The method illustrated by Jack D. Rogers draws on a 1956 paper from Welch, W. Evert. The ELSP is a mathematical model of a common issue for almost any company or industry: planning what to manufacture, when to manufacture and how much to manufacture.

The **reorder point** (**ROP**) is the level of inventory which triggers an action to replenish that particular inventory stock. It is a minimum amount of an item which a firm holds in stock, such that, when stock falls to this amount, the item must be reordered. It is normally calculated as the forecast usage during the replenishment lead time plus safety stock. In the EOQ model, it was assumed that there is no time lag between ordering and procuring of materials. Therefore the reorder point for replenishing the stocks occurs at that level when the inventory level drops to zero and because instant delivery by suppliers, the stock level bounce back.

In economics, **profit maximization** is the short run or long run process by which a firm may determine the price, input, and output levels that lead to the greatest profit. Neoclassical economics, currently the mainstream approach to microeconomics, usually models the firm as maximizing profit.

In economics, **elasticity** is the measurement of the proportional change of an economic variable in response to a change in another. It shows how easy it is for the supplier and consumer to change their behavior and substitute another good, the strength of an incentive over choices per the relative opportunity cost.

**Price elasticity of demand** is a measure used in economics to show the responsiveness, or elasticity, of the quantity demanded of a good or service to a change in its price when nothing but the price changes. More precisely, it gives the percentage change in quantity demanded in response to a one percent change in price.

The **break-even point** (BEP) in economics, business—and specifically cost accounting—is the point at which total cost and total revenue are equal, i.e. "even". There is no net loss or gain, and one has "broken even", though opportunity costs have been paid and capital has received the risk-adjusted, expected return. In short, all costs that must be paid are paid, and there is neither profit nor loss.

In economics, **marginal cost** is the change in the total cost that arises when the quantity produced is incremented by one unit; that is, it is the cost of producing one more unit of a good. Intuitively, marginal cost at each level of production includes the cost of any additional inputs required to produce the next unit. At each level of production and time period being considered, marginal costs include all costs that vary with the level of production, whereas other costs that do not vary with production are fixed and thus have no marginal cost. For example, the marginal cost of producing an automobile will generally include the costs of labor and parts needed for the additional automobile but not the fixed costs of the factory that have already been incurred. In practice, marginal analysis is segregated into short and long-run cases, so that, over the long run, all costs become marginal. Where there are economies of scale, prices set at marginal cost will fail to cover total costs, thus requiring a subsidy. Marginal cost pricing is not a matter of merely lowering the general level of prices with the aid of a subsidy; with or without subsidy it calls for a drastic restructuring of pricing practices, with opportunities for very substantial improvements in efficiency at critical points.

In macroeconomics, **Aggregate Demand** (**AD**) or **Domestic Final Demand** (**DFD**) is the total demand for final goods and services in an economy at a given time. It is often called effective demand, though at other times this term is distinguished. This is the demand for the gross domestic product of a country. It specifies the amounts of goods and services that will be purchased at all possible price levels.

In economics, a **production function** gives the technological relation between quantities of physical inputs and quantities of output of goods. The production function is one of the key concepts of mainstream neoclassical theories, used to define marginal product and to distinguish allocative efficiency, a key focus of economics. One important purpose of the production function is to address allocative efficiency in the use of factor inputs in production and the resulting distribution of income to those factors, while abstracting away from the technological problems of achieving technical efficiency, as an engineer or professional manager might understand it.

A **limit price** is a price, or pricing strategy, where products are sold by a supplier at a price low enough to make it unprofitable for other players to enter the market.

In economics and in particular neoclassical economics, the **marginal product** or **marginal physical productivity** of an input is the change in output resulting from employing one more unit of a particular input, assuming that the quantities of other inputs are kept constant.

**Cournot competition** is an economic model used to describe an industry structure in which companies compete on the amount of output they will produce, which they decide on independently of each other and at the same time. It is named after Antoine Augustin Cournot (1801–1877) who was inspired by observing competition in a spring water duopoly. It has the following features:

In marketing, **carrying cost**, **carrying cost of inventory** or **holding cost** refers to the total cost of holding inventory. This includes warehousing costs such as rent, utilities and salaries, financial costs such as opportunity cost, and inventory costs related to perishability, *shrinkage* (leakage) and insurance. Carrying cost also includes the opportunity cost of reduced responsiveness to customers' changing requirements, slowed introduction of improved items, and the inventory's value and direct expenses, since that money could be used for other purposes. When there are no transaction costs for shipment, carrying costs are minimized when no excess inventory is held at all, as in a Just In Time production system.

**Material theory** is the sub-specialty within operations research and operations management that is concerned with the design of production/inventory systems to minimize costs: it studies the decisions faced by firms and the military in connection with manufacturing, warehousing, supply chains, spare part allocation and so on and provides the mathematical foundation for logistics. The **inventory control problem** is the problem faced by a firm that must decide how much to order in each time period to meet demand for its products. The problem can be modeled using mathematical techniques of optimal control, dynamic programming and network optimization. The study of such models is part of inventory theory.

The **dynamic lot-size model** in inventory theory, is a generalization of the economic order quantity model that takes into account that demand for the product varies over time. The model was introduced by Harvey M. Wagner and Thomson M. Whitin in 1958.

The **Silver–Meal heuristic** method was composed in 1973 by Edward A. Silver and H.C. Meal. It refers to production planning in manufacturing and its purpose is to determine production quantities to meet the requirement of operations at minimum cost.

The **profit model** is the linear, deterministic algebraic model used implicitly by most cost accountants. Starting with, profit equals sales minus costs, it provides a structure for modeling cost elements such as materials, losses, multi-products, learning, depreciation etc. It provides a mutable conceptual base for spreadsheet modelers. This enables them to run deterministic simulations or 'what if' modelling to see the impact of price, cost or quantity changes on profitability.

A **Monopoly price** is set by a Monopoly. A monopoly occurs when a firm lacks any viable competition, and is the sole producer of the industry's product. Because a monopoly faces no competition, it has absolute market power, and thereby has the ability to set a monopoly price that will be above the firm's marginal (economic) cost. Since marginal cost is the increment in total required to produce an additional unit of the product, the firm would be able to make a positive economic profit if it produced a greater quantity of the product and sold it at a lower price.

The **base stock model** is a statistical model in inventory theory. In this model inventory is refilled one unit at a time and demand is random. If there is only one replenishment, then the problem can be solved with the newsvendor model.

- Taft, E. W. "The most economical production lot." Iron Age 101.18 (1918): 1410-1412.
- Gallego, G. "IEOR4000: Production Management" (Lecture 2), Columbia (2004).
- Stevenson, W. J. "Operations Management" PowerPoint slide 19, The McGraw-Hill Companies (2005).
- Kroeger, D. R. "Determining Economic Production in a Continuous Process" IIE Process Industries Webinar, IIE (2009).
- Cárdenas-Barrón, L. E. "The Economic Production Quantity derived Algebraically" International Journal of Production Economics, Volume 77, Issue 1, (2002).
- Blumenfeld, D. "Inventory" Operations Research Calculations Handbook, Florida (2001)
- Harris, F.W. "How Many Parts To Make At Once" Factory, The Magazine of Management, 10(2), 135-136, 152 (1913).

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