P a g e Q: A firm has normally distributed forecast of usage with MAD=0 units. It desires a service level, which limits the stock, out to one order cycle per year. Determine Standard Deviation (SD), if the order quantity is normally a week`s supply. Solution: SD (0) SD SD 3.4 (0). (0) SD (.)(0) SD 6.64 Q: A company centre has got four experts programmers. The centre needs four application programmers to be developed. The head of the computer centre, after studying carefully the programmer s to be developed, estimate the computer time in minutes required by the respective experts to develop the application programmers as follows. Programmers A B C D 0 00 80 90 80 90 0 0 3 0 40 0 00 4 90 90 80 90 Solution: Programmers A B C D 30 0 0 0 0 0 40 0 3 0 30 0 0 4 0 0 0 0 Programmers A B C D 30 0 0 0 0 0 40 0 3 0x 30 0 0x 4 0x 0 0x 0 Q: the cost of a new machine is Rs. 000. The maintenance cost during the nth year is given by M n = Rs.00 (n-), where n=,, 3 If the discount rate per year is 0.0, determine discount factor (v n- ) for each year.
P a g e Solution: M = 000 M n = 00 (n-) V = 0.0 (v n- ) =? Nth year M n (v n- ) 0.00 00 0.0 3 000 0.00 4 00 0.000 Or Nth year (v n- ).00 0.0 3 0.00 4 0.000 Q: Determine whether the following Transportation model has initial feasible solution? D D D 3 D 4 Supple Q x x x 3 x 4 6 Q X X X 3 X 4 8 Q 3 X 3 X 3 X 33 X 34 0 Demand 4 6 8 6 Solution: The transportation problems can be represented mathematically as a linear programming model. The Objective function in this problem is to minimize the total transportation cost given by Z = c x + c x +... + c mnx mn Subject to the restrictions: Row restrictions: x + x + x 3 + x 4 = 6 x + x + x 3 + x 4 = 8
P a g e 3 x 3 + x 3 + x 33 + x 34 = 0 Column restrictions: x + x + x 3 + x 4 = 4 x + x + x 3 + x 4 = 6 x 3 + x 3 + x 33 + x 43 = 8 x 4 + x 4 + x 34 + x 44 = 6 And x + x + x 3 + x 4 0 It should be noted that the model has feasible solutions only if a + a + a 3 + a 4 = 4+6+8+6 Or m n i=0 a i = b j j= Q: Salesman Region 3 4 6 A 0 0 0 B 0 0 0 30 4 C 3 0 40 D 0 30 30 0 E 0 30 0 30 F 3 3 0 0 0 Do next step by applying Hungarian method? Solution: Salesman Region 3 4 6 A 0x 0x 0x B 0 0 0x 30 4 C 3 0 40 D 0 30 30 0 E 0 30 0 30 F 3 3 0 0 0
P a g e 4 Here we have only three assignments. But we must have four assignments. With this maximal assignment we have to draw the minimum number of lines to cover all the zeros. Q: An oil company has 8 unit of money available for exploration of three sites. If oil is present at a site, the probability of finding it depends upon the amount allocated for exploiting the site as given below. 0 3 4 6 8 Site I 0.0 0.0 0. 0. 0.3 0. 0. 0.9.0 Site II 0.0 0. 0. 0.3 0.4 0.6 0. 0.8.0 Site III 0.0 0. 0. 0. 0.3 0. 0.8 0.9.0 The probability that the oil exits at sites I, II and III is 0.4, 0.3 and 0. respectively; we have to find the optimal allocating of money. Stage I is given below, only do stage it. Stage I Max. Z=0.4P (x ) + 0.3P (x ) Subject to: x +x +x 8 No. of boxes x 0 3 4 6 8 f (x ) 0 0 4 8 0 8 36 40 Q. a person wants to decide the constituents of a diet which will fulfill his daily requirements of protein, fats and carbohydrates at the minimum cost. The choice is to be made from four different types of foods. The yields per unit of these foods are given in the table below: Food type Yield per unit Proteins Fats carbohydrates 3 6 3 Cost per unit (Rs.) 4 4 4 40 3 8 8 4 6 4 6 Min Requirement 800 00 00 Solution:
P a g e Let x, x, x3 and x4 denote the number of units of food of type,, 3 & 4 respectively. Objective is to minimize the cost i.e. Minimize Z = 4x+40x+8x3+6x4 Constraints are on the fulfillment of the daily requirements of various constituents i.e. Proteins - 3x + 4x + 8x3 + 6x4 800 Fats - x + x + x3 + x4 00, Carbohydrates - 6x + 4x + x3 + 4x4 00. Where x, x, x3, x4 each 0 Question No: 4 ( Marks: ) Fall 0 A branch of Punjab National Bank has only one typist. Since the typing work varies in length (number of pages to be typed), the typing rate is randomly distributed approximating a Poisson distribution with mean service rate of 8 letters per hour. The letters arrive at a rate of per hour during the entire 8 hour worki9ng day. If the typewriter is valued at Rs..0 per hour, Determine Average system time. : W s= /µ-λ = /8- =/3hr=/3*60=0 min Question No: 4 ( Marks: ) An oil company has 8 unit of money available for exploration of three sites. If oil is present at a site, the probability of finding it depends upon the amount allocated for exploiting the site as given below:
P a g e 6 0 3 4 6 8 Site I 0.0 0.0 0. 0. 0.3 0. 0. 0.9.0 Site II 0.0 0. 0. 0.3 0.4 0.6 0. 0.8.0 Site III 0.0 0. 0. 0. 0.3 0. 0.8 0.9.0 The probability that the oil exits at sites I, II and III is 0.4, 0.3 and 0. respectively; we have to find the optimal allocating of money. Do stage I only. Not Attempted Question No: 43 ( Marks: ) Write the relationship between the activities.
P a g e X approches to Y X also approches to Z Y approches to Z Whether A and b might have the values between the centre points Question No: 44 ( Marks: ) For the mathematical form of a Transportation problem (T.P) i m j n min z c x () i j ij ij subject to j n j i m i x a (), i,,, m(sources) ij i x b (3), i,,, n(destinations) ij j Describe the practical significance of all the above equations(), () and (3). :
P a g e 8 The above is a mathematical formulation of a transportation problem and we can adopt the linear programming technique with equality constraints. Here the algebraic procedure of the simple method may not be the best method to solve the problem and hence more efficient and simpler streamlined procedures have been developed to solve transportation problems. Question No: 4 ( Marks: 3 ) The milk plant at a city distributes its products by trucks, located at the loading dock. It has its own fleet of trucks plus trucks of a private transport company. This transport company has complained that sometimes its trucks have to wait in line and thus the company loses money paid for a truck and driver that is only waiting. The company has asked the milk plant management either to go in for a second loading dock or discount prices equivalent to the waiting time, the following data available Average arrival rate 3 per hour Average service rate 4 per hour The transport company has provided 40%of the total number of trucks. Assuming that these rates are random according to Poisson distribution, determine a) The probability that a truck has to wait. b) The waiting time of a truck that waits. The probability that a truck has to wait. 4 4 4 4 3 The waiting time of a truck that waits.
P a g e 9 round about 40 minutes of each truck. Question No: 46 ( Marks: 3 ) A company has a machine whose cost is Rs. 30,000. Its maintenance cost and resale value at the end of different years are as given below: Years. 3 4 6 Maintenance Cost. 400 400 000 00 600 00 Resale Value 000 300 4000 000 8000 3000 Determine capital cost for each year. Question No: 4 ( Marks: 3 ) A firm produced three products. These products are processed on three different machines. The time required to manufacturer one unit of each of the three products and the daily capacities of the three machines are given in the table: Machines Time per unit (minutes) Product Product Product 3 Machine Capacity (minutes / day) M 3 440 M 4 --- 3 40 M3 3 --- 430
P a g e 0 It is required to determine the daily number of units to be manufactured for each product. The profit per unit for product, and 3 is Rs. 4, Rs. 3 and Rs. 6 respectively. It is assumed that all the amounts produced are consumed in the market. Write the constraints of above Linear Programming Problem. Step Find the key decision to be made. The key decision is to decide the extent of product,&3 to be produced as this can vary. Step Assume symbols for the extent of production. Let the extent of Product,&3 be X, X & X3. Step 3 Express the feasible alternatives mathematically in terms of variables. Feasible alternatives are those which are physically, economically and financially possible. In this example, feasible alternatives are sets of values of x, x & x3, where x,x &x3 0 since negative production has no meaning and is not feasible. Step 4 Mention the object quantitatively and express it as a linear function of variables. IN the present example, objective is to maximize the profit. i.e. Maximize Z = 4x+3x+6x3 Step Express the constraints as linear equations/inequalities in terms of variables. Here, constraints are o the machine capacities and can be mathematically expressed as x + 3x + x3 440, 4x + 0x + 3x3 40, x + x + 0x3 430. Question No: 48 ( Marks: 3 ) Express the following Transportation problem (T.P) table into algebraic form with proper objective function and non-negative constraints
P a g e D D D 3 Supply O c c c 3 O c O 3 c 3 Deman d c c 3 c 3 c 3 3 4 6 8 3 0 4x S D S 6y D 3 S S 8z 0 Question No: 49 ( Marks: ) 3 Supply 6 0 4 0 3 3 Deman d 0 0 0 Complete the above transportation Model by Vogel Approximation Method. And also find the starting basic feasible solution.
P a g e 3 Supply 6 0 4 0 3 3 Deman d 0 0 0 Cost = + + (6) + + () = This is the initial basic solution consider u = and v = and v = 3 Question No: 0 ( Marks: ) Check whether the given initial basic feasible solution is optimal or not. 6 0 4 0 3 0 0 0 0
P a g e 3 3 Supply 6 0 4 0 3 3 Deman d 0 0 0 Cost = + + (6) + + () = This is the initial basic solution consider u = and v = and v = 3 it is a n optimal solution according to if we put formula Question No: ( Marks: ) A company cost Rs. 00 operations and maintenance costs are zero for the first year and increased by Rs. 00 every year. If money is worth % every year, calculate present worth (P(r)) for each year. The resale value of the machine is negligibly small.
P a g e 4 for each year increase in money = % means to say that company significes 0 ruppes every year the company capital must be increaing as 0 * 00 Question No: ( Marks: ) Express the following linear programming problem in standard form and also construct its initial simplex table. Max Z = 3x+y Subject to constraints: x + y 4 x y x,y 0 Blank Data marks qs Mth60 30 July 03 final term paper: Q: ek bohat sari activities wali diagram di hui thi or qs ye tha
P a g e Find EFT for each activity? Q: 0 4 3 0 6 To find optimality condition we use UV multiplier process Find a) U+V b) U3+V3 Q: 3 ya mark ka tha ye qs Contractor side wali values yad nai Building: Contractor: A B C D Operate first step by optimizing row wise the above assignment model. Q: state principal of optimality (optimal policy) for dynamic programming? Q: fin EST and EFT for each activity. A B 8 6 D 0 C 0 E 4
P a g e 6 3 Q: marks 0 6 4 3 0 Find a) P3 b) P3 Using Pij= Ui + Vj Cij suppose U = 0 and U = Q: ek statement thi us me se Average Queuing length find krna thi. Q: Minimizing setup times, which are given? ( marks) Job ki values yad nai Job Job Job3 Job4 Machine 4 Machine Machine 3 Machine 4 Q: ek mark ka qs itna long tha k word pe paste krne se ek se zyada page ki just statement thi. Replacement Of Items with change in value and time
P a g e It is assumed that the maintenance cost increases with time and each cost is to be paid just in the start of the period. Let the money carry a rate of interest r per year. Thus a rupee invested now will be worth ( + r) after a year, (+r) after two years and so on. Do first step? No. of stores 3 No. of boxes 0 0 0 0 4 6 6 4 8 3 6 8 4 8 8 9 8 0 8 No. of stores 3 No. of boxes 0 0 0 0 0 4 0 4 3 0 4 0 0 0 3 Diff b/w pert n CPM?? PERT (Programme Evaluation & Review Technique) is event oriented whereas CPM (Critical Path Method) is activity oriented. In CPM based network analysis no allowance is made for the uncertainties in the duration of time involved. In CPM, times are related to costs Q: Make two steps, of rows and columns of the following table
P a g e 8 : Least ko sab me se Minus krna hay pehlay rows, then columns, to have atleast one zero in all. Markets / salesmen I II III IV A 44 80 60 B 60 6 40 C 36 60 48 48 D 6 36 40 Markets / salesmen I II III IV A 8 4 6 0 B 4 0 4 3 C 0 4 8 D 6 0 0 0 Complete the table By VOgha s method:
P a g e 9 Sol: Red is solved one 3 Supply 0 3 4 3 Demand 0 6 july 03. a branch of bank has only one typist. typing rate is randomly distributed approximating a Poisson distribution with mean service rate of 8/hour. Letter arrive rate is /hour during the entire 8 hour working day if writer is value.0 per hour determine the equipment utilization? marks and same this question is appeared as marks question. Scenario was given and we have to tell the objective function of linear programming... marks 3. Table was given n determine that transportation model has initial feasible solution... marks 4. State principle of optimality for dynamic programming... marks. Transportation model was given and one block has x we have to find the value of x... 3 marks 6. Values were given and we need to tell the capital cost for each year... 3 marks. In the context of pert and CPM summarize the project planning techniques 8. One stage problem is given find the two stage problem s Fi*(s) Xi 8 0 9 3 0 9. Table was given and asked that check initial solution is feasible or not... marks 0. Question no was again appeared as marks question. The cost of the new machine is 000. Maintenance during the nth year is given by Mn = 00 rps (n-) when n =,, 3...if discount rate per year is 0.0 calculate the present worth... marks. Graph was given and question was construct the table relation show between events and activities. mark s