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CBSE Class 12 Commerce Mathematics Unit V Linear Programming Complete Notes

 CBSE Class 12 Commerce Mathematics Unit V Linear Programming Complete Notes

CBSE Class 12 Commerce Mathematics Unit V Linear Programming : CBSE  is the goal of the Academic, Training, Innovation and Research unit of Central Board of Secondary Education is to achieve academic excellence by conceptualization policies and their operational planning to ensure balanced academic activities in the schools affiliated to the Board. The Unit strives to provide Scheme of Studies, curriculum, academic guidelines, textual material, support material, enrichment activities and capacity building programmed. The unit functions according to the broader objectives set in the National Curriculum Framework-2005 and in consonance with various policies and acts passed by the Government of India from time to time.

CBSE Class 12 Commerce Mathematics Unit V Linear Programming Complete Notes

CBSE Class 12 Commerce Mathematics Unit V Linear Programming :  Cakart team members provides here CBSE Class 12 Commerce Mathematics Unit V Linear Programming Complete Notes And Cost Complete Notes and other CBSE Class 12 Commerce Mathematics Complete Notes in pdf format. We provides you direct link for downloading CBSE Class 12 Commerce Mathematics Unit V Linear Programming  Complete Notes in pdf format. Download CBSE Class 12 Commerce Mathematics Unit V Linear Programming Complete Notes and read well.

CBSE Class 12 Commerce Mathematics Unit V Linear Programming Complete Notes

CBSE Class 12 Commerce Mathematics Unit V Linear Programming : Mathematics has since been greatly extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today. According to Mikhail B. Sevryuk, in the January 2006 issue of the Bulletin of the American Mathematical Society, “The number of papers and books included in the Mathematical Reviews database since 1940 (the first year of operation of MR) is now more than 1.9 million, and more than 75 thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical theorems and their proofs.”

Download here CBSE Class 12 Commerce Mathematics Unit V Linear Programming Complete Notes in pdf format

CBSE Class 12 Commerce Mathematics Unit V Linear Programming Complete Notes

CBSE Class 12 Commerce Mathematics Unit V Linear Programming : A linear programming problem may be defined as the problem of maximizing or minimizing a linear function subject to linear constraints. The constraints may be equalities or inequalities. In this problem there are two unknowns, and five constraints. All the constraints are inequalities and they are all linear in the sense that each involves an inequality in some linear function of the variables. The first two constraints, x1 ≥ 0 and x2 ≥ 0, are special. These are called nonnegativity constraints and are often found in linear programming problems. The other constraints are then called the main constraints. The function to be maximized (or minimized) is called the objective function. Here, the objective function is x1 + x2 .

Since there are only two variables, we can solve this problem by graphing the set of points in the plane that satisfies all the constraints (called the constraint set) and then finding which point of this set maximizes the value of the objective function. Each inequality constraint is satisfied by a half-plane of points, and the constraint set is the intersection of all the half-planes.

CBSE Class 12 Commerce Mathematics Unit V Linear Programming Complete Notes

CBSE Class 12 Commerce Mathematics Unit V Linear Programming :  Not all linear programming problems are so easily solved. There may be many variables and many constraints. Some variables may be constrained to be nonnegative and others unconstrained. Some of the main constraints may be equalities and others inequalities. However, two classes of problems, called here the standard maximum problem and the standard minimum problem, play a special role. In these problems, all variables are constrained to be nonnegative, and all main constraints are inequalities.

Large scale LP application areas

Problem areas where large LP’s arise are:

  • Pacific Basin facility planning for AT&T

The problem here is to determine where undersea cables and satellite circuits should be installed, when they will be needed, the number of circuits needed, cable technology, call routing, etc over a 19 year planning horizon (an LP with 28,000 constraints, 77,000 variables).

  • Military officer personnel planning

The problem is to plan US Army officer promotions (to Lieutenant, Captain, Major, Lieutenant Colonel and Colonel), taking into account the people entering and leaving the Army and training requirements by skill categories to meet the overall Army force structure requirements (an LP with 21,000 constraints and 43,000 variables).

  • Military patient evacuation

The US Air Force Military Airlift Command (MAC) has a patient evacuation problem that can be modelled as a LP. They use this model to determine the flow of patients moved by air from an area of conflict to bases and hospitals in the continental United States. The objective is to minimise the time that patients are in the air transport system. The constraints are:

  • all patients that need transporting must be transported; and
  • limits on the size and composition of hospitals, staging areas and air fleet must be observed.

MAC have generated a series of problems based on the number of time periods (days). A 50 day problem consists of an LP with 79,000 constraints and 267,000 variables (solved in 10 hours).

  • Military logistics planning

The US Department of Defense Joint Chiefs of Staff have a logistics planning problem that models the feasibility of supporting military operations during a crisis.

The problem is to determine if different materials (called movement requirements) can be transported overseas within strict time windows.

The LP includes capacities at embarkation and debarkation ports, capacities of the various aircraft and ships that carry the movement requirements and penalties for missing delivery dates.

One problem (using simulated data) that has been solved had 15 time periods, 12 ports of embarkation, 7 ports of debarkation and 9 different types of vehicle for 20,000 movement requirements. This resulted in an LP with 20,500 constraints and 520,000 variables (solved in 75 minutes).

  • Bond arbitrage

Many financial transactions can be modelled as LP’s, e.g. bond arbitrage, switching between various financial instruments (bonds) so as to make money. Typical problems have approximately 1,000 constraints and 50,000 variables and can be solved in “real-time”.

  • Airline crew scheduling

Here solving an LP is only the first stage in deciding crew schedules for commercial aircraft. The problem that has to be solved is actually an integer programming problem, the set partitioning problem. American Airlines has a problem containing 12,000,000 potential crew schedules (variables) – see OSL above. As such crew scheduling models are a key to airline competitive cost advantage these days (crew costs often being the second largest flying cost after fuel costs) we shall enlarge upon this problem in greater detail.

Within a fixed airline schedule (the schedule changing twice a year typically) each flight in the schedule can be broken down into a series of flight legs. A flight leg comprises a takeoff from a specific airport at a specific time to the subsequent landing at another airport at a specific time. For example a flight in the schedule from Chicago O’Hare to London Heathrow might have 2 flight legs, from Chicago to JFK New York and from JFK to Heathrow. A key point is that these flight legs may be flown by different crews.

Typically in a crew scheduling exercise aircraft types have been preassigned (not all crews can fly all types) so for a given aircraft type and a given time period (the schedule repeating over (say) a 2 week period) the problem becomes one of ensuring that all flight legs for a particular aircraft type can have a crew assigned. Note here that by crew we mean not only the pilots/flight crew but also the cabin service staff, typically these work together as a team and are kept together over a schedule.

As you probably know there are many restrictions on the hours that crews (pilots and others) can work. These restrictions can be both legal restrictions and union agreement restrictions. A potential crew schedule is a series of flight legs that satisfies these restrictions, i.e. a crew could successfully and legally work the flight legs in the schedule. All such potential crew schedules can have a cost assigned to them.

Hence for our American Airlines problem the company has a database with 12 million potential crew schedules. Note here that we stress the word potential. We have a decision problem here, namely out of these 12 million which shall we choose (so as to minimise costs obviously) and ensure that all flight legs have a crew assigned to them.

Typically a matrix type view of the problem is adopted, where the rows of the matrix are the flight legs and the columns the potential crew schedules, as below.

           Crew schedules
           1   2   3 etc ----->
Leg  A-B   0   1   1
     B-C   0   1   1
     C-A   0   0   1
     B-D   0   0   0
     A-D   1   0   0
     D-A   1   0   0 

Here a 0 in a column indicates that that flight leg is not part of the crew schedule, a 1 that the flight leg is part of the crew schedule. Usually a crew schedule ends up with the crew returning to their home base, e.g. A-D and D-A in crew schedule 1 above. A crew schedule such as 2 above (A-B and B-C) typically includes as part of its associated cost the cost of returning the crew (as passengers) to their base. Such carrying of crew as passengers (on their own airline or on another airline) is called deadheading.

LP is used as part of the solution process for this crew scheduling problem for two main reasons:

  • a manual approach to crew scheduling problems of this size is just hopeless, you may get a schedule but the cost is likely to be far from minimal
  • a systematic approach to minimising cost can result in huge cost savings (e.g. even a small percentage savings can add up to ten’s of millions of dollars)

Plainly there are people in the real world with large LP problems to solve. Informally what appears to be happening currently is that an increase in solution technology (advances in hardware, software and algorithms) is leading to users becoming aware that large problems can be tackled. This in turn is generating a demand for further improvements in solution technology.

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