Question

Question 2: (20 Marks) A) Explain and discuss with examples the stages of development of Operational research? B) Solve the following linear programming problem graphically: z = 10x+15 Subject to : 3x+6x, 560 Max X: + xy S 16 * 20 C) Discuss and explain the aim and the steps of the ?stepping stone methods of the transportation problems (Question 3: (20 Marks A) Find the starting basic feasible solution for the following Table by using 1- North west corner methods. 2- Least cost method. 3- Vogel approximation method. SUPPLY D4 D3 D2 DI S/D 2105 10 1 2 SI 380 S2 410 112914 S3 250 150 200 400 DEMAND B) Define what the decision theory means and which strategy should be used by someone wants to travel from Jeddah to Albaha as in the following Table and why? Available Transportation Cost car bus plain especial car 80 80 150 130 C) Explain with an example what Laplace criterion means under decision making theory?

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Answer 1

Question (2) ca) The stages of development of operational research so as Definition! - Operations research is the application of scientific methods and techniques and also the toals to problems in valuing the operations of Systems as to movide those in control of the systems with optimum sulutions to the problem. stages alli- (a) Formulate the problem. (6) constructing a model to represent the system. CC) acquire input data. (d) Deriving a salution from the model. (e) Testing the salution, solution. (f) Implement the (g) modifying the model, (a) formulate the problem - It is the first step in application of OR methods. In fumulating a problem, four components should be analysied. (i) The environment cii) Decision maker (ii) objectives cius alternative counses of actions and constraints, gt involves:-

2) un controllable Analysis of the physical system under Study. setting up of objectives. Identification of controllable and variables invalved in the system. measure of effectiveners. etc. construction of model - After all a is wolk. af lle we feumulation of the Moblem, it is the next step to Construct model for the for the system system under study. The proposed model is tested in the field under different environmental constraints and modified in order to Acquire input data - The main this steps, to obtain accurate and complete data because should know that the quality of quality of data determine the quality of output Deriving a solution to the problem In this the solution to the problem is determined help of the model and data input. Basically there types of mathematical techniques are generally used for this purpose. Analytical method. Numerical method. Simulation, be (d) step, by the ci) (ii) ci)

се obtained the results (f) Testing the solution- The main aim of this step is to measure its effectiveness and efficiency. The utility of the subution should he verified by comparing the comparing the result without peoplying the solution with obtained, when it is used. Implement the solution: salution : This step is wed to improving the performance of the system. know that, a poperly implemented publem obtained through application of operations research techniques results improved working conditions and gains the mgmt. suppunts. (8) modyfying the model- The main cim As we result un solution of any in af this sters are! CO) To identify the constraints that might have been omited during the original formulation ay given Roblem. Cij) To find it some of the criginal constraints inconect and need to be modified. were

CB2 solved by graphical method _ D = max. (2) 10X1 + 1542 sulyject to, 3x + 622 560 21+ 2 < 16 x X₂ 2.0 from the constraints equations ② & ③ 605 lo 2,=0 6 37,+ 642 = 60 =) 612 = 60 ) 2 = (X,Y): (0,10) 20 = 20 bo 3 2y =0 -) 321 = 60 =) 21 = (X,,22) : ( 20,0) = 16 x2 = 16 from eç. ③ x1 + x2 R=0 =) (2,22) : (0,16) 22 =0 . (14, 12) : (160) x = 16 . Now, we plut a graph between xi (inx-direction) and az (in y- direction) using these co-ordinates.

20 18 (0,16) 16 x + 2, L16 12 3x +692 660 (0,0) 8 feasible region 6 C(12,4) 4 2 (2010) AC010) 03 2 4 6 8 10 12 14 origin All points 18 D (16,0) Region - in feasible A (0,0). , B (0,10), C (12,4) D(16, 0) c point:- - 60 321+ 6X2 x+ 22 = 16 х3 672 = 60 34/+ 341 + + 312 48 342 = 12 =) =12 4 12 = X 12=4 x+ 2 = 16 xit4 = 16 X, = 16-4 = 12 %=12 ( point ☺ 18, 2*2); (124)

put الله values in objective function SZ OUT max. (2) = 10x + 1542 ALO,U) : (%, ,*2) ->Z=10X0+ 1rxu B(0,10) : (2,22) = loxo tinxio = (12,4) : (21,22) 2= 10X12+15xy = D (16,0) : (X1, 22) — ->2 = Joxlb+ 15xU max. 180 XC (12,4) 160 Result:- x = 12 x2 = 4 Ans. Max.(2) 081 x

7 CC ） STEPPING STONE METHOD The Stepping stone method check the main aim of the stepping cie. Distribution method is to optimality of the transportation Roblem after the solution of any given problem by the these methods, which are houthwest corner method, least cost method and vogel's approximation method. optimality test formula – mth-La Number of allocations. where, mo No. of rows. No- cf calumn. Consider the mateix proceed now by loul sters! ci) giving first initial the feasible solution for the given problem, and select any unoccupied cell ciie empty cells in this matrix. (i) Identify the closed path related to cell. (ii) Ansign alternate (tre) positive and (-ne) negative signs on the closed path starting with a (4) sign in the selected empty cell.

(8 move remaining cu can be (iv) one unit from a stone square into the empty cell and modity the empty cell at the other Corners of the closed path. Repeat steps lip to civ) for each of the remaining empty cells. Cvij A negative sim on the net cast change indicates that a cost reduction made by making the change of more than one negative values, poceed from the most negative values. Or cell (empty cell) (vil Subtract the smallest of the of the values from each cell at the -ve sign on the closed path and additto each square at the the sign on the closed path. (ville) In last step, prepare a new mateix with the improved solution and same steps followed until the positive values will not obtained. -X