{"id":3609,"date":"2020-02-25T05:39:10","date_gmt":"2020-02-25T05:39:10","guid":{"rendered":"https:\/\/commerceiets.com\/?p=3609"},"modified":"2020-02-25T05:39:10","modified_gmt":"2020-02-25T05:39:10","slug":"linear-programming-problem","status":"publish","type":"post","link":"https:\/\/commerceiets.com\/linear-programming-problem\/","title":{"rendered":"LINEAR PROGRAMMING PROBLEM NOTES &#8211; Operations Research for B.com\/ BBA Students"},"content":{"rendered":"\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_72 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/commerceiets.com\/linear-programming-problem\/#LINEAR_PROGRAMMING_PROBLEM\" title=\"LINEAR PROGRAMMING PROBLEM\">LINEAR PROGRAMMING PROBLEM<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/commerceiets.com\/linear-programming-problem\/#TYPES_OF_VARIABLES_IN_LINEAR_PROGRAMMING_PROBLEM\" title=\"TYPES OF VARIABLES IN LINEAR PROGRAMMING PROBLEM\">TYPES OF VARIABLES IN LINEAR PROGRAMMING PROBLEM<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/commerceiets.com\/linear-programming-problem\/#ASSUMPTIONS_OF_LINEAR_PROGRAMMING_PROBLEM\" title=\"ASSUMPTIONS OF LINEAR PROGRAMMING PROBLEM\">ASSUMPTIONS OF LINEAR PROGRAMMING PROBLEM<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/commerceiets.com\/linear-programming-problem\/#APPLICATIONS_OF_LINEAR_PROGRAMMING_PROBLEM\" title=\"APPLICATIONS OF LINEAR PROGRAMMING PROBLEM\">APPLICATIONS OF LINEAR PROGRAMMING PROBLEM<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/commerceiets.com\/linear-programming-problem\/#APPLICATIONS_IN_ENGINEERING\" title=\"APPLICATIONS IN ENGINEERING\">APPLICATIONS IN ENGINEERING<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/commerceiets.com\/linear-programming-problem\/#ADVANTAGES_OF_LINEAR_PROGRAMMING\" title=\"ADVANTAGES OF LINEAR PROGRAMMING\">ADVANTAGES OF LINEAR PROGRAMMING<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/commerceiets.com\/linear-programming-problem\/#LIMITATIONS_OF_LINEAR_PROGRAMMING\" title=\"LIMITATIONS OF LINEAR PROGRAMMING\">LIMITATIONS OF LINEAR PROGRAMMING<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/commerceiets.com\/linear-programming-problem\/#CONCLUSION\" title=\"CONCLUSION\">CONCLUSION<\/a><\/li><\/ul><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h1 class=\"has-text-align-center wp-block-heading\" id=\"linear-programming-problem\"><span class=\"ez-toc-section\" id=\"LINEAR_PROGRAMMING_PROBLEM\"><\/span><strong>LINEAR PROGRAMMING PROBLEM<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h1>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"323\" height=\"156\" src=\"https:\/\/commerceiets.com\/wp-content\/uploads\/2022\/01\/download-36.png\" alt=\"Linear programming Problem\" class=\"wp-image-4723\" srcset=\"https:\/\/commerceiets.com\/wp-content\/uploads\/2022\/01\/download-36.png 323w, https:\/\/commerceiets.com\/wp-content\/uploads\/2022\/01\/download-36-300x145.png 300w\" sizes=\"auto, (max-width: 323px) 100vw, 323px\" \/><figcaption>Linear Programming problem<\/figcaption><\/figure>\n\n\n\n<p>Linear programming problem is a powerful quantitative technique (or <a href=\"https:\/\/commerceiets.com\/operations-research\/\" target=\"_blank\" rel=\"noreferrer noopener\" aria-label=\"operational research (opens in a new tab)\">operational research<\/a> technique) designs to solve allocation problem. <\/p>\n\n\n\n<p>The term \u2018linear programming\u2019 consists of the two words <strong>\u2018Linear\u2019<\/strong> and <strong>\u2018Programming\u2019<\/strong>. The word \u2018Linear\u2019 is used to describe the relationship between decision variables, which are directly proportional. For example, if doubling (or tripling) the production of a product will exactly double (or triple) the profit and required resources, then it is linear relationship. The word \u2018programming\u2019 means planning of activities in a manner that achieves some \u2018optimal\u2019 result with available resources. A programme is \u2018optimal\u2019 if it maximises or minimises some measure or criterion of effectiveness such as profit, contribution (i.e. sales-variable cost), sales, and cost. <\/p>\n\n\n\n<p>Thus, \u2018Linear Programming\u2019\nindicates the planning of decision variables, which are directly proportional,\nto achieve the \u2018optimal\u2019 result considering the limitations within which the\nproblem is to be solved.<\/p>\n\n\n\n<p><strong>ACCORDING TO WILLIAM M. FOX<\/strong><\/p>\n\n\n\n<p>\u201cLinear\nprogramming is a planning technique that permits some objective function to be\nminimized or maximized within the framework of given situational restrictions.\u201d<\/p>\n\n\n\n<p><strong>HISTORICAL BACKGROUND<\/strong><\/p>\n\n\n\n<p>The technique of\nlinear programming was formulated by a Russian mathematician L.V. Kantorovich.\nBut the present version of simplex method was developed by Geoge B. Dentzig in\n1947. Linear programming (LP) is an important technique of operations research\ndeveloped for optimum utilization of resources.<\/p>\n\n\n\n<p><strong>TERMINOLOGY OF LINEAR PROGRAMMING<\/strong><\/p>\n\n\n\n<p><strong>LINEAR\nFUNCTION<\/strong><\/p>\n\n\n\n<p>A linear function contains terms each of\nwhich is composed of only a single, continuous variable raised to (and only to)\nthe power of 1.<\/p>\n\n\n\n<p><strong>OBJECTIVE FUNCTION<\/strong><\/p>\n\n\n\n<p>The objective function of a linear programming problem is a\nlinear function of the decision variable expressing the objective of the\ndecision maker. For example, maximisation of profit or contribution,\nminimisation of cost\/time.<\/p>\n\n\n\n<p><strong>CONSTRAINTS<\/strong><\/p>\n\n\n\n<p>The constraints indicate limitations on the resources,\nwhich are to be allocated among various decision variables. These resources may\nbe production capacity, manpower, time, space or machinery. These must be\ncapable of being expressed as linear equation (i.e. =) on inequalities (i.e.\n&gt; or&lt;; type) in terms of decision variables. Thus, constraints of a\nlinear programming problem are linear equalities or inequalities arising out of\npractical limitations.<\/p>\n\n\n\n<p><strong>NON-NEGATIVITY RESTRICTION<\/strong><\/p>\n\n\n\n<p>Non-negativity restriction indicates that all decision\nvariables must take on values equal to or greater than zero.<\/p>\n\n\n\n<p><strong>DIVISIBILITY<\/strong><\/p>\n\n\n\n<p>Divisibility means that the numerical values of the\ndecision variables are continuous and not limited to integers. In other words,\nfractional values of the decision variables must be permissible in obtaining\noptimal solution.<\/p>\n\n\n\n<p><strong>DECISION VARIABLES<\/strong><\/p>\n\n\n\n<p>The decision variables refer to the economic or physical\nquantities, which are competing with one another for sharing the given limited\nresources. The relationship among these variables must be linear under linear\nprogramming. The numerical values of decision variables indicate the solution\nof the linear programming problem.<\/p>\n\n\n\n<p><strong>FEASIBLE SOLUTION<\/strong><\/p>\n\n\n\n<p>Any\nnon-negative solution which satisfies all the constraints is known as a\nfeasible solution. The region comprising all feasible solutions is referred to\nas feasible region.<\/p>\n\n\n\n<p><strong>OPTIMAL SOLUTION<\/strong><\/p>\n\n\n\n<p>A feasible solution, which optimizes the objective function,\nis known as Optimal Solution. An optimal solution is always feasible but all\nfeasible solutions cannot be optimal.<\/p>\n\n\n\n<h3 class=\"has-text-align-center wp-block-heading\" id=\"types-of-variables-in-linear-programming-problem\"><span class=\"ez-toc-section\" id=\"TYPES_OF_VARIABLES_IN_LINEAR_PROGRAMMING_PROBLEM\"><\/span><strong>TYPES OF VARIABLES IN LINEAR PROGRAMMING PROBLEM<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p><strong>SLACK VARIABLE<\/strong><\/p>\n\n\n\n<p>Slack variables\nare defined to transform an inequality expression into an equality expression\nwith an added slack variable. The slack variable is defined by setting a lower\nbound of zero (&gt;0).<\/p>\n\n\n\n<p><strong>SURPLUS VARIABLE<\/strong><\/p>\n\n\n\n<p>A surplus\nvariable refers to the amount by which the values of the solution exceeds the\nresources utilized. These variables are also known as negative slack variables.\nIn the objective function, it carries a zero coefficient. In order to obtain\nthe equality constraint, the surplus variable is added to the greater than or\nequal to the type constraints.<\/p>\n\n\n\n<p><strong>ARTIFICIAL VARIABLE<\/strong><\/p>\n\n\n\n<p>When the\nproblems related to the mixed constraints are given and the simplex method has\nto be applied, then the artificial variable is introduced. The artificial\nvariable refers to the kind of variable which is introduced in the linear\nprogram model to obtain the initial basic feasible solution. It is utilized for\nthe equality constraints and for the greater than or equal inequality\nconstraints. There is no meaning of the artificial variables in the physical\nsense.<\/p>\n\n\n\n<h2 class=\"has-text-align-center wp-block-heading\" id=\"assumptions-of-linear-programming-problem\"><span class=\"ez-toc-section\" id=\"ASSUMPTIONS_OF_LINEAR_PROGRAMMING_PROBLEM\"><\/span><strong>ASSUMPTIONS OF LINEAR PROGRAMMING<\/strong> <strong>PROBLEM<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"323\" height=\"156\" src=\"https:\/\/commerceiets.com\/wp-content\/uploads\/2022\/01\/download-36.png\" alt=\"Linear programming problem\" class=\"wp-image-4723\" srcset=\"https:\/\/commerceiets.com\/wp-content\/uploads\/2022\/01\/download-36.png 323w, https:\/\/commerceiets.com\/wp-content\/uploads\/2022\/01\/download-36-300x145.png 300w\" sizes=\"auto, (max-width: 323px) 100vw, 323px\" \/><figcaption>Linear programming problem<\/figcaption><\/figure>\n\n\n\n<p><strong>PROPORTIONALITY<\/strong><\/p>\n\n\n\n<p>The basic assumption underlying the linear programming is\nthat any change in the constraint inequalities will have the proportional\nchange in the objective function. This means, if product contributes Rs 20\ntowards the profit, then the total contribution would be equal to 20x<sub>1<\/sub>,\nwhere x<sub>1<\/sub>&nbsp;is the number of units of the product.<\/p>\n\n\n\n<p><strong>For example,<\/strong>&nbsp;if\nthere are 5 units of the product, then the contribution would be Rs 100 and in\nthe case of 10 units, it would be Rs 200. Thus, if the output (sales) is\ndoubled, the profit would also be doubled.<\/p>\n\n\n\n<p><strong>ADDITIVITY<\/strong><\/p>\n\n\n\n<p>The assumption of additivity asserts that the total\nprofit of the objective function is determined by the sum of profit contributed\nby each product separately. Similarly, the total amount of resources used is\ndetermined by the sum of resources used by each product separately. This implies,\nthere is no interaction between the decision variables.<\/p>\n\n\n\n<p><strong>CONTINUITY<\/strong><\/p>\n\n\n\n<p>Another assumption of linear programming is that the\ndecision variables are continuous. This means a combination of outputs can be\nused with the fractional values along with the integer values.<\/p>\n\n\n\n<p><strong>For example<\/strong>, If 5<sup>2<\/sup>\/3 units of product A and\n10<sup>1<\/sup>\/3 units of product B to be produced in a week. In this case, the\nfractional amount of production will be taken as a work-in-progress and the\nremaining production part is taken in the following week. Therefore, a\nproduction of 17 units of product A and 31 units of product B over a three-week\nperiod implies 5<sup>2<\/sup>\/3 units of product A and 10<sup>1<\/sup>\/3 units of\nproduct B per week.<\/p>\n\n\n\n<p><strong>CERTAINTY<\/strong><\/p>\n\n\n\n<p>Another underlying assumption of linear programming is a\ncertainty, i.e. the parameters of objective function coefficients and the\ncoefficients of constraint inequalities are known with certainty. Such as\nprofit per unit of product, availability of material and labor per unit,\nrequirement of material and labor per unit are known and is given in the linear\nprogramming problem.<\/p>\n\n\n\n<p><strong>FINITE CHOICES<\/strong><\/p>\n\n\n\n<p>This assumption implies that the decision maker has\ncertain choices, and the decision variables assume non-negative values. The\nnon-negative assumption is true in the sense, the output in the production\nproblem cannot be negative. Thus, this assumption is considered feasible.<\/p>\n\n\n\n<p>Thus, while solving for the linear\nprogramming problem, these assumptions should be kept in mind such that the\nbest alternative is chosen.<\/p>\n\n\n\n<h3 class=\"has-text-align-center wp-block-heading\" id=\"applications-of-linear-programming-problem\"><span class=\"ez-toc-section\" id=\"APPLICATIONS_OF_LINEAR_PROGRAMMING_PROBLEM\"><\/span><strong>APPLICATIONS OF LINEAR PROGRAMMING<\/strong> <strong>PROBLEM<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>The following are the various applications of linear\nprogramming:<\/p>\n\n\n\n<p><strong>PRODUCTION\nMANAGEMENT<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\"><li><strong>Product mix:<\/strong> A company can produce several different products, each of which requires the use of limited production resources. In such cases, it is essential to determine the quantity of each product to be produced knowing its marginal contribution and amount of available resource used by it. The objective is to maximize the total      contribution, subject to all constraints.<\/li><li><strong>Production planning: <\/strong>This deals with the determination of minimum cost production plan over planning period of an item with a fluctuating demand, considering the initial number of units in inventory,      production capacity, constraints on production, manpower and all relevant cost factors. The objective is to minimize total operation costs.<\/li><li><strong>Assembly-line balancing:<\/strong> This problem is likely to arise when an item can be made by assembling different components. The process of assembling requires some specified sequence(s). The objective is to minimize      the total elapse time.<\/li><li><strong>Blending problems:<\/strong> These problems arise when a product can be made from a variety of available raw materials, each of which has a particular composition and price. The objective here is to determine the minimum cost blend, subject to availability of the raw materials, and minimum and maximum constraints on certain product constituents.<\/li><li>Trim loss When an item is made to a standard size (e.g. glass, paper sheet), the problem that arises is to determine which combination of requirements should be produced from standard materials in order to minimize the trim loss.<\/li><\/ul>\n\n\n\n<p><strong>FINANCIAL\nMANAGEMENT<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\"><li><strong>Portfolio selection:<\/strong> This deals with the selection of\n     specific investment activity among several other activities. The objective\n     is to find the allocation which maximises the total expected return or\n     minimises risk under certain limitations.<\/li><li><strong>Profit planning:<\/strong> This deal with the maximisation\n     of the profit margin from investment in plant facilities and equipment,\n     cash in hand and inventory.<\/li><\/ul>\n\n\n\n<p><strong>MARKETING\nMANAGEMENT<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\"><li><strong>Media selection:<\/strong> Linear programming technique\n     helps in determining the advertising media mix so as to maximise the\n     effective exposure, subject to limitation of budget, specified exposure\n     rates to different market segments, specified minimum and maximum number\n     of advertisements in various media. Example: Travelling salesman problem.\n     The problem of salesman is to find the shortest route from a given city,\n     visiting each of the specified cities and then returning to the original\n     point of departure, provided no city shall be visited twice during the\n     tour. Such type of problems can be solved with the help of the modified\n     assignment technique.<\/li><li><strong>Physical distribution:<\/strong> Linear programming determines\n     the most economic and efficient manner of locating manufacturing plants\n     and distribution centers for physical distribution.<\/li><\/ul>\n\n\n\n<p><strong>PERSONNEL\nMANAGEMENT<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\"><li><strong>Staffing problem:<\/strong> Linear programming is used to\n     allocate optimum manpower to a particular job so as to minimize the total\n     overtime cost or total manpower.<\/li><li><strong>Determination of equitable\n     salaries: <\/strong>Linear\n     programming technique has been used in determining equitable salaries and\n     sales incentives.<\/li><li><strong>Job evaluation and selection: <\/strong>Selection of suitable person for\n     a specified job and evaluation of job in organizations has been done with\n     the help of linear programming technique.<\/li><\/ul>\n\n\n\n<p><strong>DIET PROBLEMS<\/strong><\/p>\n\n\n\n<p>Animal feeds are produced by mixing a num\u00adber\nof ingredients each of which contains different proportions of protein,\nvitamins, minerals, etc. The animal feeds produced must meet certain\nnutritional levels as specified by the Government, at the low\u00adest cost.<\/p>\n\n\n\n<p>The producers may like to know the in\u00adcrease\nin cost when the nutritional level of the feed is increased. This problem is\nalso associated with pharmaceutical, fertilizer, lawn seed and pet food\nindustries. L.P. is an answer to such problems.<\/p>\n\n\n\n<p><strong>ADVERTISING BUDGET ALLOCATION<\/strong><\/p>\n\n\n\n<p>A firm has limited funds for advertising\nbudget allocation, e.g., T.V., radio, newspaper, magazines, bill broads,\nneon-advertisements, etc. and it would like to know how much money should be\nallocated to which medium for getting the maximum benefit. T.V. advertising may\nagain be split as day or evening shows, depending on the type of audiences. The\nfund allocation problem can be solved by L.P.<\/p>\n\n\n\n<p><strong>PROCESS SELECTION<\/strong><\/p>\n\n\n\n<p>This problem is similar to diet problems. It\ndeals with product that can be produced by using any one (or, more) of several\nprocesses, each involving dif\u00adferent proportions of various inputs. A firm\nwould like to produce the product by combining two or more processes and inputs\nin such a way that pro\u00adduction costs are minimized. L.P. can suggest an answer\nto such problems.<\/p>\n\n\n\n<p><strong>DISTRIBUTION PROBLEMS<\/strong><\/p>\n\n\n\n<p>Many firms have two or more plants where a\ncertain product is produced and several warehouses at different locations for\nstoring the products before sending them to various retail outlets.<\/p>\n\n\n\n<p>The distri\u00adbution costs also differ for\nvarious routes used from production plant to warehouse and to sales outlet.\nManagement must decide the shipping assignments, i.e., how many units should go\nfrom each sales out\u00adlet, with a minimizing view to shipping cost. Such problems\ncan be solved by transportation and L.P. techniques.<\/p>\n\n\n\n<p><strong>OIL PROCESSING AND\nTRANSPORTATION PROBLEMS<\/strong><\/p>\n\n\n\n<p>Large petroleum firms receive crude oil from\ndif\u00adferent sources which they transport to several re\u00adfineries where various\nproducts are produced jointly in the cracking process. L.P. is used to find out\nthe optimum product-mix, the best method of allocation of crude oil from\ndifferent sources to the various re\u00adfineries and transportation of products to\nsales out\u00adlets. Thus L.P. is widely used in petroleum industry.<\/p>\n\n\n\n<p><strong>PERSONNEL ASSIGNMENT PROBLEMS<\/strong><\/p>\n\n\n\n<p>These problems are similar to distribution\nprob\u00adlems. An accounting firm would like to find out the best method of\nassigning audit staff personnel to various audit activities within the audit\noffice con\u00adstraints, and simultaneously meeting the profession\u00adal and economic\nobjectives of the offices. L.P. and as\u00adsignment techniques can be used for\nsolution of such problems.<\/p>\n\n\n\n<p><strong>LONG-RANGE CAPACITY PLANNING\nFOR ELECTRIC UTILITIES<\/strong><\/p>\n\n\n\n<p>Good long-range planning models are necessary\nfor energy related complicated problems. Demand projections for future years\nand information relating to operation and investment costs should be incor\u00adporated\nin such models. John R. McNamara devel\u00adoped one such capacity planning model\nfor electric utilities based on L.P. to minimize the costs of sat\u00adisfying\nprojected demands.<\/p>\n\n\n\n<p>The model is simple, user-oriented, and can\nbe used to solve problems quickly. Besides these L.P. can be applied to various\nother fields of decision making.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"applications-in-engineering\"><span class=\"ez-toc-section\" id=\"APPLICATIONS_IN_ENGINEERING\"><\/span>APPLICATIONS IN ENGINEERING<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Engineers\nalso use linear programming to help solve design and manufacturing problems.\nFor example, in airfoil meshes, engineers seek aerodynamic shape optimization.\nThis allows for the reduction of the drag coefficient of the airfoil.\nConstraints may include lift coefficient, relative maximum thickness, nose\nradius and trailing edge angle. Shape optimization seeks to make a shock-free\nairfoil with a feasible shape. Linear programming therefore provides engineers\nwith an essential tool in shape optimization.<\/p>\n\n\n\n<p><strong>MILITARY APPLICATIONS<\/strong><\/p>\n\n\n\n<p>Military applications include the problem of selecting an air weapon system against enemy so as to keep them pinned down and at the same time minimizing the amount of aviation gasoline used. A variation of the transportation problem that maximizes the total tonnage of bombs dropped on a set of targets and the problem of community defense against disaster, the solution of which yields the number of defense units that should be used in a given attack in order to provide the required level of protection at the lowest possible cost.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"323\" height=\"156\" src=\"https:\/\/commerceiets.com\/wp-content\/uploads\/2022\/01\/download-36.png\" alt=\"Linear programming problem\" class=\"wp-image-4723\" srcset=\"https:\/\/commerceiets.com\/wp-content\/uploads\/2022\/01\/download-36.png 323w, https:\/\/commerceiets.com\/wp-content\/uploads\/2022\/01\/download-36-300x145.png 300w\" sizes=\"auto, (max-width: 323px) 100vw, 323px\" \/><figcaption>Linear programming problem<\/figcaption><\/figure>\n\n\n\n<h3 class=\"has-text-align-center wp-block-heading\" id=\"advantages-of-linear-programming\"><span class=\"ez-toc-section\" id=\"ADVANTAGES_OF_LINEAR_PROGRAMMING\"><\/span><strong>ADVANTAGES OF LINEAR PROGRAMMING<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p><strong>OPTIMUM USE\nOF RESOURCES:<\/strong> Linear programming helps in attaining the optimum use of\nproductive resources. It also indicates how a decision-maker can employ his\nproductive factors effectively by selecting and distributing (allocating) these\nresources.<\/p>\n\n\n\n<p><strong>IMPROVE\nQUALITY OF DECISIONS:<\/strong> Linear programming techniques improve the quality of\ndecisions. The decision-making approach of the user of this technique becomes\nmore objective and less subjective.<\/p>\n\n\n\n<p><strong>PRACTICAL\nSOLUTIONS:<\/strong> Linear programming techniques provide possible and\npractical solutions since there might be other constraints operating outside\nthe problem which must be taken into account. Just because we can produce so\nmany units does not mean that they can be sold. Thus, necessary modification of\nits mathematical solution is required for the sake of convenience to the\ndecision-maker.<\/p>\n\n\n\n<p><strong>RE-EVALUATION\nOF PLANS:<\/strong> Linear programming also helps in re-evaluation of a\nbasic plan for changing conditions. If conditions change when the plan is\npartly carried out, they can be determined so as to adjust the remainder of the\nplan for best results.<\/p>\n\n\n\n<p><strong>SCIENTIFIC\nAPPROACH:<\/strong> Operations Research provides a scientific approach to\nsolve the problems. All the aspects directly or indirectly related to the\nproblem are studied and optimal solution is arrived at.<\/p>\n\n\n\n<h3 class=\"has-text-align-center wp-block-heading\" id=\"limitations-of-linear-programming\"><span class=\"ez-toc-section\" id=\"LIMITATIONS_OF_LINEAR_PROGRAMMING\"><\/span><strong>LIMITATIONS OF LINEAR PROGRAMMING<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>Following are the disadvantages or limitations of linear programming:<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li><strong>SINGLE\nOBJECTIVE:<\/strong>&nbsp;Linear programming takes\ninto account a single objective only, i.e., profit maximization or cost\nminimization. However, in today&#8217;s dynamic business environment, there is no\nsingle universal objective for all organizations.<\/li><li><strong>CERTAINTY:<\/strong>&nbsp;Linear Programming assumes that the\nvalues of co-efficient of decision variables are known with certainty. Due to\nthis restrictive assumption, linear programming cannot be applied to a wide\nvariety of problems where values of the coefficients are probabilistic.<\/li><li><em>&#8220;Nothing\nis certain but death and taxes.&#8221; -Benjamin Franklin<\/em><\/li><li><strong>DIVISIBILITY:<\/strong>&nbsp;In linear programming, the decision\nvariables are allowed to take non-negative integer as well as fractional\nvalues. However, we quite often face situations where the planning models\ncontain integer valued variables. For instance, trucks in a fleet, generators\nin a powerhouse, pieces of equipment, investment alternatives and there are a\nmyriad of other examples. Rounding off the solution to the nearest integer will\nnot yield an optimal solution. In such cases, linear programming techniques\ncannot be used.<\/li><li><strong>COMPLEX:<\/strong> It is\ncomplex to determine the particular objective function<\/li><li><strong>LINEAR\nRELATIONSHIP:<\/strong> This technique is based on the hypothesis of linear\nrelations between inputs and outputs. This means that inputs and outputs can be\nadded, multiplied and divided. But the relations between inputs and outputs are\nnot always clear. In real life, most of the relations are non-linear.<\/li><li><strong>WRONG\nASSUMPTIONS:<\/strong> This technique presumes perfect competition in product\nand factor markets. But perfect competition is not a reality.<\/li><li><strong>CONSTANT\nREURNS:<\/strong> The LP technique is based on the hypothesis of constant\nreturns. In reality, there are either diminishing or increasing returns which a\nfirm experiences in production.<\/li><li><strong>COMPLICATED\nTECHNIQUE:<\/strong> It is a highly mathematical and complicated technique.\nThe solution of a problem with linear programming requires the maximization or minimization\nof a clearly specified variable. The solution of a linear programming problem\nis also arrived at with such complicated method as the simplex method which\ncomprises of a huge number of mathematical calculations.<\/li><li>Even if a particular objective function is laid down, it\nmay not be so easy to find out various technological, financial and other\nconstraints which may be operative in pursuing the given objective.<\/li><li>Given a Specified objective and a set of constraints it\nis feasible that the constraints may not be directly expressible as linear\ninequalities.<\/li><li>Mostly, linear programming models present trial and error\nsolutions and it is difficult to find out really optimal solutions to the\nvarious economic complexities.<\/li><\/ul>\n\n\n\n<h3 class=\"has-text-align-center wp-block-heading\" id=\"conclusion\"><span class=\"ez-toc-section\" id=\"CONCLUSION\"><\/span><strong>CONCLUSION<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>Linear programming has turned out to be a highly useful tool of analysis for the business executives. It is being increasingly made use of in theory of the firm, in managerial economics, in inter regional trade, in general equilibrium analysis, in welfare economics and in development planning.<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td><strong>Also Study<\/strong><\/td><td><strong>Also Study<\/strong><\/td><td><strong>Also Study<\/strong><\/td><\/tr><tr><td><a href=\"https:\/\/commerceiets.com\/operations-research\/\">Operations Research<\/a><\/td><td><a href=\"https:\/\/commerceiets.com\/linear-programming\/\">Linear Programming<\/a><\/td><td><a href=\"https:\/\/commerceiets.com\/assignment\/\">Assignment<\/a><\/td><\/tr><tr><td><a href=\"https:\/\/commerceiets.com\/linear-programming-problem\/\">Linear programming problems<\/a><\/td><td><a href=\"https:\/\/commerceiets.com\/linear-programming-applications\/\">Linear programming applications<\/a><\/td><td><a href=\"https:\/\/commerceiets.com\/linear-programming-applications\/\">Models of Operations research<\/a><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<div class=\"wp-block-buttons is-layout-flex wp-block-buttons-is-layout-flex\">\n<div class=\"wp-block-button\"><a class=\"wp-block-button__link\" href=\"https:\/\/www.linkedin.com\/in\/kajal-mahajan-7549b2197\/\" target=\"_blank\" rel=\"noreferrer noopener\">CONNECT ON LINKEDIN<\/a><\/div>\n<\/div>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"266\" src=\"https:\/\/commerceiets.com\/wp-content\/uploads\/2019\/04\/cropped-Presentation-resize-1024x266.jpg\" alt=\"Linear Programming Problem\" class=\"wp-image-787\" srcset=\"https:\/\/commerceiets.com\/wp-content\/uploads\/2019\/04\/cropped-Presentation-resize-1024x266.jpg 1024w, https:\/\/commerceiets.com\/wp-content\/uploads\/2019\/04\/cropped-Presentation-resize-300x78.jpg 300w, https:\/\/commerceiets.com\/wp-content\/uploads\/2019\/04\/cropped-Presentation-resize-768x199.jpg 768w, https:\/\/commerceiets.com\/wp-content\/uploads\/2019\/04\/cropped-Presentation-resize-1536x398.jpg 1536w, https:\/\/commerceiets.com\/wp-content\/uploads\/2019\/04\/cropped-Presentation-resize.jpg 1920w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><figcaption>Linear programming problem<\/figcaption><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Linear programming problem<\/p>\n","protected":false},"author":2,"featured_media":3610,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[228],"tags":[],"class_list":["post-3609","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-operations-research"],"_links":{"self":[{"href":"https:\/\/commerceiets.com\/wp-json\/wp\/v2\/posts\/3609","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/commerceiets.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/commerceiets.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/commerceiets.com\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/commerceiets.com\/wp-json\/wp\/v2\/comments?post=3609"}],"version-history":[{"count":0,"href":"https:\/\/commerceiets.com\/wp-json\/wp\/v2\/posts\/3609\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/commerceiets.com\/wp-json\/wp\/v2\/media\/3610"}],"wp:attachment":[{"href":"https:\/\/commerceiets.com\/wp-json\/wp\/v2\/media?parent=3609"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/commerceiets.com\/wp-json\/wp\/v2\/categories?post=3609"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/commerceiets.com\/wp-json\/wp\/v2\/tags?post=3609"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}