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Mg211.1 Operational Research Techniques Notes

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MG211: Operational Research Methods

MG211.1: OPERATIONAL RESEARCH TECHNIQUES Lecture 1: Introduction - Topics, History and Applications Operational Research: (1) The application of scientific methods to make better decisions and improve the effectiveness of operational, decisions and management (2) The science of efficient allocation and usage of scarce resources
- Development fostered by computer revolution in mid-twentieth century
- Comprises of both theoretical methods and application methods
- Key step in application is modelling: constructing mathematical representations of realworld problems that accurately represent essential features
- Phases of an OR Study (Quantitative; Qualitative): (1) Define problem and gather relevant information (2) Formulate mathematical model to represent problem (3) Develop methods to obtain solutions (4) Model validation and refinement (5) Preparing for ongoing application of the model as prescribed by management (6) Implement Historical Background
- Originated in military operations during World War 2 in
- Emerged in multiple scientific communities simultaneously: UK/US/Soviet Union
- Application of mathematical methods resulted in multiple policies contradicting military common sense which turned out to be successful (i.e. Blackett's Circus vs German UBoats: increased successful sinkings from 2% to 45%; considering whether to organise defence in large number of small convoys or small number of large convoys)
- Most fundamental tool of OR is linear programming (LP) which includes The Simplex Method (most heavily used optimisation algorithm)
- After war industrial boom led to increase in size of corporations and production --->
triggered demand of systematic modelling to increase efficiency (particularly regarding computers in the Age of the Internet and Electronic commerce in the 1990s) Application Context: Google Search/Online Shopping
- MARKOV CHAIN MODEL: Google PageRank - Identifies and ranks the most relevant queries using a circular definition (i.e. a page is important if it is linked by important pages)
- MATCHING MODEL: Google earns revenue through ad allocation; to match ads with slots Google uses complex matching models which finds the best assignment satisfying certain requirements e.g. if searching for phones ads for websites selling phones (other applications include school admissions, job market, kidney transplantation)
- INVENTORY MODEL: When clicking on phone ad redirects to O2 shop which lets you know if the product you are looking at is in stock using inventory models (help keep enough stock satisfy most customers without storing excessive amounts)
- DELIVERY: If product is bought and needs to be delivered, OR used to find shortest path and optimal route for van to deliver all packages (links to travelling salesman problem)
- PRICING: Uses game theory (how to set the prices to make the most profit and beat competitors)
- Other applications involved in queueing (queueing theory), printed circuit manufacturing (travelling salesman problem), routing mobile traffic through networks

MG211: Operational Research Methods

Lecture 2: Deterministic Inventory Management PRIMARY OBJECTIVE: Minimise Cost (not only possible objective - Toyota Just In Time) Assume full knowledge of expected demand Variables:

* Sequence of Time Periods - Models where demands & costs change over time

* How Demand (d) Given - Deterministic/stochastic

* Cost of Ordering - Represented by function C(z); simplest case linear (cz) but can also include fixed costs (K)

* Holding Cost - Generally assumed linear; assessed on period-by-period basis

* Shortage Cost - Loss incurred from unsatisfied demands

* Review Mode - Continuous (orders placed any time) or periodic (new orders decided at fixed time period)

* Lead Time - Time between ordered place and goods arrive EOQ Model Use model to determine optimal levels of Q (order quantity) Assumptions:

* Deterministic demand (constant rate d)

* Cost of ordering z > 0 units is K + cz

* Holding cost is h per time period

* No backlogging (planned shortages)

* Continuous review mode - inventory replenished any time

* Lead time = 0

Key Equations: Length of Cycle: t = Q/d Order Cost per Cycle: K + cQ Holding Cost per Cycle = hQ2/2d Cost Per Unit Time = Kd/Q + dc + hQ/2 Optimum Value: Q* = [?]2Kd/h Optimum Length of Time Cycle: t* = Q*/d " " " " "
= [?]2K/dh

Stock level
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EOQ Model With Planned Shortages Relax assumption of no planned shortage; allow shortage cost p per unit per period Optimal replenishing policy defined by optimal batch size (Q) and inventory level just after new batch arrives and delayed demands are satisfied (S); Q S satisfied with delay

2!/$

Time

Key Equations: Order Cost per Cycle: K + cQ Holding Cost per Cycle = hS2/2d Shortage Cost Per Cycle: p(Q-S)2/2d Cost Per Unit Time = Kd/Q + dc + hS2/2Q + p(Q-S)2/2Q Q* = [?](2dK/h * p+h/p) S* = [?](2dK/h * p/p+h) Optimum Length of Time Cycle: t* = Q*/d " " " "
= [?](2K/dh * p+h/p)

MG211: Operational Research Methods

EOQ Model for Serial Two Echelons (Simplest multiechelon system) Echelon: Every stage in procedure between production and warehouse Assumptions:

* Deterministic demand (constant rate d)

* Cost of ordering z > 0 units is K + cz

* Holding cost is h per time period

* No backlogging (planned shortages)

* Continuous review mode - inventory replenished any time

* Lead time = 0

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