PLEASE ANSWER ALL EXCERCISES 1-5

The model uses the following variables: In this application, we set up a mathematical model for determin- ing the total cost required to set up a training program. Then we use calculus to find the time interval between training programs that produces the minimum total cost. The model assumes that the demand for trainees is constant and that the fixed cost of training a batch of trainees is known. Also, it is assumed that people who are trained, but for whom no job is readily available, will be paid a fixed amount per month while waiting for a job to open up. D = demand for trainees per month; N = number of trainees per batch; C = fixed cost of training a batch of trainees; C2 = variable cost of training per trainee per month; Cz = salary paid monthly to a trainee who has not yet been given a job after training; m = time interval in months between successive batches of trainees; The total cost per batch is the sum of the training cost per batch, G + mDtC2, and the cost of keeping trainees without a proper job, given by equation (1). Since we assume that a batch of trainees is trained every m months, the total cost per month, Z(m), is given by mlm - 1) DCZ C + mDtC 2 Z(m) = + t = length of training program in months; Z(m) = total monthly cost of program. The total cost of training a batch of trainees is given by G + Nt2. However, N = mD, so that the total cost per batch is C + mDtC2. After training, personnel are given jobs at the rate of D per month. Thus, N D of the trainees will not get a job the first month, N 2D will not get a job the second month, and so on. The N D trainees who do not get a job the first month produce total costs of (N DC3, those not getting jobs during the second month produce costs of (N 2D)C3, and so on. Since N = mD, the costs during the first month can be written as (N DC3 = (mD DC; = (m 1)DC3, m m C = + DtC2 + DC; DC(";)) m Exercises 1. Find Z'(m). while the costs during the second month are (m 2)DC3, and so on. The total cost for keeping the trainees without a job is thus (m 1)DC3 + (m 2)DC3 + (m 3)DC3 + ... + 2DC; + DC3, 2. Solve the equation Z'(m) = 0. As a practical matter, it is usually required that m be a positive integer. If m does not come out to be an integer, then m and m", the two integers closest to m, must be chosen. Calculate both Z(mt) and Z(m)); the smaller of the two provides the optimum value of Z. 3. Suppose a company finds that its demand for trainees is three per month, that a training program requires 12 months, that the fixed cost of training a batch of trainees is $15,000, that the mar- ginal cost per trainee per month is $100, and that trainees are paid $900 per month after training, but before going to work. Use your result from Exercise 2 to find m. which can be factored to give DC3[(m 1) + (m 2) + (m 3) + ... + 2 + 1]. The expression in brackets is the sum of the terms of an arithmetic sequence discussed in most algebra texts. Using formulas for arith- metic sequences, we can show that the expression in brackets is equal to m(m 1)/2, so we have 4. Since m is not an integer, find m+ and m . 5. Calculate Z(mt) and Z(m ). mm 1) DC3 2 (1) as the total cost for keeping jobless trainees. 6. What is the optimum time interval between successive batches of trainees? How many trainees should be in a batch? On pages 733 and 734 in you textbook, read through the derivation and complete the corresponding exercises for Case Study 12 A Total Cost Model for a Training Program. Note: The arithmetic sequence used in the derivation is a way of calculating the sum of the first m integers: m(m-1) 1 + 2 + 3 + + (m 1) + m 2 So, for example, the sum of the first 79 integers is 1+ 2+ 3+ + 77 + 78 + 79 79(791) 2 3081 Another example, the sum of the first 1086 integers is 1+2+3+. ...+ 1084 + 1085 + 1086 = 1086(1085) 2 589155 The model uses the following variables: In this application, we set up a mathematical model for determin- ing the total cost required to set up a training program. Then we use calculus to find the time interval between training programs that produces the minimum total cost. The model assumes that the demand for trainees is constant and that the fixed cost of training a batch of trainees is known. Also, it is assumed that people who are trained, but for whom no job is readily available, will be paid a fixed amount per month while waiting for a job to open up. D = demand for trainees per month; N = number of trainees per batch; C = fixed cost of training a batch of trainees; C2 = variable cost of training per trainee per month; Cz = salary paid monthly to a trainee who has not yet been given a job after training; m = time interval in months between successive batches of trainees; The total cost per batch is the sum of the training cost per batch, G + mDtC2, and the cost of keeping trainees without a proper job, given by equation (1). Since we assume that a batch of trainees is trained every m months, the total cost per month, Z(m), is given by mlm - 1) DCZ C + mDtC 2 Z(m) = + t = length of training program in months; Z(m) = total monthly cost of program. The total cost of training a batch of trainees is given by G + Nt2. However, N = mD, so that the total cost per batch is C + mDtC2. After training, personnel are given jobs at the rate of D per month. Thus, N D of the trainees will not get a job the first month, N 2D will not get a job the second month, and so on. The N D trainees who do not get a job the first month produce total costs of (N DC3, those not getting jobs during the second month produce costs of (N 2D)C3, and so on. Since N = mD, the costs during the first month can be written as (N DC3 = (mD DC; = (m 1)DC3, m m C = + DtC2 + DC; DC(";)) m Exercises 1. Find Z'(m). while the costs during the second month are (m 2)DC3, and so on. The total cost for keeping the trainees without a job is thus (m 1)DC3 + (m 2)DC3 + (m 3)DC3 + ... + 2DC; + DC3, 2. Solve the equation Z'(m) = 0. As a practical matter, it is usually required that m be a positive integer. If m does not come out to be an integer, then m and m", the two integers closest to m, must be chosen. Calculate both Z(mt) and Z(m)); the smaller of the two provides the optimum value of Z. 3. Suppose a company finds that its demand for trainees is three per month, that a training program requires 12 months, that the fixed cost of training a batch of trainees is $15,000, that the mar- ginal cost per trainee per month is $100, and that trainees are paid $900 per month after training, but before going to work. Use your result from Exercise 2 to find m. which can be factored to give DC3[(m 1) + (m 2) + (m 3) + ... + 2 + 1]. The expression in brackets is the sum of the terms of an arithmetic sequence discussed in most algebra texts. Using formulas for arith- metic sequences, we can show that the expression in brackets is equal to m(m 1)/2, so we have 4. Since m is not an integer, find m+ and m . 5. Calculate Z(mt) and Z(m ). mm 1) DC3 2 (1) as the total cost for keeping jobless trainees. 6. What is the optimum time interval between successive batches of trainees? How many trainees should be in a batch? On pages 733 and 734 in you textbook, read through the derivation and complete the corresponding exercises for Case Study 12 A Total Cost Model for a Training Program. Note: The arithmetic sequence used in the derivation is a way of calculating the sum of the first m integers: m(m-1) 1 + 2 + 3 + + (m 1) + m 2 So, for example, the sum of the first 79 integers is 1+ 2+ 3+ + 77 + 78 + 79 79(791) 2 3081 Another example, the sum of the first 1086 integers is 1+2+3+. ...+ 1084 + 1085 + 1086 = 1086(1085) 2 589155