Models

Most operations research studies involve the construction of a mathematical model. The model is a collection of logical and mathematical relationships that represents aspects of the situation under study. Models describe important relationships between variables, include an objective function with which alternative solutions are evaluated, and constraints that restrict solutions to feasible values.

Although the analyst would hope to study the broad implications of the problem using a systems approach, a model cannot include every aspect of a situation. A model is always an abstraction that is of necessity simpler than the real situation. Elements that are irrelevant or unimportant to the problem are to be ignored, hopefully leaving sufficient detail so that the solution obtained with the model has value with regard to the original problem.

Models must be both tractable, capable of being solved, and valid, representative of the original situation. These dual goals are often contradictory and are not always attainable. It is generally true that the most powerful solution methods can be applied to the simplest, or most abstract, model.

We provide in this section, a description of the various types of models used by operations research analysts. The division is based on the mathematical form of the model. All the models described here are solved with Excel add-ins described in the Computation section of this site. In some cases, the methods used to solve a model are described in the Methods section. Student exercises for creating models are in the Problems section. Additional models related to problems arising in Operations Management and Industrial Engineering are in the OM/IE section.

	Operations research (OR) is a discipline explicitly devoted to aiding decision makers. This section reviews the terminology of OR, a process for addressing practical decision problems and the relation between Excel models and OR.
	Linear Programming	A typical mathematical program consists of a single objective function, representing either a profit to be maximized or a cost to be minimized, and a set of constraints that circumscribe the decision variables. In the case of a linear program (LP) the objective function and constraints are all linear functions of the decision variables. At first glance these restrictions would seem to limit the scope of the LP model, but this is hardly the case. Because of its simplicity, software has been developed that is capable of solving problems containing millions of variables and tens of thousands of constraints. Countless real-world applications have been successfully modeled and solved using linear programming techniques.
	Network Flow Programming	The term network flow program describes a type of model that is a special case of the more general linear program. The class of network flow programs includes such problems as the transportation problem, the assignment problem, the shortest path problem, the maximum flow problem, the pure minimum cost flow problem, and the generalized minimum cost flow problem. It is an important class because many aspects of actual situations are readily recognized as networks and the representation of the model is much more compact than the general linear program. When a situation can be entirely modeled as a network, very efficient algorithms exist for the solution of the optimization problem, many times more efficient than linear programming in the utilization of computer time and space resources.
	Integer Programming	Integer programming is concerned with optimization problems in which some of the variables are required to take on discrete values. Rather than allow a variable to assume all real values in a given range, only predetermined discrete values within the range are permitted. In most cases, these values are the integers, giving rise to the name of this class of models. Models with integer variables are very useful. Situations that cannot be modeled by linear programming are easily handled by integer programming. Primary among these involve binary decisions such as yes-no, build-no build or invest-not invest. Although one can model a binary decision in linear programming with a variable that ranges between 0 and 1, there is nothing that keeps the solution from obtaining a fractional value such as 0.5, hardly acceptable to a decision maker. Integer programming requires such a variable to be either 0 or 1, but not in-between. Unfortunately integer programming models of practical size are often very difficult or impossible to solve.

Nonlinear Programming	When expressions defining the objective function or constraints of an optimization model are not linear, one has a nonlinear programming model. Again, the class of situations appropriate for nonlinear programming is much larger than the class for linear programming. Indeed it can be argued that all linear expressions are really approximations for nonlinear ones. Since nonlinear functions can assume such a wide variety of functional forms, there are many different classes of nonlinear programming models. The specific form has much to do with how easily the problem is solve, but in general a nonlinear programming model is much more difficult to solve than a similarly sized linear programming model.
	Dynamic Programming	Dynamic programming (DP) models are represented in a different way than other mathematical programming models. Rather than an objective function and constraints, a DP model describes a process in terms of states, decisions, transitions and returns. The process begins in some initial state where a decision is made. The decision causes a transition to a new state. Based on the starting state, ending state and decision a return is realized. The process continues through a sequence of states until finally a final state is reached. The problem is to find the sequence that maximizes the total return. The models considered here are for discrete decision problems. Although traditional integer programming problems can be solved with DP, the models and methods are most appropriate for situations that are not easily modeled using the constructs of mathematical programming. Objectives with very general functional forms may be handled and a global optimal solution is always obtained. The price of this generality is computational effort. Solutions to practical problems are often stymied by the "curse of dimensionally" where the number of states grows exponentially with the number of dimensions of the problem.
	Stochastic Programming	The mathematical programming models, such as linear programming, network flow programming and integer programming generally neglect the effects of uncertainty and assume that the results of decisions are predictable and deterministic.  This abstraction of reality allows large and complex decision problems to be modeled and solved using powerful computational methods.  Stochastic programming explicitly recognizes uncertainty by using random variables for some aspects of the problem. With probability distributions assigned to the random variables, an expression can be written for the expected value of the objective to be optimized. Then a variety of computational methods can be used to maximize or minimize the expected value. This page provides a brief introduction to the modeling process.
	Combinatorial Optimization	The most general type of optimization problem and one that is applicable to most spreadsheet models is the combinatorial optimization problem. Many spreadsheet models contain variables and compute measures of effectiveness. The spreadsheet user often changes the variables in an unstructured way to look for the solution that obtains the greatest or least of the measure. In the words of OR, the analyst is searching for the solution that optimizes an objective function, the measure of effectiveness. Combinatorial optimization provides tools for automating the search for good solutions and can be of great value for spreadsheet applications.
	Stochastic Processes	In many practical situations the attributes of a system randomly change over time. Examples include the number of customers in a checkout line, congestion on a highway, the number of items in a warehouse, and the price of a financial security, to name a few. When aspects of the process are governed by probability theory, we have a stochastic process.  The model is described in part by enumerating the states in which the system can be found.  The state is like a snapshot of the system at a point in time that describes the attributes of the system. The example for this section is an Automated Teller Machine (ATM) system and the state is the number of customers at or waiting for the machine. Time is the linear measure through which the system moves. Events occur that change the state of the system. For the ATM example the events are arrivals and departures. In this section we describe the basic ideas associated with modeling a stochastic process that are useful for both Discrete and Continuous Time Markov Chains.
	Discrete Time Markov Chains	Say a system is observed at regular intervals such as every day or every week. Then the stochastic process can be described by a matrix which gives the probabilities of moving to each state from every other state in one time interval. Assuming this matrix is unchanging with time, the process is called a Discrete Time Markov Chain (DTMC). Computational techniques are available to compute a variety of system measures that can be used to analyze and evaluate a DTMC model. This section illustrates how to construct a model of this type and the measures that are available.
	Continuous Time Markov Chains	Here we consider a continuous time stochastic process in which the duration of all state changing activities are exponentially distributed. Time is a continuous parameter. The process satisfies the Markovian property and is called a Continuous Time Markov Chain (CTMC). The process is entirely described by a matrix showing the rate of transition from each state to every other state. The rates are the parameters of the associated exponential distributions. The analytical results are very similar to those of a DTMC. The ATM example is continued with illustrations of the elements of the model and the statistical measures that can be obtained from it.
	Simulation	When a situation is affected by random variables it is often difficult to obtain closed form equations that can be used for evaluation. Simulation is a very general technique for estimating statistical measures of complex systems. A system is modeled as if the random variables were known. Then values for the variables are drawn randomly from their known probability distributions. Each replication gives one observation of the system response. By simulating a system in this fashion for many replications and recording the responses, one can compute statistics concerning the results. The statistics are used for evaluation and design.

Operation Research -Pendahuluan

OPERATION RESEARCH (OR)
Perkembangan Operation research
Digunakan tahun 1940 oleh Mc Closky dan Trefthen disuatu kota di Inggris
digunakan oleh pemimpin militer Inggris untuk mencari cara-cara yang efisien untuk menggunakan alat yang baru ditemukan untuk menghadapi serangan udara
Setelah perang, keberhasilan kelompok peneliti operasi-operasi dibidang militer menarik perhatian para industriawan yang mencari penyelesaian masalah-masalah yang rumit
Akhirnya, pada tahun lima puluhan, di Inggris dan di Amerika,tehinik-tehnik program linier dan dinamik ditemukan dan diperluas
Pada saat ini OR mulai mendapat pengakuan sebagai pelajaran yang bermanfaat di Perguruan Tinggi dan materi menjadi makin banyak dan penting bagi mahasiswa

Arti Operation Research
Adalah memutuskan secara ilmiah bagaimana merancang dan menjalankan sistem manusia-mesin dengan yang terbaik, yang biasanya membutuhkan alokasi sumber daya yang langka

Model dalam OR
Model adalah abstraksi atau penyederhanaan realistis sistem yang komplek dimana hanya komponen-komponen yang relevan atau factor-faktor yang dominan dari masalah yang dianalisa diikutsertakan

Model dapat diklasifikasikan menurut jenisnya
Iconic Model
Analogue Model
Mathematical Model

Tahap-tahap Dalam OR
Merumuskan Masalah
Pertama kali suatu difinisi persoalan yang tepat harus dirumuskan. Dalam perumusan masalah ini ada tiga pertanyaan penting yang harus dijawab
Variabel Keputusan
Tujuan (objective)
Kendala (constraint)
2. Pembentukan model
sesuai dengan difinisi persoalan, pengambil keputusan menentukan model yang paling cocok untuk mewakili sistem, karena jika model yang dihasilkan cocok dengan salah satu model matematik yang biasa maka solusinya dengan mudah diperoleh
3. Mencari penyelesaian masalah
Pada tahap ini bermacam-macam tehnik dan metode solusi kuantitatif memasuki proses
4. Validasi Model
5. Penerapan hasil akhir

Ciri-ciri OR
1. OR merupakan pendekatan kelompok antar disiplin untuk mencari hasil yang optimum
2. OR menggunakan tehnik penelitian ilmiah untuk mendapatkan solusi optimum
3. OR hanya hanya memperbaiki kualitas solusi

Kelemahan OR
1. Perumusan masalah dalm suatu program OR adalah suatu tugas yang sulit
2. Jika organisasi mempunyai beberapa tujuan yang bertentangan maka organisasi tidak dapat mencapai yang terbaik secara serempak
3. Suatu hub yang non linier yang diubah menjadi linier dengan program linier dapat menggganggu solusi yang disaranka

LINIER PROGRAMMING
Linier Programming merupakan suatu model yang dipergunakan untuk pemecahan masalah pengalokasian sumber-sumber yang terbatas secara optimal
LP mencakup perencanaan kegiatan-kegiatan untuk mencapai suatu hasil yang optimal yaitu suatu hasil yang mencerminkan tercapainya sasaran tertentu paling baik diantara alaternatif-alternatif yang mungkin dengan mempergunakan funsi linier
Di dalam model LP dikenal 2 macam fungsi yaitu
1. Fungsi tujuan (objectiv function) adalah fungsi yang menggambarkan tujuan permasalahan LP yang terkait dengan pengaturan secara optimal sumber daya-sumberdaya, untuk memperoleh keuntungan mak atau biaya yang minimal dinyatakan dengan Z
2. Fungsi batasan (constraint function) merupakan bentuk penyajian secara matematis batasan-batasan kapasitas yang tersedia yang dialokasikan secara optimal keberbagai kegiatan

Asumsi LP
1. Proportionality : naik turunnya nilai Z dan penggunaan sumber atau fasilitas yang tersedia akan berubah secara proporsional
2. Additivity : nilai tujuan tiap kegiatan tidak saling mempengaruhi atau setiap kenaikkan nilai Z yang diakibatkan oleh kenaikkan suatu kegiatan dapat ditambahkan tanpa mempengaruhi bagian nilai Z yang diperoleh dari kegiatan lain
3. Divisibility : Output yang dihasilakn oleh setiap kegiatan dapat berupa bilangan pecahan
4. Deterministic : semua parameter yang ada pada model LP dapat diperkirakan dengan pasti.

Contoh soal 1
Perusahaan sepatu “Ideal” membuat dua macam sepatu. Macam pertama merek X1, dengan sol dari karet, dan macam kedua merek X2, dengan sol dari kulit. Untuk membuat sepatu-sepatu itu perusahaan memiliki 3 macam mesin. Mesin 1 khusus membuat sol dari karet, mesin 2 khusus membuat sol dari kulit, dan mesin 3 membuat bagian atas sepatu dan melakukan assembeling bagian atas dengan sol.
Setiap lusin sepatu merek X1 mula-mula dikerjakan di mesin 1 selama 2 jam, kemudian tanpa melalui mesin 2 terus dikerjakan di mesin 3 selam 6 jam.
Sedangkan untuk sepatu merek X2 tidak diproses di mesin 1, tetapi pertama kali dikerjakan di mesin 2 selama 3 jam kemudian di mesin 3 selam 5 jam
Jam kerja maksimum setiap hari untuk mesin 1 = 8 jam, mesin 2 = 15 jam, dan mesin 3 = 30 jam. Sumbagan terhadap laba untuk setiap lusin sepatu merek X1 = Rp. 30.000,00 sedangkan merek X2 = Rp. 50.000,00.
Masalahnya adalah menentukan berapa lusin sebaiknya sepatu merek X1 dan merek X2 dibuat agar bisa memaksimumkan laba.

Langkah-Langkah metode Grafik
1. Menentukan fungsi tujuan dan memformulasikannya dalam bentuk matematis
2. Mendifinisikan batasan-batasan yang berlaku dan memformulasikannya dalam bentuk matematis
3. Menggambarkan masing-masing fungsi batasan dalam satu sistem salib sumbu
3. Mencari titik yang paling menguntungkan dihubungkan dengan fungsi tujuan

Pacar Baruku.

Inventory WH MANAGEMENT

Facility Management

Housekeeping is not just for homes. The environment in your warehouse reflects your expectations from vendors and workers. A sloppy warehouse in disrepair shows the business does not really care how efficiently or safely the work is done. As a result, workers cut corners and do as little as possible to walk away with a paycheck. After all, if the person with the greatest stake in the business does not care, why should they?
Even older warehouses can be kept in good working order and neatened up. You should have workers responsible for cleaning up at shift changes and be certain the building is in sound working condition. Visual reminders to employees about cleanliness and safety help to show them you care about running a safe and efficient operation. Experienced businessmen will tell you that no sloppy warehouse has efficient and motivated workers and no bright, clean operation tolerates sloppy workers.

Slotting Optimization

The places you choose to store stock within the warehouse makes a huge different in picking time, accuracy and safety. By creating a picking or slotting profile in your warehouse, you can ensure efficient operations and give your business the ability to easy adapt and change to market trends in ordering.  If your slots are too small, you will be replacing stock more frequently than necessary. Too large, and you will waste space and making your workers travel farther to pick orders.
When planning your picking profile, first consider the items that come in and out of your warehouse the fastest. Ensure you have allocated slots for these items that make receiving, picking and shipping faster. Obviously, the slots must be set up in a way that maximizes your ability to store and move such items in relation to their size and weight. They should be easy to access with all necessary worker safety gear nearby.
When initiated from the beginning, slotting helps your business to evolve as it grows. You can set up the appropriate hardware and shelving in advance. Otherwise, you may need to set aside time to reorganize your warehouse and invest in new storage solutions. Clearly, this is not the best option for your business, so get it right from the beginning if at all possible.
There is software available that uses the science of product slotting to help you get the most from your warehouse space. By using the measurements of a product and its order frequency, you can calculate the best locations in your warehouse. The software calculates and compares storage combinations until you come up with the optimal layout for your warehouse space. You can then change input in comparison with market trends to reconfigure as necessary.

Taking Advantage of Technology

In the old days, warehouses were run through individual order and picking slips that were sorted by hand. One worker would highlight items and makes notes for the picker, which would then be used to locate and pack orders. This method took an enormous amount of energy and employee resources, creating a bottle neck in operations.
The modern warehouse uses various technologies for optimizing efficiency. This can be as simple as ensuring a computerized picking system or as complicated as using robotic means to pick orders. Many companies now use voice systems to direct warehouse floor employees in all activities including equipment checks and order picking.
Technology benefits the warehouse and the entire business by improving speed and accuracy. Voice technology is the latest trend in warehouse management, focused on keeping workers safe and productive. Voice technology allows order pickers to work hands free. Instead of holding a piece of paper in one hand and driving a fork lift with the other, workers are able to keep both eyes on the warehouse floor, dramatically reducing warehouse traffic accidents.
Some warehouses have doubled efficiency by using this technology. Not only does this technology direct employee activities, it tracks inventory, eliminating the need for barcodes and scanning. Not only are the workers more productive, they are happier, resulting in a 50% reduction in turnover. In the future, voice technology will direct stowage and replenishment as well. It may even be used in cycle counting, receiving and yard management.

Labor Management

To keep things running smoothly, you must have the right employees for the job. An effective warehouse supervisor is needed to coordinate receiving, stowage, picking and shipping. There is a fine line to walk in balancing speed and efficiency with worker safety. Injuries damage morale and the company’s bottom line. It is not enough to keep employees safe, they must also be kept happy to prevent turnover.
Supervisors must also understand the aspects of your operation dealing with point of sale and supplier relations. Otherwise, they will not be able to initiate procedures in the warehouse that can benefit other aspects of your business. Good customer service starts at the warehouse, making your warehouse supervisor an important foundation to successful business relations with your customers.
The supervisor must know his subordinates jobs as well. He must be able to do all tasks that other employees perform so that he can train new employees and optimize operations for long standing employees. It is important that your supervisors are provided with structured training materials, manuals and software to teach proper safety and handling procedures to workers. Informal on the job training is more costly to efficiency and safety in the long run.
Instituting a long-term training and development program for both supervisors and subordinates allows businesses to reduce turnover. Employees trained under such programs are more satisfied, capable and efficient. Developing such a program will pay for itself in lower turnover, higher productivity and fewer work injuries.
The warehouse is like the human heart, taking in products and pumping them to where they are needed. When warehouse productivity slips, the entire organization is effected. By paying attention to the facility itself, the contents in it and the people running the operation, you will ensure the life of your business continues to thrive.

Material Handling

Moving and Handling Material

The moving and handling of materials must be done with the proper equipment by experienced and trained professionals. Using the wrong equipment or letting just anyone try to move and store materials can lead to accidents and slow down production progress. Equipment that is used must be big enough to safely handle the load being transported. Attention has to be paid to height, weight, and leverage.  being used must be of a size and have the power to handle the load safely.
Experience is a necessary in lifting and moving materials around a business or job site. An operator needs to have a working
knowledge of how to stack items and where to store them so that they are not in the way. In a retail business, you don’t want to place any times that customers might run into, trip over, or otherwise hurt themselves. In construction or warehouse settings, one must always think safety and have the ability to react quickly.
Experienced operators and handlers  should plan every lift and move. They must make certain that their path is free of all obstacles and pedestrians. If line of sight is difficult, you should use someone as a spotter to help guide you.
Storage of materials is a part of material handling and very important. must not create a hazard. Storage areas can easily create a hazard and slow down production of stored improperly. Areas should be kept free of scattered materials that may cause someone to trip and fall. Hazardous materials should be carefully stored so as not to cause fires or pose a threat to employee health. You should also be mindful of pests such as rodents that can get into stored materials and cause damage. When stacking materials for storage, keep in mind such factors as the  height and weight of the combined stacks, the condition of the containers, and how accessible the materials are to the business that uses them.

Material handling, in its most basic of terms, is the moving of materials from point A to point B. Every business involves some form of material handling. It may be the moving of crates around a warehouse or boxes of paper from the storage closet to the office. Material handling is found in many different fields and industries such as construction, manufacturing, shipping, research, and retail. There are many methods used to handle materials and different equipment depending on the type of materials that need to be moved. Utilizing efficient handling and storage of materials is a vital part of any industry as it provides a continuous flow of materials and reduces the stress of labor. So what does material handling involve? Are there any safety concerns? What kind of equipment is used? Here is a closer look at material handling.

Why Is Material Handling Important

Material handling is important because it is a reliable means to transport goods and materials to areas where they are needed. They help keep production flowing. Without proper material handling, production slows down. There are many companies out there that specialize in providing material handling services for any type of job or industry.
Safety is also an important aspect of material handling. The methods and equipment used are designed to help make tasks easier and (if safety guidelines are observed) to help decrease injuries while on the job. Accidents can easily occur from unsafe or improperly handled equipment and materials. Workers frequently cite the weight and bulkiness of objects being lifted as major contributing factors to their injuries, one of the most common of which is back injuries. In 1990, back injuries resulted in 400,000 workplace accidents. The majority of these types of injuries occurred from body movement such as bending and lifting. In addition, other accidents can occur from such things as falling objects that were not stacked properly to items that were not stored properly.

Material Handling Methods and Equipment

There is not one piece of equipment that is designed to carry out all forms of material handling. It takes many different kinds of equipment that all perform different functions. Some of them do have some overlapping functions but most have only one purpose. Here are some of the more common types of material handling equipment: cranes, slings, moving trucks, forklifts, pallet jacks, hand dollys, conveyors, trailers, storage bins, pallets, and storage containers.
What is more is that sometimes material handling involves landscaping, excavating, or demolition. In this case, the materials would be things such as dirt, sand, rocks, broken masonry, or debris. This is also classified as materials handling and can be just as important as moving finished products from a warehouse. To handle and move these materials, the proper equipment that is going to be used will be tractors, bulldozers, backhoes, cranes, and gravel trucks.

Safety And Training Programs

There are many safety and education training programs involving proper materials handling. These programs center on such topics as safety, engineering, equipment training, handling of hazardous materials, and storage. The content of the training should emphasize factors that will contribute to successfully moving materials, keeping production flowing, and reducing workplace hazards. Training and safety programs should educate employees to the dangers of improper lifting and how to avoid unnecessary physical stress and strain. Training programs should also instruct workers on the proper use of various equipments.

Stock indicators

In order book the stock level bar is shown for every item in the order. The bar displays information about current stock level in relation to the minimum stock level and current stock level less quantity in orders in relation to minimum stock level.

ottom arrow shows percentage of items in stock less items in orders against minimum stock level * 4 If minimum stock level is not defined the system will show the amount of stock you have, and the amount available. So in the example below there are 10 in stock and -100 available.

My Inventory stock control fields explained:

• The stock level fields will be visible on My Inventory screen only if you have selected location in Show Location.
• If you have multiple locations, you can review and change stock levels for each location by clicking + next to the stock item record. This will display a sub grid with stock levels for each location.
• Level column - shows current stock level for the selected location. This is physical stock before processing any outstanding open orders.
• Stock Value - is dynamically calculated value that automatically keeps track of the cost of your stock. This is worked out from single unit cost i.e. Stock Value / Stock Level. When you acquire new stock through a purchase order the system will increase the stock value based on the value specified in the purchase order. When you sell or make any adjustments to the stock the system automatically recalculates your actual stock value based on the unit cost. Note that Unit Cost is not a simply a Purchase Price value. Unit value is calculated based on the actual cost of stock rather than fixed value. For example if you purchased 10 Mailing Bags today for $0.10 each your stock value will be $1.00, making each unit cost $0.10. So when you sold one, the stock value will be changed to $0.90. If you then purchased another 10 Mailing Bags but paid $0.15 for each, your stock value becomes $2.40 ($0.90 + 10 * $0.15) where each unit costs around $0.12, ($2.40 / 19). In other words stock value is calculated based on mean averaged value of all units.
• Minimum Level - threshold which indicates the minimum stock level which will cause the system to prompt you to reorder stock.
• Due - number of items due to be delivered from the supplier.
• Available - (read only) number of items available. This is calculated field showing stock level minus number of items in open orders.
• Not Tracked flag - not tracked switch allows you to instruct the system to disregard any stock tracking functions for this item. This is also used to tell the system to calculate stock level dynamically when this stock item is part of a composition (i.e. the stock level for this item will be calculated from stock level of child items). When Not Tracked is set (Red) the minimum level is set to -1. To set or unset Not Tracked double click on the cell.

To make adjustments to the stock level

• find the product you want to make adjustment in the My Inventory grid,
• double click on the Stock Level cell (or start typing in the cell)
• To apply the changes hit enter – DO NOT click away from the cell – it will result in the error.

Simplify your online order management – and increase sales

Linnworks Order Management and Stock Control system automates your order management, saving you time and money, and making selling online through multiple channels more efficient, cost-effective and profitable. Why waste time on manual tasks, when your entire order management process can be left to Linnworks?

• All your orders from multiple channels are downloaded into one central place
• Fulfilment is handled at the click of a button – including invoice printing, shipping labels, picking and packaging.
• Works with eBay, Amazon, Magento, PlayTrade, Pixmania and  more – virtually all  e-commerce websites, including bespoke platforms.

Order management has never been so simple, easy, fast – or so affordable. By automating the order management process, you reduce manpower costs, improving your profit margins and reducing your cost base.
You also provide more integrated and consistent customer service. Even small firms can harness the power of automated systems to impress customers, and become more efficient and responsive.

How automated order management works

Linnworks software integrates all your sales across multiple sites – including eBay, Amazon, Magento and additional channels, such as PlayTrade and Pixmania. Linnworks system integrates with X-Cart, osCommerce, ZenCart and many more website carts.
When an order arrives, you can choose to print invoices, shipping labels, packing slips, and pick lists for multiple orders at the click of a button.
Or you can specify to have the order forwarded automatically to your drop shipper, with barely any involvement on your part.
The system automatically handles all the notifications and updates that are needed.

Keep track of the whole order management lifecycle

Linnworks manages every aspect of the order management lifecycle from initial order to despatch and beyond. Save time and improve customer satisfaction by automatically marking  orders as shipped on your selling channels and uploading shipping tracking:

• Create automated customer email notifications – send out despatch notes with PDF invoices attached
• Submit tracking numbers and shipping times back to the channel
• Send bespoke email notifications for every step of the order management process – such as shipping times and delivery confirmations
• Automate management of returns and resends, exchanges and customer notes
• Increase customer loyalty and grow sales by sending automated emails and newsletters.

Linnworks even supports despatch console for barcoded invoices and allows you to process orders by scanning invoice barcodes. It supports picklist barcode, packing lists, scanning product UPC/EAN numbers or off-the-screen processing.

Cut the paperwork down to size and tame the chaos of online order management

Linnworks system puts all your order management information in one place, available to easily view and export. You can easily create order notes and carry out audit trails, and recalculate tax.
It’s simple to sort and filter open orders, group orders together and print invoices and shipping labels for all orders in bulk. You can even create conditional invoices and pick lists for complete flexibility.
If your business expands beyond online sales, then Linnworks also provides integration with telesales, direct orders and an integrated EPOS (electronic point of sale) system.

Stock Chart

A stock control chart is a graphical illustration of a simple approach to stock management over time. This ‘saw tooth’ shaped diagram is normally shown as if sales were steady throughout each month. Whilst this oversimplifies the situation for many businesses, the principles can be adapted to most situations.
The key features and terms are:

• Maximum stock level – this is the maximum amount of stock a business would wish to hold. This could represent enough stock for a month or a week, it might be as much as the warehouse has space for, or it might depend on the order size needed to qualify for a quantity discount – known as the Economic Order Quantity (EOQ). On the diagram below, the maximum stock level is 600 units, and the usual order quantity is 500 units

• Re-order level – this acts as a trigger point, so that when stocks fall to this level, the next order should be placed. This helps take account of fluctuations in sales levels over time. When an order is placed, there is a lead time that the supplier needs to meet that order. Ideally this new order will arrive just before stocks fall below the minimum stock level. On the diagram below, 300 units

• Lead time – the amount of time between placing the order and receiving the stock On the diagram below, just under two weeks

• Minimum stock level – this is the minimum amount of product the business would want to hold in stock. Assuming the minimum stock level is more than zero, this is known as buffer stock – see below. On the diagram below, 100 units

• Buffer stock – an amount of stock held as a contingency in case of unexpected orders so that such orders can be met and in case of any delays from suppliers.

 

Location, Location , Location!

You might have heard people say that location is the most important thing for a business. Then, the next most important is ... location, and so on.
For many businesses, getting the right location can make the difference between success and failure. Can think of a shop or restaurant near where you live that has closed down, maybe because it was in the wrong place?
There are lots of different reasons why location is important to a business and location matters to some businesses much more than it does to others.
Here are some reasons why location matters.
Labour

• Workers must be available locally, or must be willing to travel to work at the business.
• These workers must have the right skills.
• If there is high unemployment locally, you might find it easier to recruit workers, and maybe you won’t have to pay them as much as you would elsewhere.
• But if there is high unemployment, local people may not have as much to spend with your business.
• Often a location becomes a centre for related industries - Staffordshire for potteries, Sheffield for steel, and the local people have particular skills.

Land/buildings

• The right amount and type of land and buildings must be available.
• For some businesses, you need a lot of space - perhaps your business is noisy or creates fumes and needs to be well away from where people live.
• Some businesses need to be near their customers, or to their suppliers.

Transport and communications links

• Your workers need to be able to travel to work.
• You might need to be able to transport materials and products in and out of your business.
• Telephone, postal and Internet services might be better in cities than in the countryside.

Natural resources

• Primary industries need to be sited near to natural resources.
• Because of the costs of transport of raw materials, secondary businesses may also be sited close to resources that are important to their businesses.

Customers

• Every business needs to be able to reach its customers.
• For a retail shop, you might want potential customers to be walking past all the time.
• An Internet business might be able to locate almost anywhere!

Language

• As businesses become more global, you need people who can speak the same language as your customers. This is one reason why, for example, India has been successful in attracting call centres and software development from the UK and North America.

Image

• Some businesses need to be in a location that suits their image.
• Remember, though, high class locations tend to have high rents!

Competitors

• In some cases, you might want to be the only business of your type nearby - perhaps this would be good for a petrol station or a news agent.
• Other businesses cluster together - restaurants in Soho or Chinatown, fashion shops and jewellers on Bond Street.

The Normal Distribution By Ongki

2. The Normal Distribution

The normal distribution holds an honored role in probability and statistics, mostly because of the central limit theorem, one of the fundamental theorems that forms a bridge between the two subjects. In addition, as we will see, the normal distribution has many nice mathematical properties. The normal distribution is also called the Gaussian distribution, in honor of Carl Friedrich Gauss, who was among the first to use the distribution.

The Standard Normal Distribution

A random variable Z has the standard normal distribution if it has the probability density function ϕ given by

ϕ(z)=12π−−−√e−12z2,z∈R

ϕ is a probability density function.

Proof:

Let c=∫∞−∞e−12z2dz. We need to show that c=2π−−√. That is, 2π−−−√ is the normalzing constant for the function z↦e−12z2. The proof uses a nice trick:

c2=∫∞−∞e−12x2dx∫∞−∞e−12y2dy=∫∞−∞∫∞−∞e−12(x2+y2)dxdy

We now convert the double integral to polar coordinates: x=rcos(θ), y=rsin(θ) where r∈[0,∞) and θ∈[0,2π). So, x2+y2=r2 and dxdy=rdrdθ. Thus

c2=∫2π0∫∞0re−12r2drdθ

Substituting u=r2/2 in the inner integral gives ∫∞0e−udu=1 and then the outer integral is ∫2π01dθ=2π. Thus, c2=2π and so c=2π−−√.

The standard normal density function ϕ satisfies the following properties:

1. ϕ is symmetric about z=0.
2. ϕ is increasing on (−∞,0) and decreasing on (0,∞).
3. The mode occurs at z=0.
4. ϕ is concave upward on (−∞,−1) and on (1,∞) and is concave downward on (−1,1).
5. The inflection points of ϕ occur at z=±1.
6. ϕ(z)→0 as z→∞ and as z→−∞.

Proof:

These results follow from standard calculus. Note that ϕ′(z)=−zϕ(z). This differential equation helps simplify the computations.

In the Special Distribution Simulator, select the normal distribution and keep the default settings. Note the shape and location of the standard normal density function. Run the simulation 1000 times, and note the agreement between the empirical density function and the true density function.

The standard normal distribution function Φ, given by

Φ(z)=∫z−∞ϕ(t)dt=∫z−∞12π−−−√e−12z2dz

and its inverse, the quantile function Φ−1, cannot be expressed in closed form in terms of elementary functions. However approximate values of these functions can be obtained from the special distribution calculator, and from most mathematics and statistics software. Indeed these functions are so important that they are considered special functions of mathematics.

The standard normal distribution function Φ satisfies the following properties:

1. Φ(−z)=1−Φ(z) for z∈R
2. Φ−1(p)=−Φ−1(1−p) for p∈(0,1)
3. Φ(0)=12, so the median is 0.

Proof:

Part (a) follows from the symmetry of ϕ. Part (b) follows from part (a). Part (c) follows from part (a) with z=0.

In the special distribution calculator, select the standard normal distribution.

1. Note the shape of the density function and the distribution function.
2. Find the first and third quartiles.
3. Compute the interquartile range.

Use the special distribution calculator to find the quantiles of the following orders for the standard normal distribution:

1. p=0.001, p=0.999
2. p=0.05, p=0.95
3. p=0.1, p=0.9

Moments

The mean and variance of the standard normal distribution are

1. E(Z)=0
2. var(Z)=1

Proof:

Of course, by symmetry, if Z has a mean, the mean must be 0, but we have to argue that the mean exists. Actually it's not hard to compute the mean directly. Note that

E(Z)=∫∞−∞z12π−−√e−z2/2dz=∫0−∞z12π−−√e−z2/2dz+∫∞0z12π−−√e−z2/2dz

The integrals on the right can be evaluated explicitly using the simple substitution u=z2/2. The result is E(Z)=−1/2π−−√+1/2π−−√=0. For part (b), note that

var(Z)=E(Z2)=∫∞−∞z2ϕ(z)dz

Integrate by parts, using the parts u=z and dv=zϕ(z)dz. Thus du=dz and v=−ϕ(z). Note that zϕ(z)→0 as z→∞ and as z→−∞. Thus, the integration by parts formula gives var(Z)=∫∞−∞ϕ(z)dz=1.

Many important properties of the normal distribution are most easily obtained using the moment generating function.

If Z has the standard normal distribution then Z has moment generating function

m(t)=E(etZ)=e12t2,t∈R

Proof:

Note that

E(etZ)=∫∞−∞etz12π−−√e−z2/2dz=∫∞−∞12πexp(−12z2+tz)dz

We complete the square in z to get −12z2+tz=−12(z−t)2+12. Thus we have

E(etZ)=e12t2∫∞−∞12π−−√exp[−12(z−t)2]dz

In the integral, if we use the simple substitution u=z−t then the integral becomes ∫∞−∞ϕ(u)du=1. Hence E(etZ)=e12t2,

The characteristic function of Z is χ(t)=m(it)=E(eitZ)=e−t2/2. Thus, the standard normal distribution has the curious property that the characteristic function is a multiple of the PDF.
The moment generating function can be used to give another proof that Z has mean 0 and variance 1. More generally, we can compute all of the moments of Z, which we know must exist since the moment generating function is finite for all t∈R.

For n∈N,

1. E(Z2n)=(2n)!/(n!2n)
2. E(Z2n+1)=0

Proof:

The result follows from repeated differentiation of the MGF. Recall that E(Zk)=m(k)(0). Of course, the odd order moments must be 0 by symmetry.

The following exercise gives the skewness and kurtosis of the normal distribution.

If Z has the standard normal distribution then

1. skew(Z)=0
2. kurt(Z)=3

Proof:

Since Z has mean 0 and variance 1, skew(Z)=E(Z3)=0 and kurt(Z)=E(Z4)=4!/(2!22)=3.

Because of the last result, (and the use of the normal distribution as a standard), the excess kurtosis of a random variable is defined to be the ordinary kurtosis minus 3. Thus, the excess kurtosis of the normal distribution is 0.

The General Normal Distribution

The general normal distribution is the location-scale family associated with the standard normal distribution. Specifically, suppose that μ∈R and σ∈(0,∞) and that Z has the standard normal distribution. Then X=μ+σZ has the normal distribution with location parameter μ and scale parameter σ. The basic properties of the density function and distribution function follow easily from general results for location scale families.

The normal distribution with location parameter μ and scale parameter σ has probability density function f given by

f(x)=1σϕ(x−μσ)=12π−−−√σexp[−12(x−μσ)2],x∈R

Proof:

This follows from the change of variables formula corresponding to the transformation x=μ+σz.

The normal density function f satisfies the following properties:

1. f is symmetric about x=μ.
2. f is increasing on (−∞,μ) and is decreasing on (μ,∞)
3. The mode occurs at x=μ.
4. f is concave upward on (−∞,μ−σ) and on (μ+σ,∞) and is concave downward on (μ−σ,μ+σ).
5. The inflection points of f occur at x=μ±σ.
6. f(x)→0 as x→∞ and as x→−∞.

Proof:

These properties follow from the corresponding properties of ϕ.

In the special distribution simulator, select the normal distribution. Vary the parameters and note the shape and location of the density function. With your choice of parameter settings, run the simulation 1000 times and note the apparent convergence of the empirical density function to the true probability density function.

Let F denote the distribution function for the normal distribution with location parameter μ and scale parameter σ, and as above, let Φ denote the standard normal distribution function.

The normal distribution function F satsifies the following properties:

1. F(x)=Φ(x−μσ) for x∈R.
2. F−1(p)=μ+σΦ−1(p) for p∈(0,1).
3. F(μ)=12 so the median occurs at x=μ.

Proof:

Part (a) follows since X=μ+σZ. Parts (b) and (c) follow from (a).

In the special distribution calculator, select the normal distribution. Vary the parameters and note the shape of the density function and the distribution function.

Moments

As the notation suggests, the location and scale parameters are also the mean and standard deviation, respectively.

If X has the normal distribution with location parameter μ and scale parameter σ then

1. E(X)=μ
2. var(X)=σ2

Proof:

This follows from the representation X=μ+σZ and basic properties of expected value and variance.

If X has the normal distribution with location parameter μ and scale parameter σ then X has moment generating function

E(etX)=exp(μt+12σ2t2)t∈R

Proof:

This follows from the representation X=μ+σZ and basic properties of expected value:

E(etX)=E(etμ+tσZ)=etμE(etσZ)=etμe12t2σ2=etμ+12σ2t2

The central moments of X can be computed easily from the moments of the standard normal distribution. The ordinary (raw) moments of X can be computed from the central moments, but the formulas are a bit messy.

If X has the normal distribution with mean μ and standard deviation σ, then for n∈N,

1. E[(X−μ)2n]=(2n)!σ2n/(n!2n)
2. E[(X−μ)2n+1]=0

All of the odd central moments of X are 0, a fact that also follows from the symmetry of the probability density function.

In the special distribution simulator select the normal distribution. Vary the mean and standard deviation and note the size and location of the mean/standard deviation bar. With your choice of parameter settings, run the simulation 1000 times and note the apparent convergence of the empirical moments to the true moments.

The following exercise gives the skewness and kurtosis of the normal distribution.

If X has the normal distribution with mean μ and standard deviation σ then

1. skew(X)=0
2. kurt(X)=3

Proof:

The skewness and kurtosis of a variable are defined in terms of the standard score, so these results follows form the corresponding reults of Z in Theorem 10.

Transformations

The normal family of distributions satisfies two very important properties: invariance under linear transformations and invariance with respect to sums of independent variables. The first property is essentially a restatement of the fact that the normal distribution is a location-scale family.

Suppose that X is normally distributed with mean μ and variance σ2. If a∈R and b∈R∖{0}, then a+bX is normally distributed with mean a+bμ and variance b2σ2.

Proof:

The MGF of a+bX is

E(et(a+bX))=etaE(e(tb)X)=etaeμ(tb)+σ2(tb)2/2=e(a+bμ)t+b2σ2t2/2

which we recognize as the MGF of the normal distribution with mean a+bμ and variance b2σ2.

In particular

1. If X has the normal distribution with mean μ and standard deviation σ then Z=X−μσ has the standard normal distribution.
2. If Z has the standard normal distribution and if μ∈R and σ∈(0,∞) are constants, then X=μ+σZ has the normal distribution with mean μ and standard deviation σ.

Recall that in general, if X is a random variable with mean μ and standard deviation σ>0, then Z=(X−μ)/σ is the standard score of X. Thus, if X has a normal distribution then the standard score Z has a standard normal distribution.

Suppose that X1 and X2 are independent random variables, and that Xi is normally distributed with mean μi and variance σ2i for i∈{1,2}. Then X1+X2 is normally distributed with

1. E(X1+X2)=μ1+μ2
2. var(X1+X2)=σ21+σ22

Proof:

The MGF of X1+X2 is the product of the MGFs, so

E{exp[t(X1+X2)]}=exp(μ1t+σ21t2/2)exp(μ2t+σ22t2/2)=exp[(μ1+μ2)t+(σ21+σ22)t2/2]

which we recognize as the MGF of the normal distribution with mean μ1+μ2 and variance σ21+σ22.

The result of the previous exercise generalizes to a sum of n independent, normal variables. The important part is that the sum is still normal; the expressions for the mean and variance are standard results that hold for the sum of independent variables generally.

Suppose that X has the normal distribution with mean μ and variance σ2. The distribution is a two-parameter exponential family with natural parameters (μσ2,−12σ2), and natural statistics (X,X2).

Computational Exercises

Suppose that the volume of beer in a bottle of a certain brand is normally distributed with mean 0.5 liter and standard deviation 0.01 liter.

1. Find the probability that a bottle will contain at least 0.48 liter.
2. Find the volume that corresponds to the 95th percentile

Answer:

Let X denote the volume of beer in liters

1. P(X>0.48)=0.9772
2. x0.95=0.51645

A metal rod is designed to fit into a circular hole on a certain assembly. The radius of the rod is normally distributed with mean 1 cm and standard deviation 0.002 cm. The radius of the hole is normally distributed with mean 1.01 cm and standard deviation 0.003 cm. The machining processes that produce the rod and the hole are independent. Find the probability that the rod is to big for the hole.

Answer:

Let X denote the radius of the rod and Y the radius of the hole. P(Y−X<0)=0.0028

The weight of a peach from a certain orchard is normally distributed with mean 8 ounces and standard deviation 1 ounce. Find the probability that the combined weight of 5 peaches exceeds 45 ounces.

Answer:

Let X denote the combined weight of the 5 peaches, in ounces. P(X>45)=0.0127

A Further Generlization

In some settings, it's convenient to consider a constant as having a normal distribution (with mean being the constant and variance 0, of course). This convention simplifies the statements of theorems and definitions in these settings. Of course, the formulas for probability density and distribution functions do not hold for a constant, but the other results involving moments, the moment geneating function, and transformations in Theorems 21 and 23 are still valid, and of course Theorem 23 would hold for all a and b.