Submit Hard copy / Soft copy by 4th Nov, 2014Email: basant.mishra16@gmail.comAssignment (Semester III) B.Tech1) Ozark Farms uses at least 800 lb of special feed daily. The special feed is a mixture of corn and soybean meal with the following compositions:Ib per Ib of feedstuffFeedstuffProteinFiberCost ($/Ib)

Corn0.90.020.30Soybean meal0.600.060.90

The dietary requirements of the special feed are at least 30% protein and at most 5% fiber. Ozark Farms wishes to determine the daily minimum-cost feed mix.2) A company produces two products, A and B. The sales volume for A is at least 80% of the total sales of both A and B. However, the company cannot sell more than 100 units of A per day. Both products use one raw material, of which the maximum daily availability is 240 lb. The usage rates of the raw material are 2 lb per unit of A and 4 lb per unit of B. Total profit units for A and B are $20 and $50, respectively. Determine the optimal product mix for the company.3) Wild West produces two types of cowboy hats. A type 1 hat requires twice as much labor time as a type 2. If the all available labor time is dedicated to Type 2 alone, the company can produce a total of 400 Type 2 hats a day. The respective market limits for the two types are 150 and 200 hats per day. The profit is $8 per Type 1 hat and $5 per Type 2 hat. Determine the number of hats of each type that would maximize profit.4) Top Toys is planning a new radio and TV advertising campaign. A radio commercial costs $300 and a TV ad costs $2000.A total budget of $20,000 is allocated to the campaign. However, to ensure that each medium will have at least one radio commercial and one TV ad, the most that can be allocated to either medium cannot exceed 80% of the total budget. It is estimated that the first radio commercial will reach 5000 people, with each additional commercial reaching only 2000 new ones. For TV, the first ad will reach 4500 people and each additional ad an additional 3000. How should the budgeted amount be allocated between radio and TV?5) John must work at least 20 hours a week to supplement his income while attending school. He has the opportunity to work in two retail stores. In store 1, he can work between 5 and 12 hours a week, and in store 2 he is allowed between 6 and 10 hours. Both stores pay the same hourly wage. In deciding how many hours to work in each store, John wants to base his decision on work stress. Based on interviews with present employees, John estimates that, on an ascending scale of 1 to 10, the stress factors are 8 and 6 at stores 1 and 2, respectively. Because stress mounts by the hour, he assumes that the total stress for each store at the end of the week is proportional to the number of hours he works in the store. How many hours should 10hn work in each store?6) An industrial recycling center uses two scrap aluminum metals, A and B, to produce a special alloy. Scrap A contains 6% aluminum, 3% silicon, and 4% carbon. Scrap B has 3% aluminum, 6% silicon, and 3% carbon. The costs per ton for scraps A and Bare $100 and $80, respectively. Title specifications of the special alloy require that (1) the aluminum content must be at least 3% and at most 6%, (2) the silicon content must lie between 3% and 5%, and (3) the carbon content must be between 3% and 7%. Determine the optimum mix of the scraps that should be used in producing 1000 tons of the alloys.

Consider the following LP:7) Max Z= x1+3x2 s.t, x1 + x2 2-x1 + x2 4x1 Unrestricted x2 0a) Determine all the basic feasible solutions of the problem.b) Use direct substitution in the objective function to determine the best basic solution.

8) Consider the following LP:Max Z= 2x1+3x2 s.t, x1 + 3x2 63x1 + 2x2 6x1 , x2 0 c) Determine all the basic solutions and classify them as feasible and infeasible.d) Use direct substitution in the objective function to determine the best basic solution.

9) Consider the following LP:Maximize z = Xl s.t, 5x1+x2 = 46x1+x3 = 83x1+x4 = 3x1 ,x2, x3 ,x4 0 (a) Solve the problem by inspection (do not use the Gauss-Jordan row operations), and justify the answer in terms of the basic solutions of the simplex method. (b) Repeat (a) assuming that the objective function calls for minimizing z = Xl.10) Solve by Two phase method;Min Z= 4x1+x2 s.t, 3x1+x2 = 34x1+3x2 6x1+2x2 4x1,x2 011) Solve by Big M method;Min Z= 4x1+x2 s.t, 3x1+x2 = 34x1+3x2 6x1+2x2 4x1,x2 0

12) The following tableau represents a specific simplex iteration. All variables are nonnegative. The tableau is not optimal for either maximization or a minimization problem. ll1Us, when a non basic variable enters the solution it can either increase or decrease z or leave it unchanged, depending on the parameters of the entering non basic variable.Basic Xl X2 X3 X4 X5 X6 X7 X8 SolutionZ 0 -5 0 4 -1 -10 0 0 620X8 0 3 0 -2 -3 -1 5 1 12X3 0 1 1 3 1 0 3 0 6XI 1 -1 0 0 6 -4 0 0 0

(a) Categorize the variables as basic and non basic and provide the current values of all the variables. (b) Assuming that the problem is of the maximization type, identify the non basic variables that have the potential to improve the value of z. If each such variable enters the basic solution, determine the associated leaving variable, if any, and the associated change in z. Do not use the Gauss-Jordan row operations.13) Solve the given LPP using Simplex method a) Max Z=3x1+2x2 s.t, 2x1 + x2 23x1 + 4x2 12x1,x2 0

b) Max Z=2x1+4x2 s.t, x1 + 2x2 5 x1 + x2 4 x1,x2 0

14) JOBCO produces two products on two machines. A unit of product 1 requires 2 hours on machine 1 and 1 hour on machine 2. For product 2, a unit requires 1 hour on machine 1 and 3 hours on machine 2. The revenues per unit of products 1 and 2 are $30 and $20, respectively. The total daily processing time available for each machine is 8 hours. Determine the dual prices and their feasibility ranges ( both machine)

15) Wild West produces two types of cowboy hats. A Type 1 hat requires twice as much labor time as a Type 2. If all the available labor time is dedicated to Type 2 alone, the company can produce a total of 400 Type 2 hats a day. The respective market limits for the two types are 150 and 200 hats per day. The revenue is $8 per Type 1 hat and $5 per Type 2 hat.(a) Use the graphical solution to determine the number of hats of each type that maximizes revenue.(b) Determine the dual price of the production capacity (in terms of the Type 2 hat) and the range for which it is applicable.(c) If the daily demand limit on the Type 1 hat is decreased to 120, use the dual price to determine the corresponding effect on the optimal revenue.(d) What is the dual price of the market share of the Type 2 hat? By how much can the market share be increased while yielding the computed worth per unit?16) Consider the Reddy Mikks Problem and answer the following questions ( Write LPP )a) Determine the range for the ratio of the unit revenue of exterior paint to the unit revenue of interior paint.b) If the revenue per ton of exterior paint remains constant at $5000 per ton, determine the maximum unit revenue of interior paint that will keep the present optimum solution unchanged.c) If for marketing reasons the unit revenue of interior paint must be reduced to $3000, will the current optimum production mix change?17) Consider the following LP:Min Z= 4x1+x2 s.t, 3x1 + x2 = 34x1 + 3x2 6 x1 + 2x2 4x1 , x2 0 The starting solution consists of artificial X4 and Xs for the first and second constraints and slack X6 for the Third constraint. Using M = 100 for the artificial variables, the optimal tableau is given as

Basic XI X2 X3 X4 Xs x6 Solution Z 0 0 0 -98.6 -100 -.2 3.4XI 1 0 0 .4 0 -.2 .4Xl 0 1 0 -.2 0 .6 1.8X3 0 0 1 1 -1 1 1.0

Write the associated dual problem and determine its optimal solution

18) Write the dual of the given LPP Max Z=8x1+10x2+5x3 s.t, x1 - x3 42x1 + 4x2 12x1 + x2 + x3 23x1 + 2x2 - x3 = 8 x1,x2,x3 0

19) Consider the following LP: Max Z= 5x1+ 2x2+ 3x3 s.t, x1 + 5x2 + 2x3 b1 x1 - 5x2 - 6x3 b2 x1,x2,x3 0 The following optimal tableau corresponds to specific values of b1 and b2:Basic Xl X2 X3 X4 Xs SolutionZ 0 a 7 d e 1500Xl 1 b 2 1 0 30X5 0 c -8 -1 1 10

Determine the following:(a) The right-hand-side values, b1 and &b2(b) The optimal dual solution, (c) The elements a, b, C, d, e.20) Consider the following LP: Max Z=2x1+ x2 s.t, 3 x1+x2 - x3 = 34x1 + 3x2 x4 = 6x1 + 2x2 + x5 = 3 x1,x2,x3,x4,x5 0Compute the entire simplex tableau associated with the following basic solution and check its optimality and feasibility.Basic variables = { x1,x2,x5} and Inverse 3/5-1/5 0 -4/5 3/5 0 1-1 121) Consider the following LP: Min Z= 3x1+ 2x2+ x3 s.t, 3x1 + x2 + x3 3 -3x1 +3x2 + x3 6 x1 + x2 + x5 3 x1,x2,x3 0Solve using dual simplex method.

22) TOYCO assembles three types of toys-trains, trucks, and cars-using three operations. The daily limits on the available times for the three operations are 430,460, and 420 minutes, respectively, and the revenues per unit of toy train, truck, and car are $3, $2, and $5, respectively. The assembly times per train at the three operations are 1, 3, and 1 minutes, respectively. The corresponding times per train and per car are (2,0,4) and (1,2,0) minutes (a zero time indicates that the operation is not used).a) Suppose that TOYCO wants to expand its assembly lines by increasing the daily capacity of operations 1,2, and 3 by 40% to 602, 644, and 588 minutes, respectively. How would this change affect the total revenue?b) In the TOYCO model, suppose that the company has a new pricing policy to meet the competition. The unit revenues under the new policy are $2, $3, and $4 for train, truck, and car toys, respectively. How is the optimal solution affected?23) Chemistry lab uses raw material I and II to produce two cleaning solutions A and B. The daily availabilities of raw materials I and II are 150 and 145 units, respectively. One unit of solution A consumes 0.5 unit of raw material I and 0.6 unit of raw material II, and one unit of solution B uses 0.5 unit of raw material I and 0.4 unit of raw material II. The profits per unit of solutions A and B are $8 and $10, respectively. T he daily demand for solution A lies between30 and 150 units, and that for solution B between 40 and 200 units. Find the optimal production amounts of A and B.24) Solve the following LPP using two phase method Max Z= 2x1+5x2 s.t, 3x1+2x2 6 2x1+x2 2x1 ,x2 0

25) Define convex set. Show that the Convex half plane Q= {CX Z } is convex. 26) Show that the set Q= {x1, x2 : x1 1 ,or x2 2} is not convex.

27) Prove that intersection of two convex sets is convex28) Write short notes on the application of operation research in decision making.

29) Define each of the following a) Feasible solution;b) Basic feasible solution;c) Degeneracy;d) Alternate Optima;e) Unbounded solution;f) Convex Set;g) Convex Polyhedron

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