Lecture

Markov Chain: Long-Run Average Costs

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Description

This lecture covers the Markov Law of Large Numbers (LLN) applied to an irreducible Markov chain with transition matrix P and invariant distribution π. It discusses the first return time to a state, reward functions, and the long-run average costs in an (S, S) inventory model. The proof of the Markov LLN implies that the expectation of the reward converges to a limit. The lecture also explores the dynamics of the inventory level based on demand, the cost of lost customers, and the unique invariant distribution. It concludes with the implications of the Markov LLN on the long-run average holding cost and the cost of lost customers.

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Related concepts (32)

Inventory theory

Material theory (or more formally the mathematical theory of inventory and production) 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.

Inventory control

Inventory control or stock control can be broadly defined as "the activity of checking a shop's stock". It is the process of ensuring that the right amount of supply is available within a business. However, a more focused definition takes into account the more science-based, methodical practice of not only verifying a business's inventory but also maximising the amount of profit from the least amount of inventory investment without affecting customer satisfaction.

Inventory

Inventory (American English) or stock (British English) refers to the goods and materials that a business holds for the ultimate goal of resale, production or utilisation. Inventory management is a discipline primarily about specifying the shape and placement of stocked goods. It is required at different locations within a facility or within many locations of a supply network to precede the regular and planned course of production and stock of materials.

Inventory management software

Inventory management software is a software system for tracking inventory levels, orders, sales and deliveries. It can also be used in the manufacturing industry to create a work order, bill of materials and other production-related documents. Companies use inventory management software to avoid product overstock and outages. It is a tool for organizing inventory data that before was generally stored in hard-copy form or in spreadsheets.

Newsvendor model

The newsvendor (or newsboy or single-period or salvageable) 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.

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