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.