Special Cases In Linear Programming Dr. T Kachwala

Transportation Technique Slide 2 Transportation Technique Introduction Nature of Transportation Problem A transportation problem is a special type of linear programming problem and typically involves situations where goods are required to be transferred from some sources or plants to some destinations or markets at a minimum cost. Although such problems can be formulated and solved as linear programming problems, obtaining such solution is very inefficient because of large computational effort involved in that. In a typical transportation problem, A matrix is given where sources are given row wise, destinations are indicated column-wise and unit cost of transportation from each source to each destination is provided. Also indicated is the supply at each source and demand at every market.

Transportation Technique Slide 3 Transportation Technique Introduction (Continued) Objective Function The objective of the Transportation Model is to minimize the transportation Cost in an application of Logistics or distribution of goods or product from different sources within the constraints of the respective capacities to different destinations within the constraints of the respective demands   Balanced and Unbalanced Problems   If total demand is equal to total supply, the problem is called balanced transportation problem and if the two do not match, it is called unbalanced.

Transportation Technique Slide 4 Transportation Technique Nature of the Problem The following example explains the nature of the Transportation Problem: A fertilizer company has 3 units located in three different locations. The company has to cater to markets in eastern, western, southern and northern region. Cost of transportation per ton of fertilizer from specific factories to particular markets was compiled first. Capacities of various factories and requirements from different markets were duly estimated. These are given in the table below: --------------------------------------------------------------------------------------------------------------------------- Location Transportation Cost (Rs.)  Supplies of factory (Per ton to marketing region) (capacities)  North South East West tons  F1 410 395 400 435 90 F2 460 305 380 345 100 F3 300 375 455 405 50 Demand 60 50 85 45 240 tons  Determine the optimum schedule of shipment for the transportation problem (i.e. how many tons of Fertilizer should be transported from which factory to which region such that the total cost is minimum?)

Assignment Technique Introduction Slide 5 An assignment problem is a special class of linear programming problem. The objective of an assignment problem is to determine the optimal assignment of given tasks to a set of workers that they can perform with varying efficiency, in terms of time taken, cost, amount of sales and so on. Thus, if there are n tasks to perform and an equal number of persons who can do them, in varying times which are known, the algorithm seeks to assign the jobs to persons in such a manner that each person gets one job and the total time in which all jobs can be done is the minimum. The algorithm works in varied situations wherein pairings are sought to be made.

Slide 6 Assignment Technique Objective Function: Optimum Assignment (or Allocation) of Jobs to Facilities   Example : Six contractors submitted quotation for six projects. It was decided that one contractor should be given one project as otherwise it was feared that the time for completion & quality of workmanship will be affected. The estimates given by each of them on all the contracts in thousands of rupees are given below: ----------------------------------------------------------------------------------------- Contractor Quotation for project (Rs. in thousands)   I II III IV V VI A 41 72 39 52 25 51 B 22 29 49 65 81 50 C 27 39 60 51 32 32 D 45 50 48 52 37 43 E 29 40 39 26 30 33 F 82 40 40 60 51 30 Determine the optimum allocation of projects to the contractors and the corresponding total cost.

Assignment Technique Assignment problems can be solved by : Slide 7 Assignment Technique Introduction (continued) Assignment problems can be solved by : Completely enumerating all possibilities and choosing the best one. Drafting and solving the problem as a linear (integer) programming problem. Drafting and solving the problem as a transportation problem. Hungarian assignment method (HAM).