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.
Scheduling is the process of arranging, controlling and optimizing work and workloads in a production process or manufacturing process. Scheduling is used to allocate plant and machinery resources, plan human resources, plan production processes and purchase materials.
EOQ applies only when demand for a product is constant over the year and each new order is delivered in full when inventory reaches zero. There is a fixed cost for each order placed, regardless of the number of units ordered. There is also a cost for each unit held in storage, commonly known as holding cost, sometimes expressed as a percentage of the purchase cost of the item.
Demand is the quantity of a good that consumers are willing and able to purchase at various prices during a given period of time.
We want to determine the optimal number of units 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, though this number can be determined from the other parameters.
The single-item EOQ formula finds the minimum point of the following cost function:
Total Cost = purchase cost or production cost + ordering cost + holding cost
To determine the minimum point of the total cost curve, calculate the derivative of the total cost with respect to Q (assume all other variables are constant) and set it equal to 0:
Solving for Q gives Q* (the optimal order quantity):
Q* is independent of P; it is a function of only K, D, h.
The optimal value Q* may also be found by recognising that
where the non-negative quadratic term disappears for which provides the cost minimum
Economic order quantity = = 400 units
Number of orders per year (based on EOQ)
If we check the total cost for any order quantity other than 400(=EOQ), we will see that the cost is higher. For instance, supposing 500 units per order, then
Similarly, if we choose 300 for the order quantity then
This illustrates that the economic order quantity is always in the best interests of the firm.
An important extension to the EOQ model is to accommodate quantity discounts. There are two main types of quantity discounts: (1) all-units and (2) incremental.Here is a numerical example:
In order to find the optimal order quantity under different quantity discount schemes, one should use algorithms; these algorithms are developed under the assumption that the EOQ policy is still optimal with quantity discounts. Perera et al. (2017)establish this optimality and fully characterize the (s,S) optimality within the EOQ setting under general cost structures.
In presence of a strategic customer, who responds optimally to discount schedule, the design of optimal quantity discount scheme by the supplier is complex and has to be done carefully. This is particularly so when the demand at the customer is itself uncertain. An interesting effect called the "reverse bullwhip" takes place where an increase in consumer demand uncertainty actually reduces order quantity uncertainty at the supplier.
Several extensions can be made to the EOQ model, including backordering costsand multiple items. Additionally, the economic order interval can be determined from the EOQ and the economic production quantity model (which determines the optimal production quantity) can be determined in a similar fashion.
A version of the model, the Baumol-Tobin model, has also been used to determine the money demand function, where a person's holdings of money balances can be seen in a way parallel to a firm's holdings of inventory.
Malakooti (2013)has introduced the multi-criteria EOQ models where the criteria could be minimizing the total cost, Order quantity (inventory), and Shortages.
A version taking the time-value of money into account was developed by Trippi and Lewin.
Another important extension of EOQ model is to consider items with imperfect quality. Salameh and Jaber (2000) are the first to study the imperfect items in an EOQ model very thoroughly. They consider an inventory problem in which
the demand is deterministic and there is a fraction of imperfect items in the lot and are screened by the buyer and sold by them at the end of the circle at discount price.Imperfect quality items have also been considered in a decentralized supply chain and the problem has also been studied with game theoretical models.
Recently an interesting similarity between EOQ of Melon picking and fuel injection in Gasoline Direction Injection has been proposed.
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 physics and mathematics, the heat equation is a partial differential equation that describes how the distribution of some quantity evolves over time in a solid medium, as it spontaneously flows from places where it is higher towards places where it is lower. It is a special case of the diffusion equation.
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.
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 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.
In physics, the Planck charge, denoted by , is one of the base units in the system of natural units called Planck units. It is a quantity of electric charge defined in terms of fundamental physical constants.
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:
Nondimensionalization is the partial or full removal of units from an equation involving physical quantities by a suitable substitution of variables. This technique can simplify and parameterize problems where measured units are involved. It is closely related to dimensional analysis. In some physical systems, the term scaling is used interchangeably with nondimensionalization, in order to suggest that certain quantities are better measured relative to some appropriate unit. These units refer to quantities intrinsic to the system, rather than units such as SI units. Nondimensionalization is not the same as converting extensive quantities in an equation to intensive quantities, since the latter procedure results in variables that still carry units.
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.
Constant elasticity of substitution (CES), in economics, is a property of some production functions and utility functions.
The newsvendormodel 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 production quantity 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. 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 actuarial science and applied probability ruin theory uses mathematical models to describe an insurer's vulnerability to insolvency/ruin. In such models key quantities of interest are the probability of ruin, distribution of surplus immediately prior to ruin and deficit at time of ruin.
The direct-quadrature-zerotransformation or zero-direct-quadraturetransformation is a tensor that rotates the reference frame of a three-element vector or a three-by-three element matrix in an effort to simplify analysis. The DQZ transform is the product of the Clarke transform and the Park transform, first proposed in 1929 by Robert H. Park.
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.
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.