Preview; Full text Operations Research, S. In order to derive full benefits, continuity research must be maintained. Its PDF is Account More Details Show full review. Sixth Edition. Professor, Amity Bussines School.
Amity University Uttar Pradesh, Noida. Behaviors on. Sharma org and D. Toggle navigation pdf Book free download. Search Result for "operations research by jk sharma pdf" List of ebooks and manuels about "operations research by jk sharma pdf" Free PDF ebooks user's guide, manuals, sheets about "operations research by jk sharma pdf" ready for download.
Copyright Disclaimer: All books are the property of their respective owners. Parameters These are constants in the functional relationships, Parameters can either be deterministic or probabilistic in nature. A deterministic parameter is one whose value is assumed to occur with certainty. Once mathematicel model of the problem hus been formulated, the next step isto solve it, that is, to obtain numerical values of decision variables.
Obtaining these values depends on the specific form or type, of mathematical model. Solving the model requires the use of various methematical tools and numerical procedures. Although using this rule may not work if everyone in the shortest line requires extra time, in general, it is not a bad rule to follow. These methods are used when obtaining optimal solution is either very time consuming or the model is too complex.
But if the objective function of eny or all of the constraints cannot be expressed 2s a system of linear equalities or inequalities, the allocation problem is classified s a non-linear programming problem. If the decision variables in the linear programming problem depend on chance the problem is called a stochastic programming problem. If resources such as workers, machines or salesmen have to be assigned to perform a certain number of activities such as jobs or territories on a one-to-one basis so as to minimize total time, cost or distance involved in performing a given activity, such problems are classified as assignment problems.
But ifthe activities require more than one resource and conversely ifthe resources can be used for more than one activity, the allocation problem is classified as a trarsporaation problem. The main objective is to minimize the sum of three contlicting inventory costs: The cost of holding or carrying extra inventory, the cost of shortage or delay in the delivery of items when it is needed and the cost of ordering or Set-up.
These are also useful in dealing with quantity discounts and selective inventory control. Constructing a mode! These models are classified according to several factors such as number of competitors, sum of loss and gain, and the type of siratezy which would yield the best or the worst outcomes. These techniques improve project coordination and enable the efficient use of resouress. Network methods are also used to determine time-cost trade-off, resource allocation and help in updating activity time.
For example, in case of an automobile, the user has his own measure of effectiveness. So there will net be one single optimal answer for everyone, even if each automobile gives exactly the same service.
The method starts by dividing a given problem into stages or sub-problems and then solves those sub-problems sequentially until the solution to the original problem is obtained. The model, while dealing with such systems, describes transitions in terms of transition probabilities of various states. Linear programming also helps in the re-evaluation ofa basic plan for changing conditions. Ifeondivions change when the plan is partly carried out, they can be determined so as to adjust the remainder of the plan for best results.
These are given below: 1. Linear programming trea's all relationships among decision variables as linear. However, generally, neither the objective functions nor the constraints in real-life situations conceming business and industrial problems are linearly related to the variables, 2 While solving an LP model, there is no guarantee that we will get integer valued solutions. For example, in finding out how many men and machines would be required to perform a particular job.
Rounding off the solution to the nearest integer will not yield an optimal solution. In such cases, integer programming is used to ensure integer value to the decision variables 3. The linear programming model does not take into consideration the effec of time and uncertainty. Thus, the LP model should be defined in sucha way that any change due to internal as well as external factors can be incorporated. Sometimes large-scale problems can be solved with linear programming techniques even when the assistance of a computer is available.
For this, the main problem ean be divided into several smal problems and each one of them can be solved separately 5. Parameters appearing in the model are assumed to be constant but in real-life situations, they are frequently neither known nor constant.
In such cases, instead of the LP model, a goal programming model is used to get satisfactory values of these cbjectives 2. Efficient production patterns can be specified by a linear programming model under regional, land resources and rational demand constraints.
Linear programming can be applied in agricultural planning, e. The applications are also used for varying the transportation in such a way that it maximizes the total tonnage of bombs dropped on a set of targets and takes care of the problem of community defence against disester, the solution of which yields the number of defence units that should be used in a given attack in order to provide the required level of protection at the lowest possible cost.
In such cases, it is essential to determine the quantity of each product to be produced, knowing its marginal contribution and amount of available resourc: used by it. The company is obliged to produce daily atleast 10 thousand gallons of A and 8 quintals of B. Each unit of C, costs the company Rs 5 in wages and Rs 5 in material, while each of , costs the company Rs 25 in wages and Rs 15 in material.
First, at the beginning of period , the company has an initial belance of Rs 4, cash plus bank credit plus collections from past credit sales Second, the company has available in each period 2, hours of machine time and 1, hours of assembly time. The production of each C, requires 3 hours of machine time and 2 hours of assembly time, whereas the production of each C, requires 2 hours of machine time and 3 hours of assembly time.
Formulate this problem as an LP model so as to maximize the total profit to the company. Formulate this problem as an LP model to maximize toval profit. Ifany of these man-hours are not needed, some members of the firm woulduse them to work on a neighbouring farm for Rs 2 per hour during September — May and Rs 3 ner hour during June — August. Cash income can be obtained from the three main crops and two types of livestock: airy cows and laying hens.
No investment funds are needed for the crops. However, each cow will require an investment outlay of Rs 3, and each hen will require Rs In additon each cow will also require 15 acres of land, man-hours curing the summer Each cow will produce a net annval cash income of Rs 3, for the farm.
The corresponding figures for each hen are: no acteage, 0. Formulate this problem as an LP model to maximize net annual cash income. No more than 40 doctors can start their five working days on the same day. Formulate this problem as an LP model to minimize the number of doctors employed by the hospital. The training programme lasts for one month, From past experience it has been found that out of the ten trainees hired, only seven complete the programme successfully and the rest are released.
The company's requirement for machining for the next three months is as follows: January , February and March There are trained machinists available at the beginning of the year. Assuming thatthe items preduced can be sold, which department needs to be expanded for increasing proits? Formulate this problem as an LP model. MBA, ind has the following However the ratios of the number of units produced must be equal to , Assume that the profit per unit of models I and I is Ais 60, Fs 40 and As , respectively.
Formulate this problem as an LP mocel to determine the number of urits of fact product that will the maximize amount 0!
When preparing the budget, it was found that the limitations on capacity were represented by the folowng woek! Assume that the company can sell any quantiy of either product. The product can be manufactured during regular time at 2 cos of Rs 16 per unit produced, or during overtime at cost of RS 20 per unit. Remark Ifa constraint to which the objective function is parallel does not form the boundary of the feasible region, the multiple solutions will not exist.
This type of a constraint is called redundant constraint, i. This means that when one or more decision variable values and the value of the objective function maximization case are permitted to increase Ynpaunded. This is known as en te soutton an unbounded solution. It is important here to note that there is a difference between a feasible region vatable and the ing unbounded and an LP problem being unbounded.
It is possible that in a particular problem the Prfit can be made Teale region may be urbounded but LP problem may not be unbounded, The general cause for au unbounded LP St me mavimizaton problem is an improper formulation of the real-life problem. LP problem Example 3. Solution Plot oa graph each constraint by frst treating it as a linear equation in the same way as discussed earlier.
Then use the inequality condition of each constraint to mark the feasible region as shown in Fig. That is, both the variables x, and x, can be made arbitrarily large and according the value of objective function Z will also increase.
Thus, the problem has an unbounded solution Example 3. The solution space is shaded and is bounded jrom below. It is noted here that the shaded convex region solution space is unbounded from above. For example, the point 2, 3 lies in the region and the function value at this point is 12 which is more than 8.
Thus, both the variables x, and x, can be made arbitrarily large and accordingly the value of Z will also increase. Hence, the problem has an unbounded solution. Example 3. The solution space is shown shaded and is bounded from below. This shaded convex region solution spsce is unbounded from above. Since the given LP problem isof maximization, these exists a number of points in the solution space where the value of objective function is much more than Hence, the unique value of Z cannot be found as it occurs at infinity only.
Each unt of type A requires 39 Of siver and 1g of gold while B requites 19 of slver and 2g Of gold. The company can produce 9g of siver and 8g of oc. The ABC company has been a producer of picture tubes for television sets and of certain printed cicuis for radios. Formulate this prodlem as an LP model so as to determine the optimal producton mix of AM-FM radios that will maximize profts. Thoso parte aro then sent to one ofthe two divisions for beng assembly into the final product.
Dvision 1 ie used for product A, and Divison 2 for product B. Product A requires 40 unis of raw material land 10 hours of machine processing line. The capabilites of the two assembly dvisons dung the petiod are 6 and 9 Units, respectively. The eight-day seven-right package includes the round- tip fare, surface transporaion, boarding and lodging end Selected tour options. The charter tro is restrcted to [persons and past experience Indates that there will not be any problem in arranging passangers.
The following table summarizes the estimated prices of the three packages and the Corresponding expenses of the travel agent. The travel agent hhag hired an srera for a Mat foe of is 2,00, for tha entra trip. Ww The maximum numter of deluxe packages aralatie in any aircraft is restricted to The book may be bourd by either clath oF hard paper. Each ciot-bound book sold contibutes Fs 24 towerds the profit and each paperbound book contributes AS The total avaiable time for binding is 80 hours.
Aller considering a number of market surveys, it Is predcted that the cbth-cover sales will be arything more than 10, copies, but the paperback sales will not be more than The feed mix, Fetiex, requis at least twive as much wheat as barley.
Wheat costs Rs 1. Barley costs Fs 1. Multiplex sels at Rs 1. Formulatethis problem asa LP problem to determine the product mix so as to maximize the profs.
On October 1st, a company received a contract to supply 8, unis of a specialzed product. The terms of contract requires that units be shipped in the north of October; 3, units in November and 2, units in December. The monthly storage cost is Re 1.
Formulate this protiem as a linear programming problem so as to minimize the total cos A emall-scale manufacturer has preducton facilites.
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