MSTE-002 Solved Assignment 2021

MSTE-002: Industrial Statistics-II

All questions are compulsory. Answer in your own words.

1. State whether the following statements are true or false and also give the reason in support of your answer.                                                                                        
   1. 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.
   2. 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.
   3. The Set S ={(x, y) : 0 ≤ y ≤ 5 when 0 ≤ x ≤ 2 and 3 ≤ y ≤ 5 when 2 ≤ x ≤ 7 } is not a convex set.
   4. The solution to a transportation problem with 3-rows (supplies) and 3-columns (destinations) is feasible if number of positive allocations is 6.
   5. 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.                                                
2. Solve the following LPP using simplex method:

          Maximize Z = 10 x₁ + x₂ + 2x₃           Subject to the constraints:

              4x₁ + x₂ – 6x₃ = 7

              6x₁+ x₂ – 4x₃ ≤ 5

              3 x₁ – x₂ – x₃ ≤ 0

              x₁, x₂, x₃ ≥ 0                                                                                                                

3. Let X is the advertisement expenditures (in Lakh Rs.) and Y is the sales (in Lakh Rs.). Let the data are

Estimate the parameters and find the estimated linear equation. Whether the advertisement

influences the sale of product? Test and comment on the goodness of fit of the model. 

4. a) The production department for a company requires 3600 kg of raw material for manufacturing a particular item per year. It has been estimated that the cost of placing an order is Rs 36 and the cost of carrying inventory is 25 percent of the investment in the inventories. The price is Rs10 per kg. The purchase manager wishes to determine an ordering policy for raw material. 
5. b) Arrivals at telephone booth are considered to be Poisson with an average time of 10 minutes between are arrival and the next. The length of phone call is assumed to be distributed exponentially with mean 3 minutes.
   1. What is the probability that a person arriving at the booth will have to wait?
   2. The telephone department will install a second booth when convinced that an arrival would expect waiting for at least 3 minutes for phone call. By how much should the flow of arrivals increases in order to justify a second booth? 

• What is the average length of the queen that forms from time to time?

1. What is the probability that it will take him or her more than 10 minutes altogether to wait for the phone and complete his or her call?

5. The annual sales revenue (in lakhs of Rs) of a product as a function of sales force (number of salesmen) and annual adverting expenditure (in Thousands of Rs) for the past 10 years are summarised in following table:

Obtain a regression model to forecast the annual sales revenue of the product using Matrix

        Method.                                                                                                                                    

6. A solicitors’ firm employs typists on hourly price-rate basis for their daily work. There are five typists and their charges and speed are different. According to an earlier understanding only one job is given to one typist and the typist is paid for a full hour even if he works for a fraction of an hour. Find the least cost allocation for the following data:

7. a) Obtain seasonal Indices by the “Moving average” method from the following data:

(b)         For the following Auto regressive model 

                           X_t = 0.7X_t−1 −0.4X_t−2 +a_t

1. i) Verify whether the series is Stationary
2. ii) Obtain ρ_k : k =1,2,3,4 and 5

             Plot the Correlogram.                                                                                                         

8. a) A company has three production factories S₁, S₂ and S₃ with production capacity of 7, 9 and 18 units (in 100 s) per week of a product, respectively. These units are to be shipped to four warehouses D₁, D₂, D₃ and D₄ with requirements of 5, 8, 7 and 14 units (in 100’s) per week, respectively. The transportation costs (in rupees) per unit between factories to ware houses are given in table below.

Obtain the initial basic solution using LC Method and also obtain the optimum solution

              using MODI method.                                                                                                         

1. b) Twenty-five successive observations on a stationary time series are given as follows:

              30, 33, 32, 27, 25, 28, 29, 31, 35, 34, 38, 31, 23, 24, 34, 36, 29, 32, 38, 27, 22, 29, 20, 40,

              Calculate r₁, r₂, ….., r₁₀ and plot the correlogram.