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OR methodology consists of five steps. They are defining the problem, constructing the model, solving the model, validating the model, and implementing the result. Definition: The first and the most important step in the OR approach of problem solving is to define the problem. Construction: based on the problem definition, you need to identify and select the most appropriate model to represent the system. Solution: After deciding on an appropriate model, you need to develop a solution for the model and interpret the solution in the context of the given problem.
Validation: A model is a representation of a system, however, the optimal elution must work towards improving the system’s performance. Implementation: You need to apply the optimal solution obtained from the model to the system and note the improvement in the performance of the system. Techniques and tools of operation research: Linear programming: You can use linear programming to find a solution for optimizing a given objective. The objective may be to maximize profit or to minimize cost.
Inventory control methods: The production, purchasing, and material managers are always confronted with questions, such as when To buy, and how much to keep in stock. Goal programming: In linear programming , you take a single objective function and consider all other factors as constraints. Queuing model: The queuing theory is based on the concept of probability. There is no minimization or minimization of an objective function. Transportation model: The transportation model is an important class of linear programs. The supply at each source and the demand at each destination are known.
In additional to the above, there are tools such as the sequence model, the assignment model, and network analysis, which you will learn in detail in the later units. Q. 2: a. Explain the steps involved in linear programming problem formulation. Discuss in brief the advantages of linear programming. B. Alpha Limited produces & sells two different products under the brand names black & white. The profit per unit on these products in RSI. 50 & RSI. 40 respectively. Both the products employ the same manufacturing process which has a fixed total capacity of 50,000 man-hours.
As per the estimates of the marketing research department of Alpha Limited, there is a market demand for maximum 8,000 units of Black & 10,000 units of white. Subject to the overall demand, the products can be sold o produce one unit of white, formulate the model of linear. NAS: (A)The steps involved in the formation of linear programming problem are as follows: Step 1 Identify the Decision Variables of interest to the decision maker and express them as XSL,XX, step 2 Ascertain the Objective of the decision maker whether he wants to minimize or to maximize.
Step 3 Ascertain the cost (in case of minimization problem) or the profit (in case of minimization problem) per unit of each of the decision variables. Step 4 Ascertain the constraints representing the maximum availability or minimum ointment or equality and represent them as less than or equal to type inequality or greater than or equal to (2) type inequality or ‘equal to’ (z) type equality respectively. Step 5 Put non-negativity restriction as under: x] 20, J ; = 1, 2 …. N (non-negativity restriction) Step 6 Now formulate the LIP problem as under: Maximize (or Minimize) Z = call + Cox ….. Nix Subject to constraints: alls +awake, ANSI + ax + awake, . … Al Nix bal (Maximum availability) …. Annex be (Minimum commitment) …. Annex = be (Equality) mammal + mamas, …. Annex bum XSL; xx CNN O (Non-negativity restriction) where, ] -?Decision Variables I. E. Quantity of Jet variable of interest to the decision maker. C] -?Constant representing per unit contribution (in case of Minimization Problem) or Cost (in case of Minimization Problem) of the Jet decision variable. Ail -?Constant representing exchange coefficients of the Jet decision variable in the tit constant. I -?Constant representing tit constraint requirement or availability. Discuss in brief the advantages of linear programming: Linear programming technique helps in making the optimum utilization of productive resources. The quality of decisions may also be improved by linear programming sequences. Linear programming technique provides practically applicable solutions because there might be other constraints operating outside the problem. In production processes, high lighting of bottlenecks is the most significant advantage of this technique. B) Solve the following transportation problem using Vogue’s approximation method. Factories Distribution Centers Supply 3 2 7 6 50 5 4 25 Requirements 20 15 NAS:(A) If the basic feasible solution of a transportation problem with m origins and n destinations has fewer than m + n – 1 positive xii (occupied cells), the problem is said o be a degenerate transportation problem. This also means that a feasible solution to a transportation problem can have only m+ n-l positive component, otherwise, the solution will degenerate.
The basic solution to an m-origin, n destination transportation problem can have at most m+n-l positive basic variables (non-zero), otherwise the basic solution degenerates, it follows that whenever the number of basic cells is less than m+n-l, the transportation problem is a degenerate one. This value may be thought of as an infinitely small amount, having no direct bearing on 1. At the initial solution 2. During the testing of the optimal solution To resolve degeneracy, we make use of an artificial quantity (d). The quantity d is assigned to that unoccupied cell, which has the minimum transportation cost.
This can be achieved either y inspection or by following some simple rules. We can begin by imagining that the transportation table is blank that is initial Xii = O. (B) First find the penalty cost, naively the difference between the smallest and next smallest costs in each row and column. The VAMP assigns penalties to bad choices by assigning to each row (respectively, column) the penalty equal to the difference of the two mallets cost coefficients in that row (respectively, column). Row Penalty 3-2=1 4-2=2 column penalty = 3-2=1 5-2=3 4-2=2 5-3=2 Now again calculation the penalty. -3=3 Column penalty = 3-2=1 4-2=2 5-3=2 Final table of Vogue’s approximation method The total cost of this initial Vogue’s approximation model solution is 2*40 + 7*10 + 7*60 + 4*10 + 5*15 = 80+70+420+40+75 = RSI. 685 Q. 4: A. Explain the steps Hungarian method. Differentiate between Transportation and Assignment problem. B. Find the optimal assignment of four Jobs and four machines when the cost of assignment is given by the following table: 2 14 MI 10 9 8 NAS:(A) The following steps are adopted to solve an AP using the Hungarian method algorithm.
Step 1 :Prepare row ruled matrix by selecting the minimum values for each row and subtract it from the other element of the row. Step 2:Prepare column reduced matrix by subtracting minimum value of the column from the other values of that column. Step 3:Assign zero row wise if there is only one zero in the row and cross(X) or cancel other zeros in that column. Step 4:Assign column wise if there is only one zero in the column and cross other zeros in that row. Step 5:Repeat steps 3 4 till all zeros are either assigned or crossed. If the number of assignments is equal to number of rows present, if not, proceed to steps.
Step 6:Mark the unassigned rows. Look for crossed zero in that row. Mark the column containing the crossed zero. Mark the row containing assigned zero. Repeat this process till all the makings are done. Step 7:Draw a straight line through unmarked rows and marked column. The number of straight line drawn will be equal to the number of assignments made. Step 8:Select the minimum. Subtract it from the uncovered elements. Add it at the point of intersection of lines. Leave the rest as is. Prepare a new table. Step 9:Repeat steps 3 to 7 till optimum assignment is obtained. Step 10:Repeat steps 5 to 7 till number if allocation = number of rows.
Transportation Assignment This problem contains specific demand and requirement in columns and rows. The demand and availability in each column or row is one. Total demand must be equal to the total Availability. It is a square matrix. The no of rows must be equal to the no of columns. The optimal solution involves the following conditions M+N-I M rows N columns The optimal solutions involves one assignment in each row and each column. There s no restriction in the number of allotments in any row or column. There should be only one allotment in each row and each column.
It is a problem of allocating multiple resources to multiple markets. It is a problem of allocation resources to Job. (B) Applying Hungarian method Step 1: Row reduced matrix Step 2: Column reduced matrix Since the number of assignment is 4. MI took-7 MM took-1 MM to 12-3 14 Total cost Q. 5: Define Simulation. Explain the Simulation procedure. Discuss the use of Simulation with an example. NAS: Simulation: Simulation is the process of defining a model of a real system and conducting experiments with this model for the purpose of understanding the behavior for the operation of a system.
Using simulation, an analyst can introduce the constants and variables related to the problem, set up the possible courses of action and establish criteria which act as measures of effectiveness. Simulation procedure: In the previous section, you learnt why simulation technique is applied. You will now learn the methodology of simulation. The methodology developed for simulation process by follow as: Step-I Identify and clearly define the problem. Step-2 List the statement of objective of the problem. Step-3 Formulate the variables that influence the situation and an exact or probabilistic description of their possible values or states.
Step-4 Obtain a consistent set of values for the variables, for example: a sample of probabilistic variable, random sampling technique may be used. Step-5 Use the sample obtained in step 4 to calculate the value of the decision criterion, by actually following the relationships among the variables for each of the alternative decisions. Step-6 Repeat steps 4 and 5 until a sufficient number of samples are available. Step-7 the ablate the various values of the decision criterion and choose the best policy.
Discuss the use of Simulation with an example: Discuss below, Economics A mathematical model of the economy, having been fitted to historical economic data, is used as a proxy for the actual economy; proposed values of government spending, taxation, open market operations, etc. Are used as inputs to the simulation Simulation is an important feature in engineering systems or any system that involves many processes. Finance In finance, computer simulations are often used for scenario planning. Q. 6: Explain the following: a. Integer programming model b. PERT and CPM c. Operating Characteristics of a Queuing System
NAS: A. Integer programming model: These methods may be used when one or more of the variables can take only integral values. Examples are the number of trucks in a fleet, the number of generators in a power house, etc. This problem can be classified into three categories: Pure Integer programming problem: all decision variables are restricted to integer values. Mixed integer programming problem: Here some, but not all, of the decision variables are restricted to integer values. Zero-one integer programming problem: all decision variables are restricted to integer values of O and 1 .
Airline decides on the umber of flights to operate in a given sector must be an integer or whole number amount. Other examples: The number of aircraft purchased this year, The number of machines needed for production, The number of trips made by a sales person, The number of police officers assigned to the night shift. Some Facts: Integer variables may be required when the model represents a one time decision (not an ongoing operation). Integer Linear Programming (LIP) models are much more difficult to solve than Linear Programming (LIP) models. Algorithms that solve integer linear models do not provide valuable sensitivity analysis results.
B. PERT and CPM: Network scheduling is a technique used for planning, scheduling and monitoring large projects. Such large projects are very common in the field of construction, maintenance, computer system installation, research and development design, etc. Projects under network analysis are broken down into individual tasks, which are arranged in a logical sequence by deciding as to which activities should be performed simultaneously and which others sequentially. In this Project scheduling and PERT-CPM in operation management, we will study about the basic different teen PERT and CPM.
We will study about PERT and CPM network components, determination of floats and project management. Project management has evolved as a new field and project management. PERT and CPM are basically time-oriented methods in the sense that they both lead to the determination of a time schedule. C. Operating Characteristics of a Queuing System: a queuing model has the following operation characteristics which enables us to understand and efficiently manage a queue: Queue length: The number of customers in the waiting line reflects one of the two conditions.
Number of customers in system: The number of customers in queue and also those being served in the queue relates to the service efficiency and capacity. Waiting time in queue: Long Waiting times may indicate a need to adjust the services rate of the system or change the arrival rate of customers. Waiting time in system: The total problems with customers, sever efficiency or capacity. Service facility utilization: The collective utilization of the service facilities reflects the percentage of time the facilities are busy. A queuing system is said to be in transient state when its operating characteristics are dependent upon time.