Trasportation problem

Applications of OR Techniques in Transportation.

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What is OR ?

• Operations Research (OR) has been variously described as the “science of use”, “quantitative common sense ”, “scientific approach

to decision making problems”, etc. But only a few are widely accepted.

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• OR is a scientific approach to solving problems for scientific management. --- H.M.Wagner
• OR is a scientific knowledge through interdisciplinary team effort for the purpose of determining the best utilization of limited resources.

--- H.A.Taha

• OR is art of giving bad answers to problems which otherwise have worse answers.

--- T.L.Saaty

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OR Models in Transportation

• Transportation models for Cargo transport.
• Assignment models for Flight crew, Flight scheduling.
• Queue models for Probabilistic routines at maintenance, counters.
• Replacement models for decision making about effective life, spares purchase, overhaul –maintenance.

Transportation Model

• Transportation Model deals with transportation of products from various sources/origin to several sinks/destination. In general, let there be ‘m’ sources S₁, S₂, .. , S_m with A_i (i = 1, 2, …., m) available supplies or capacity at each source ‘i’, to be allocated to among ‘n’ destinations D₁, D₂, …, D_n with B_j (j = 1, 2, ….., n) specified requirements at each destination ‘j’. Let C_ij be the cost of shipping one unit from source ‘i’ to destination ‘j’, problem is to determine the transportation schedule so as to minimize the total cost of transportation.

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Mathematical formulation.

Minimize

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Subject to

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For feasible solution to exist, it is necessary that total supply equals total requirement (Rim condition), i.e.,

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Feasible & optimal solution Algorithms

• Feasible solution
• North-west corner rule
• Least cost method
• Vogel’s Approximation method (VAM)

(very close to optimal solution)

• Optimize the feasible solution by MODI’s method

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Example Problem in TM

Cargo units have to be air lifted by there aircraft from three airports and dropped to five destinations. The quantities that can be carried in one trip by these aircraft to each of these destinations are given. The total number of trips that the aircraft can make to destinations are also given. Find the number of trips each aircraft should make to each destination so that the total quantity of cargo transported is a maximum.

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Assignment Problem

The assignment problem is a special case of the TM in which the objective is to assign a number of origins to the equal number of destinations at a minimum cost (or maximum profit ). The basic assumption that the optimum solution to an assignment problem

remains unaltered if a constant is added / subtracted to / from

any row or column of the effective matrix.

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An airline, operating seven days a week, serves three cites A, Band C according to the schedule shown in the table. The layover cost per stop is roughly proportional to the square of the layover time. How should plane be assigned the flights so as o minimize the total layover cost ?

Inventory Models

• Inventory is a stock of any kind that the industry would like to have to promote smooth and efficient management of business. The stock may be spares, fuel, consumables etc.
• The stock is a dead capital which is unavoidable to smooth and efficient management of the business, at the same time over-stocking eats into the profit, which cannot be tolerated in Airline industry where the capital investments are huge. Hence optimum stock levels are to be maintained.
• Types of inventory models

Deterministic models, Probabilistic models.

Probabilistic Model

• Some of the spare parts of the aircraft engine cost Rs.1,00,000 each. These parts can be only be ordered together with the aircraft engine. If not purchased with aircraft engine these parts cannot be available on need. Suppose that the loss of Rs. 1,00,00,000 is suffered for each spare that is needed when none is available in the stock. Further suppose that the probabilities that the spare will be needed as replacement during the life-term of the class of aircraft engine is given. How many spares should be procured ?

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• Spare required 0 1 2 3 4 5
• Probability 0.9488 0.04 0.01 0.001 0.0002 0.0000

Thank You !