Solve the following LPP using graphical method:

Maximize Z = 3x₁ + 2x₂

Subject to the Constraints:

-2x₁ + x₂ ≤ 1

x₁ ≤ 2

x₁ + x₂ ≤ 3

x1, x₂ ≥0

Variations which occur due to natural forces and operate in a regular and periodic manner over a span of less than or equal to one year are termed as cyclic variations

The solution to a transportation problem with 3-rows (supplies) and 3-columns (destinations) is feasible if number of positive allocations is 6

The Set $ ={(x, y) : 0 ≤ y ≤ 5 when 0 ≤ x≤ 2 and 3≤ y≤ 5 when 2≤ x ≤ 7 } is not a convex set.

If the coefficient of determination is 0.833, the number of observations and independent variables are 12 and 3, respectively, then Adjusted R² will be 0.84.

If the arrival rate is 12 per hour and service rate is 4 per hour, then the probability of no customer in queue is 0.3.

Twelve samples of 4 LED bulbs were selected at regular intervals from a LED bulbs manufacturing company. If bulbs have mean life equal to 2000 hours, it is considered satisfactory. The SD of life of the bulbs is expected to be 520 hours. On testing the samples, the failure times (in hours) were recorded and given below:

i) Prepare control chart for mean when the mean life and SD of the life of the LED bulbs are known and draw the conclusion.

ii) If mean and SD of the life of the LED bulbs are to be unknown, then prepare the control charts for mean and variability. If process is out of control then calculate the revised control limits.

iii) If specification limits as the 2000±SD, then find the process capability. Does it appear that the manufacturing process is capable of meeting the specification requirements?

A system having Weibull failure distribution with pdf as

i) Compute the reliability function of the system,

ii) Find the hazard rate,

iii) Find the MTTF, and

iv) What is the life of the system if reliability of 0.90 is desired?

A system has eight independent components and reliability block diagram of it shown blow.

Components 1, 2 and 3 are not identical and at least two component of this group must be available for system success. The reliability of component 1 to 8 (for a mission 1 year) is given below

R₆ = R₇ = R₈ = 0.80 ,R₁ = 0.60  , R₂ = 0.40, R₃ = 0.50 and R₄ = 0.60. Find reliability of the system.

Two breakfast food manufacturers X and Y are competing for an increase market share. The payoff matrix, shown in the following table, describes the increase in the market share for X and decrease in the market share of Y.

(i) Check whether saddle point exit or not.

(ii) If saddle point does not exit then determine optimal strategies for both the manufacturers and value of the game.

The wave function of a particle in an infinite square well of width 2a (-a to a) is given by,

Determine   x² for the particle.

A person has two independent investments A and B, but he can undertake only one at a time due to certain constraints. He can choose A first and then stop, or if A is successful, then take B or vice-versa. The probability of success of A is 0.6 while for B it is 0.4. Both investments require an initial capital outlay of Rs. 10,000 and both return nothing if the venture is unsuccessful. Successful competition of A will return Rs. 20,000 (over cost) and successful completion of B will return Rs. 24,000 (over cost). Draw an appropriate decision tree and determine the best strategy.

An office supply company ordered a lot of 400 printers. When the lot arrives the company inspector will randomly inspect 12printers. If more than three printers in the sample are nonconforming, the lot will be rejected. If fewer than two printers are non-conforming, the lot will be accepted. Otherwise, a second sample of size 10 will be taken. Suppose the inspector finds two non-conforming printers in the first sample and two in the second sample. Also AQL and LTPD are 0.05 and 0.10 respectively. Let incoming quality be 4%.

i) What type of acceptance plan is used here?

ii) What is the probability of accepting the lot at the first sample?

iii) What is the probability of accepting the lot at the second sample?

A particle of mass m and zero energy has a wave function  , where N is a constant. Determine the potential energy V(x) for the particle.

A leather bag manufacturing company supplies bags in lots of size 200 to a buyer. A sample of 2 begs is drawn and the corresponding lot is accepted if and only if both begs are nondefective. The company and the buyer decide that AQL = 0.04 and LTDD = 0.10. If there are 15 defective bags in each lot, compute the

(i) Probability of accepting the lot,

(ii) Producer’s risk and consumer’s risk,

(iii) AOQ, if the rejected lots are screened and all defective bags are replaced by nondefectives, and

(iv) Average total inspection.

A restaurant produces fresh burgers for its customers every day. The company is known for supplying fresh burgers and never uses burgers prepared on the previous day. Demand for burgers is uncertain, preparation capacity is limited, and the restaurant has the option of producing 0, 100, 200, 300 and 400 burgers every day. It has been estimated that the cost of producing each burgers pack is Rs.15. Each burger is sold for Rs. 20. Prepare a payoff matrix when 0, 100, 200, 300 or 400 demands of the burgers turn up on any given day. Prepare an opportunity loss table and hence find the optimum strategy.

A company manufactures water pumps. The quality control inspector of the company takes a sample of 100 water pumps at regular intervals. The numbers of defective pumps for 15 samples are given below:

Use the data to construct a suitable chart. Observe the results and comment on the control of the process as indicated by the chart

Determine the normalization constant A for the wave function:

(a) Twenty pieces of different length of cloth contained 2, 4, 1, 3, 5, 4, 2, 7, 3, 5, 2, 2, 4, 5, 6, 4, 2, 1, 2, 4 defectsrespectively. To check the process is under control with respect to the number of defects, we should use p-chart.

(b) If density function of the time to failure of an appliance is   

then reliability of the appliance for two years will be 0.444.

(c) If the probabilities are associated with the occurrence of different states of nature, then the situation is known as decision making under uncertainty.

(d) If there are 1% defectives notes of Rs.500 in a lot of 10000 notes of Rs 500, then the lot quality will be 99%.

(e) A system has four components connected in parallel configuration with reliability 0.2, 0.5, 0.4, 0.6. To improve the reliability of the system, we have to replace the weakest component to the more reliable component.

Show that the wavefunction ψ(x) = N sin kx + iN cos kx is an eigen function of the momentum operator and determine its eigen value.