## Description

1. Write a function/program in Matlab/Python to solve n/2 (n jobs, 2 machines) shop floor scheduling [20]

problem using Johnson’s algorithm. An n × 2 matrix in which an element at row ’i’ and column

’j’ depicts the duration of job ‘j’ on machine ‘i’ (see Fig. 1 below as an example) should serve as

an input to your function/program.

Your program should be able to output:

i The optimal sequence of jobs

ii The idle time ‘Xi’ on machine ‘B’ for all jobs

iii Total idle time on machine B

iv Total processing time

Include a well-commented code/script. Test your code with the example shown in Figure 1.

Figure 1: n/2 example problem with n = 5

Figure 2: Project activities

2. Given the project activities with their respective duration (in days) as shown in the Table (Fig. 2) [7]

below, find the critical path and the total duration of the project. Also, find float (or slack) for

each activity, i.e., how much the activity can be delayed without delaying the project. Show all

the necessary steps.

3. During manufacturing of cylindrical liner, samples are collected each hour to measure the critical [8]

dimension, i.e., the inner diameter. The data is shown in the Table below. Each sample consists

of 5 parts chosen at random. Construct a well-labeled x¯ and R control chart to determine if the

process is in-control. Identify the deviations outside the control limit, remove them from the data

and revise your control chart.

Figure 3: Cylindrical liner manufacturing process data

4. A cricket ball manufacturing company is asked to make balls with diameter (D) 2.83 inches. [15]

However, due to variations/imperfections in the manufacturing process, the actual diameter of the

balls made is uniformly distributed over the range of 2.75 inches to 2.90 inches. In other words,

the probability distribution function (PDF) is uniform/rectangular in the range (2.75,2.90).

i Assumming a quadratic loss function with zero loss to company at ball diameter of 2.83

inches, and loss of 10 Rs. for a ball diameter of 2.90 inches; find average loss per ball to the

company.

ii Suppose the loss function is not quadratic. In fact, the balls with diameter between 2.80-2.86

inches are perfectly acceptable to the customer, and can be sold for profit of 100 Rs/ball. If

the ball is oversized (D > 2.86), it can be sold, but at a smaller profit of 10 Rs/ball.

However,

if the ball is undersized (D < 2.80), it needs to be discarded with a loss of 50 Rs/ball. Now,

what is the expected profit (or loss) per ball to the company?