{"id":3617,"date":"2020-02-25T05:56:45","date_gmt":"2020-02-25T05:56:45","guid":{"rendered":"https:\/\/commerceiets.com\/?p=3617"},"modified":"2020-02-25T05:56:45","modified_gmt":"2020-02-25T05:56:45","slug":"linear-programming-applications","status":"publish","type":"post","link":"https:\/\/commerceiets.com\/linear-programming-applications\/","title":{"rendered":"LINEAR PROGRAMMING APPLICATIONS &#8211; Notes of Operations Research"},"content":{"rendered":"\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_65 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 \" >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-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/commerceiets.com\/linear-programming-applications\/#LINEAR_PROGRAMMING_APPLICATIONS\" title=\"LINEAR PROGRAMMING APPLICATIONS\">LINEAR PROGRAMMING APPLICATIONS<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/commerceiets.com\/linear-programming-applications\/#TERMINOLOGY\" title=\"TERMINOLOGY \">TERMINOLOGY <\/a><\/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-applications\/#TYPES_OF_VARIABLES\" title=\"TYPES OF VARIABLES\">TYPES OF VARIABLES<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/commerceiets.com\/linear-programming-applications\/#ASSUMPTIONS\" title=\"ASSUMPTIONS \">ASSUMPTIONS <\/a><\/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-applications\/#LINEAR_PROGRAMMING_APPLICATIONS-2\" title=\"LINEAR PROGRAMMING APPLICATIONS\">LINEAR PROGRAMMING APPLICATIONS<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"has-text-align-center wp-block-heading\" id=\"linear-programming-applications\"><span class=\"ez-toc-section\" id=\"LINEAR_PROGRAMMING_APPLICATIONS\"><\/span><strong>LINEAR PROGRAMMING APPLICATIONS<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Linear programming 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\nconsists of the two words \u2018Linear\u2019 and \u2018Programming\u2019. The word \u2018Linear\u2019 is used\nto describe the relationship between decision variables, which are directly\nproportional. For example, if doubling (or tripling) the production of a\nproduct will exactly double (or triple) the profit and required resources, then\nit is linear relationship. The word \u2018programming\u2019 means planning of activities\nin a manner that achieves some \u2018optimal\u2019 result with available resources. A\nprogramme is \u2018optimal\u2019 if it maximises or minimises some measure or criterion\nof effectiveness such as profit, contribution (i.e. sales-variable cost),\nsales, 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<h2 class=\"has-text-align-center wp-block-heading\" id=\"terminology\"><span class=\"ez-toc-section\" id=\"TERMINOLOGY\"><\/span><strong>TERMINOLOGY <\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\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<h2 class=\"has-text-align-center wp-block-heading\" id=\"types-of-variables\"><span class=\"ez-toc-section\" id=\"TYPES_OF_VARIABLES\"><\/span><strong>TYPES OF VARIABLES<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"960\" height=\"720\" src=\"https:\/\/commerceiets.com\/wp-content\/uploads\/2020\/02\/TYPES-OF-VARIABLES.jpg\" alt=\"TYPES OF VARIABLES\" class=\"wp-image-3642\" srcset=\"https:\/\/commerceiets.com\/wp-content\/uploads\/2020\/02\/TYPES-OF-VARIABLES.jpg 960w, https:\/\/commerceiets.com\/wp-content\/uploads\/2020\/02\/TYPES-OF-VARIABLES-300x225.jpg 300w, https:\/\/commerceiets.com\/wp-content\/uploads\/2020\/02\/TYPES-OF-VARIABLES-768x576.jpg 768w\" sizes=\"(max-width: 960px) 100vw, 960px\" \/><\/figure>\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\"><span class=\"ez-toc-section\" id=\"ASSUMPTIONS\"><\/span><strong>ASSUMPTIONS <\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\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<h2 class=\"has-text-align-center wp-block-heading\" id=\"linear-programming-applications\"><span class=\"ez-toc-section\" id=\"LINEAR_PROGRAMMING_APPLICATIONS-2\"><\/span><strong>LINEAR PROGRAMMING<\/strong> <strong>APPLICATIONS<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>The following are the various linear programming applications:<\/p>\n\n\n\n<p><strong>PRODUCTION\nMANAGEMENT<\/strong><\/p>\n\n\n\n<ul><li><strong>Product mix:<\/strong> A company can produce several\n     different products, each of which requires the use of limited production\n     resources. In such cases, it is essential to determine the quantity of\n     each product to be produced knowing its marginal contribution and amount of\n     available resource used by it. The objective is to maximize the total\n     contribution, subject to all constraints.<\/li><li><strong>Production planning: <\/strong>This deals with the determination\n     of minimum cost production plan over planning period of an item with a\n     fluctuating demand, considering the initial number of units in inventory,\n     production capacity, constraints on production, manpower and all relevant\n     cost factors. The objective is to minimize total operation costs.<\/li><li><strong>Assembly-line balancing:<\/strong> This problem is likely to arise when\n     an item can be made by assembling different components. The process of\n     assembling requires some specified sequence(s). The objective is to minimize\n     the total elapse time.<\/li><li><strong>Blending problems:<\/strong> These problems arise when a\n     product can be made from a variety of available raw materials, each of\n     which has a particular composition and price. The objective here is to\n     determine the minimum cost blend, subject to availability of the raw\n     materials, and minimum and maximum constraints on certain product\n     constituents.<\/li><li>Trim\n     loss When an item is made to a standard size (e.g. glass, paper sheet),\n     the problem that arises is to determine which combination of requirements\n     should be produced from standard materials in order to minimize the trim\n     loss.<\/li><\/ul>\n\n\n\n<p><strong>FINANCIAL\nMANAGEMENT<\/strong><\/p>\n\n\n\n<ul><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><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><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<p><strong>APPLICATIONS IN ENGINEERING<\/strong><\/p>\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-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 applications\" 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=\"(max-width: 1024px) 100vw, 1024px\" \/><figcaption>LINEAR PROGRAMMING APPLICATIONS<\/figcaption><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>LINEAR PROGRAMMING APPLICATIONS Linear programming is a powerful quantitative technique (or operational research technique) designs to solve allocation problem. The term \u2018linear programming\u2019 consists of the two words \u2018Linear\u2019 and \u2018Programming\u2019. 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&#8230;<\/p>\n<p class=\"more-link-wrap\"><a href=\"https:\/\/commerceiets.com\/linear-programming-applications\/\" class=\"more-link\">Read More<span class=\"screen-reader-text\"> &ldquo;LINEAR PROGRAMMING APPLICATIONS &#8211; Notes of Operations Research&rdquo;<\/span> &raquo;<\/a><\/p>\n","protected":false},"author":2,"featured_media":3618,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[228],"tags":[],"_links":{"self":[{"href":"https:\/\/commerceiets.com\/wp-json\/wp\/v2\/posts\/3617"}],"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=3617"}],"version-history":[{"count":0,"href":"https:\/\/commerceiets.com\/wp-json\/wp\/v2\/posts\/3617\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/commerceiets.com\/wp-json\/wp\/v2\/media\/3618"}],"wp:attachment":[{"href":"https:\/\/commerceiets.com\/wp-json\/wp\/v2\/media?parent=3617"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/commerceiets.com\/wp-json\/wp\/v2\/categories?post=3617"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/commerceiets.com\/wp-json\/wp\/v2\/tags?post=3617"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}