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Business Decision Analysis — Linear Programing Model Essay Example
 Category:Management
 Document type:Math Problem
 Level:Undergraduate
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Table of Content
Linear Programming
12
LINEAR PROGRAMMING
Table of Contents
1. Introduction 3
2. Literature review 5
3. Methodology 11
4. Implementation 13
5. Discussion and conclusion 15
6. Reference 17

Introduction
We have utilized derivatives to help discover the minimums and maximums of a few capacities given by conditions. However, it is improbable that somebody will just give one a capacity and request that he or she locates its extreme data of values [ CITATION CHV93 l 1033 ]. All the more normally, somebody will depict an issue and ask assistance in minimizing or maximizing something: for example what is the biggest volume bundle which the warehouse will take? What is the fastest approach to get goods produced? What is the minimum costly approach to fulfill some errand?
Optimizing is very critical to production of goods and services in any firm, whether service providing and/ or goods producing companies. There are factors which have to be considered in the market for any good or service. One of the factors include price elasticity. This is how the customer behave when the price changes for goods. Steel production industry is not exceptional. Raw materials, production process and labor are other factors which are also to consider.
It is difficult to put all thiese into consideration without a guiding principle. Linear programming makes matters simple since it can be done with the assistance of electronic devices such as computers. However, the constrain must be known and the mathematical models must be developed first. This makes its use limited to the literate people only.
For CEL, the electricity and the time of production is very critical. In the linear programming modelling in this decisionmaking process, the following formulae and terms will be used. CEL having been using the old methods in most of its process including ordering of materials, will realize increased profits once a systematic and more advanced approach is applied. Apparently, these factors have been affecting the supply of raw materials,

Capacity of suppliers

Market liquidity

The expected prices in the future and
Some of the supplied raw materials for CEL include;

Local scrap

Imported scrap

High carbon (HC)
The above mentioned raw materials require different processing methods and different amount of power as well.
The table below shows how raw materials mentioned above are to be optimized.
Table 1 : List of raw materials and they percentage use
Rate/Ton (in rupees) 
Recovery 
Minimum per Batch (% of Raw Materials) 
Maximum per Batch (% of Raw Materials) 
Maximum per Month (ton) 

Local Scrap 

Imported Scrap 

The two major resources used in the production process of the steel heavily depend on the weight of the materials. As for the electricity, the more the materials are present for melt, the more power will be required to do the job. As described, there has been a strong correlation between the raw materials and the power required as shown in the equation below.
Power per batch (kWh) = (700 *weight of raw material in batch in tons) +1,200 Equation 1
Concerning time required in the processing of the raw materials, a strong correlation has been found between the weight of the raw material and time required for melting them. This time does not include the maintenance time required and the stop over during the charging time. The equation below shows the relationship between the weight of the batch and time required.
Time required to process one batch (hours) =0.2 + (0.3 * weight of raw material in batch in tons) Equation 2
Salary and many other consumables should be considered in the final equation for cost calculation for accurate value in the product optimization.

Literature review
Indian economy
In a third world upcoming nation like India, the many factors, for example, labor, capital and raw materials and so on are valuable and rare. The development planner, is consequently confronted with the issue of rare resource allocation to meet the different contending demands and various objectives that are clashing. The traditional and normal methods can never again be used in the changed conditions for tackling this issue and are henceforth quickly losing their significance in the present economy. Thus, the planner in Indian nation are ceaselessly and continually looking for profoundly goal and result situated strategies for sound and appropriate basic method of making decisions which can be highly effective at all levels of monetary and economic planning. There are however choices that can never be programmed. Such choices comprise of production increment, plant area, diversifying product line, plant expansion, redesign and modernization and so on. There are decisions that can be planned. Such decisions include, planning, budgeting, substitution, transportation procurement upkeep and so on.
In these current times, various new and better strategies, methods, tools and apparatuses have been created by the business analysts everywhere throughout the world. Every one of these discoveries shape the basis of operations research. Some of these outstanding operations inquired about systems have been effectively connected in Indian circumstances. For example, forecasting in business, stock models — probabilistic and deterministic, dynamic programming, integer programming, Goal programming, Linear Programming and so forth[ CITATION CHV93 l 1033 ]. Concerning the Indian economy, the areas where linear programming and business analysis is utilized is as shown in the list below;

Plan formulation

Railways

Agriculture sector

Aviation centers

Commercial institutions

Process industries

Steel industry and corporate houses

Plan Formulation.
In the detailing of the nation’s five year development plans, the econometric models and Linear Programming techniques are being utilized as a part of different and diverse areas, for example, transportation, grain storage planning, multilevel planning, urban district, state and levels systems.
2. Railways.
Indian Railways, which the publicsector employer has effectively connected the technique of Linear Programming in different key regions.
For instance, the area of Rajendra Bridge near the Ganges connecting South Bihar and North Bihar in region of Mokama was in preference to other by the assistance of Linear Programming.
3. Agriculture planning.
Linear Programming methodology is as a method widely utilized as a part of agribusiness too. It has been attempted on a small scale for the crop rotation, crop mixing, cash crops planning and so forth
4. Aviation Industry.
National airlines are additionally utilizing Linear Programming in the determination of aircraft allocation and determination of course. This has been made conceivable by the utilization of PC technology situated at the home office. The method of linear programming has been helpful in this sector to reduce congestion and to improve resource allocation.
5. Commercial Institutions.
The business organizations and individual traders are likewise utilizing Linear Programming methods for cost diminishment and benefit boost. The oil refineries are utilizing this system for making viable and ideal mixing or blending choices and for the improvement of products already finished.
6. Process Industries.
Different processes, for example, paint industry settles on choices relating to the determination of the product blend and areas of storage with the assistance of Linear Programming procedures. This scientific system is by and large broadly utilized by very many corporations, for example, TELCO for choosing what forging and castings to be fabricated in process plants and what ought to be obtained from other suppliers.
7. Steel Industry.
The significant steel plants are utilizing Linear Programming systems for deciding the ideal blend of the final product, for example; sheets, plates, bars, rounds and billets.
Corporate Houses.
Enormous corporate houses, for example, Hindustan Lever utilize these method for the appropriation of consumer goods all through the Indian nation. Linear Programming methodology is additionally utilized for capital planning choices, for example, the choice of one project from the man projects at the table. Capital is very important in any business when it comes to investments. If the capital is not properly planned, the company is likely to experience losses and hence the company becomes bankrupt
Other places where linear programming is applied in the world is as discussed below.
Military applications.
This is the methodology that was applied in the Second World War. This was most organized event in the history of mankind. During air strike the best points and places are selected using linear programming. Least fuel has to be used and so is the least weapon with maximum damage on the side of the enemy[ CITATION CHV93 l 1033 ].
Agriculture
The agriculture of a nation or a region is effectively planned using the method mentioned above. Food shortage is a real threat to many nations especially in African regions. With proper use of linear programming this problem has reduced as seen in the Kenyan agriculture system. With linear programming, government agencies have managed to allocate scarce resources such as capital, raw materials, labor and other factors of production in a manner that maximum profit is achieved.
Environmental protection
With linear programming alternative methods of handling waste products are evaluated so that the strict measures laid down by governments and NGOs can be satisfied. Also, alternative sources of energy can be evaluated in addition to air cleaner designs and recycling papers.
Allocation of facilities
Public recreation facilities such as ark and community halls, health care facilities and other social facilities are located using linear programming. Drug control and Public expenditure are effectively planned and executed with the help of linear programming.
Production
Many manufacturing companies such as CEL are faced with many problems such as this. This is so because they do produce many products and use many raw materials as well. It in such a problem where the cost needs to minimized and the profits maximized in whatever manner possible. Line balancing in the production companies where assembling is done at different stages. In assembling processes the best sequence is selected to reduce the total time that elapses during the assembling process.
Mixing and blending
Mixing different raw materials at different proportions gives different blends. The raw materials are mixed in different ratios and also at different costs. So as to come up with the cheapest blend using the available raw materials, linear programming models are called upon. The available raw materials operate at different constrains and that is why the linear programming comes into handy. Different products also has restriction on the constituent raw materials.
Transport and transshipment
Optimal distribution systems are developed by use of linear programming models. This is so in all companies that supply goods to very many parts for example the export process. With this programming, the minimum cost are realized unlike when the company would just ferry goods with no plan. This ensures that the markets nearer and those far off are served well.
Portfolio selection
It has proved to very hard for manager to select the best investments out of the many that are available. This selection involves careful evaluation of all the present options without overlooking anything. And since the factors to be considered in those many options are equally many, the linear programming comes at handy since it can be computed on the computer. The expected return, the level of risk and the capital available are just but some of the most critical things to be evaluated.
Contract and profit planning
Since cash at hand, stock and maximization of use of the available machines and other facilities, planning is necessary and hence linear programming.
There endless sectors in the Indian economy where linear programing is used. However, the above are just few illustrations.
Limitations of linear programming
This programming as the name suggest, it is only applicable to the relations that are linear. Not all equations are linear, some of them are hyperbolic.
The constrains and values must be well known before one proceeds to use linear programming. If these value changes during the study period, whole lot of other models have to be made again. For example
ax^{2}+bx+c = 0 where a # 0 such equation cannot be solved.

Methodology
The linear objective in this case would be to minimize different constrains and maximize on others.

There is need to minimize power consumptions while still melting many batches

There is need to reduce the amount of time while still melting many batches
The factors to consider here are the salaries, consumable raw materials, time, the batch weight, the set selling pricing for each batch, power and the cost of raw materials.
The following are mathematical models derived from the information provided;
Time used (T)
Equation 3
)W T = 0. 2 + ( 0.3 *
Electricity (p)
P= 1200 + ( 700 * W ) Equation 4
Where W is the total weight of batch
Cost of power
Equation 5
CP = P* T * 4.3
Minimization
Equation 6
Z = a +b + c + d + e + f + g
Where a is the Tasla
B is the Pig iron
C is the Local scrap
D is the Rangeen
E is the Sponge
F is the imported scrap
G is the High carbon (HC)
The total cost will be the cost of raw materials, power, labor and other consumables.
Total cost (T C)
T C = cost of raw materials (R m) + electricity (E) + salaries (S) + consumables (C)
T C =Rm + E + S + C Equation 7
T C < 29000
Rate/Ton (in rupees) 
Recovery 
Minimum per Batch (% of Raw Materials) 
Maximum per Batch (% of Raw Materials) 
Maximum per Month (ton) 

Local Scrap 

Imported Scrap 

Table 2: From the above table we can get the maximum and minimum of each selection.
Minimum per batch
Z = a (0%)+b (0%)+ c(10%) + d(15%) + e(0%) + f(0%) + g(5%)
= c(10%) + d(15%) + g(5%) Equation 8
Maximum of each per batch
4000kg = a(50%) +b(25%) + c(50%) + d(80%) + e(80%) + f(20%) + g (10%) Equation 9

Implementation
The above mathematical models can be implemented to give the best choices for minimum cost and maximum profits.
So solving the equations above;
Taking the weight of batch as 1 ton
Time used (T)
T = 0. 2 + ( 0.3 * 1000)
= 300.2 hours per ton
Electricity (p)
P= 1200 + ( 700 * W )…
= 1200 + ( 700 * 1000)
Where W is the total weight of batch
Cost of power
CP = P* * 4.3
= 701200 * 4.3
= 2945040
So the cost power per kg
Cost per kg = cost per ton / 1000kg
= 2945040/1000
= INR 2945.04 / kg
When the minimum weight per batch of the constituent component
Z = a (0%)+b (0%)+ c(10%) + d(15%) + e(0%) + f(0%) + g(5%)
= c(10%) + d(15%) + g(5%)
= 1000(10%) + 1000(15%) + 1000(5%)
= 100+ 150 + 50
When the maximum of each is used
= a(50%) +b(25%) + c(50%) + d(80%) + e(80%) + f(20%) + g (10%)
= 1000(50%) +1000(25%) + 1000(50%) + 1000(80%) + 1000(80%) + 1000(20%) + 1000(10%)
= 500+ 250+ 500 + 800 + 800+ 200 + 100
Total cost ( T C)
T C = cost of raw materials ( R m) + electricity (E) + salaries (S) + consumables (C)
T C =Rm + E + S + C
= Rm + INR 2945.04 / kg + INR2,000 /1000kg
= Rm + INR 2945.04 / kg + INR2 /1kg
= Rm + INR 2947.04/ kg
T C < 29000
Rm + INR 2947.04/ kg < 29000
Apparently, the total cost would be the sum of cost of the raw materials and the cost of batch. That cost as shown should be less than INR 29000 so that the CEL Company would realize profits.
The mass of each as shown should be less than 3150kg and more than 300kg.

Discussion and conclusion
The regulation of 4000kg per batch has limited the profits. This is so because the weight is limited and so is the constituent raw materials.
The profit will reduce if the at least one unit of each raw material is to be used. This would be reduced by at least INR 2947.02 / kg
For the cost to remain down all costs must come down. From the description CEL has a capacity to cut down all costs as seen in the implementation of the mathematical models.
On the side of the raw materials, some are very important to be done away with. Their content must be very high in the ratio. Such includes the local crap, pig iron and high carbon. As shown in the table they have different rates of discoveries with the high carbon being the least. The percentage of the Tasla can be reduced to less than 1% due to its high cost of transport. Imported crap on the other hand, despite being of very high quality and very high discovery percentage, its ratio in the compositions has to be kept low to cut on the cost.
Imported crap is supplied in higher quantity compared to others. This supply can be kept high but its constituent in the composition be kept at its lowest 0%.
With that the company can keep its cost at its lowest and the profits at its peak as shown by the linear programming technique. Labor and other consumables tend to remain constant all through since the weight of the batch remains the same while the finished product may vary depending on the composition of the constituent’s steel.
It can now be concluded that with the linear programing is a useful tool in planning of almost every project for maximization of the profits available and minimization of the costs to be incurred.
The total cost must be less than INR 29000 if the company is to realize profits. If the cost would exceed INR 29000 the company would realize losses which would lead to closure of the company.

Reference
CHVÁTAL, V. (1983). Linear programming
doi:10.1109/iti.2006.1708531Top of Form28th International Conference on Information Technology Interfaces, 2006. doi:10.1109/iti.2006.1708531Bajalinov, E., & Racz, A. (2006). Scaling problems in linearfractional programing. 28th International Conference on Information Technology Interfaces, 2006. doi:10.1109/iti.2006.1708531Bajalinov, E., & Racz, A. (2006). Scaling problems in linearfractional programing. 28th International Conference on Information Technology Interfaces, 2006. doi:10.1109/iti.2006.1708531Bajalinov, E., & Racz, A. (2006). Scaling problems in linearfractional programing. 28th International Conference on Information Technology Interfaces, 2006. doi:10.1109/iti.2006.1708531 Bajalinov, E., & Racz, A. (2006). Scaling problems in linearfractional programing. 28th International Conference on Information Technology Interfaces, 2006. doi:10.1109/iti.2006.1708531 Bajalinov, E., & Racz, A. (2006). Scaling problems in linearfractional programing. 28th International Conference on Information Technology Interfaces, 2006. doi:10.1109/iti.2006.1708531 Bajalinov, E., & Racz, A. (2006). Scaling problems in linearfractional programing. 28th International Conference on Information Technology Interfaces, 2006. doi:10.1109/iti.2006.1708531 Bajalinov, E., & Racz, A. (2006). Scaling problems in linearfractional programing. 28th International Conference on Information Technology Interfaces, 2006. doi:10.1109/iti.2006.1708531 Bajalinov, E., & Racz, A. (2006). Scaling problems in linearfractional programing. 28th International Conference on Information Technology Interfaces, 2006. doi:10.1109/iti.2006.1708531 Bajalinov, E., & Racz, A. (2006). Scaling problems in linearfractional programing. 28th International Conference on Information Technology Interfaces, 2006. doi:10.1109/iti.2006.1708531 Bajalinov, E., & Racz, A. (2006). Scaling problems in linearfractional programing. 28th International Conference on Information Technology Interfaces, 2006. doi:10.1109/iti.2006.1708531 Bajalinov, E., & Racz, A. (2006). Scaling problems in linearfractional programing. 28th International Conference on Information Technology Interfaces, 2006. doi:10.1109/iti.2006.1708531 Bajalinov, E., & Racz, A. (2006). Scaling problems in linearfractional programing. 28th International Conference on Information Technology Interfaces, 2006. doi:10.1109/iti.2006.1708531 Bajalinov, E., & Racz, A. (2006). Scaling problems in linearfractional programing. 28th International Conference on Information Technology Interfaces, 2006. doi:10.1109/iti.2006.1708531 Bajalinov, E., & Racz, A. (2006). Scaling problems in linearfractional programing. 28th International Conference on Information Technology Interfaces, 2006.(14), 234. doi:10.2307/3964507 Bajalinov, E., & Racz, A. (2006). Scaling problems in linearfractional programing. 116
Science News,Steen, L. A. (1979). Linear Programing: Solid New Algorithm.
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