Industrial Applications. Linear programming is extensively used to solve a variety of industrial problems. In each of these applications, the general objective is to determine a plan for production and procurement in the time period under consideration. It is necessary to satisfy all demand requirements without violating any of the constraints. few examples of industrial applications are as follows.
(a) Product Mix-Problem. An industrial concern has available to itself a certain productive capacity of its various manufacturing process and has the opportunity to utilize this capacity to manufacture various products. Generally different products will have different unit costs as well as selling prices and therefore will have different unit profits. The problem is the selection of the optimal product mix to make best use to machine and man hours available while maximizing the firm's profit.
(b) Production Scheduling. The linear programming technique may often be used in situations where several products can (with varying efficiency) be made on each of several different machines the problem is to decide on a program which will maximize output minimize cost, or produce some other criterion of efficiency. For example if 20 jobs are to be arranged between 5 machines, there are a million of possible arrangements since the machines could be different in their performances. Thus a very large choice remains for the scheduling officer who can only consider a fraction of these as a result he may well overlook an unexpected optimum. A different kind of production scheduling problem arises where for instance a manufacturer has to meet fluctuating (e.g., seasonal) demand but is faced either with high production costs during peak demand periods or high inventory costs if he keeps production steady in slack periods during which surpluses are stored until required later on. Here again there are many different schedules which could satisfy requirements, but the problem is to find the correct balance between the conflicting costs of output fluctuation and excessive inventories.
Similarly linear programming is useful for allocating operators to machines products to machines clerks to particularly tasks and in providing optimal connection between production centers and wholesalers.
(C) Productions Smoothing. An industrial concern can solve the problem of scheduling its production (or procurement) over a number of future time periods with the total span being considered the planning horizon with the help of linear programming approach.
(d) Blending Problems. These problems are likely to arise when a product can be made from a variety of available raw materials of various composition and prices. The manufacturing process involves (mixing-some if these materials in varying quantities ot make a product) conforming to given specifications. The supply of raw materials and specifications serve as constraints in obtaining the minimum cost material blend. The solution would state the number of units of each raw material which are to be blended to make one unit of product.
These types of problems occur frequently in the petroleum industry (such as blending crude oil to produce different octane gasoline's), chemical industry (such as blending chemicals to produce fertilizers). And food industry (such as blending input ingredients to produce soft drinks soups and so on). In most of these applications management may decide how much of each resource to purchase in order to satisfy product specifications and product demands at minimum cost.
(e). transportation problems. Using transportation technique of linear programming we can determine the distribution system that will minimize total shipping cost from several warehouses to various market locations.
(f) Production distribution problems. These problems occur when the products needed by the various destinations in a transportation problem do not list in finished form but rather must be manufactured at the sources before shipment. The sources may have different production costs. The problem then is to minimize cost by deciding what is to be produced at each source and where the goods are to be shipped.
(g) Trim Loss. In a number of manufacturing situations, products are made in standard sizes (e.g., paper steel sheet and glass) while orders are received for materials in various shapes sizes and quantities. The problem is to determine which combination of requirements should be produced from standard materials in order to keep trim loss to a minimum.
(h) Linear programming is also used by oil refineries to determine the optimal mix of products to ne produced by the refinery during a given period. Given the restraints of the capacities of the various manufacturing units together with the types of residual fuels available and the market prices of finished products the model can be solved to indicate the quantity of each product that should be manufactured if profits are to be maximized.
(i) Communication Industry. LP methods are used in solving problems involving facilities for transmission switching relaying, etc.
(j) Rail Road Industry. An LP model for optimal programming of railway freight and train movements has been formulated to handle scheduling problems as fond at large terminal switching rail points. The constraints of the model were based on the methods of hiring and paying the trainmen, the scheduling of shipment and the capacity limitations of rail road, the objective being to minimize the total crew and engine expenses.
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