Bazaraa Mokhtar S. Sherali Hanif D. Shetty C. M. - Solutions manual to accompany Nonlinear programming: theory and algorithms

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ISBN 978-1-118-76237-0

1.1 In the figure below, xmin and xmax denote optimal solutions for Part (a) and Part (b), respectively.

1.2 a. The total cost per time unit (day) is to be minimized given the storage limitations, which yields the following model:

Note that the last two terms in the objective function are constant and thus can be ignored while solving this problem.

b. Let Sj denote the lost sales (in each cycle) of product j, j = 1, 2. In this case, we replace the objective function in Part (a) with F(Q1, Q2, S1, S2), where F(Q1, Q2, S1, S2) = F1(Q1, S1) + F2(Q2, S2), and where

This follows since the cycle time is , and so over some T days, the number of cycles is . Moreover, for each cycle, the fixed setup cost is kj, the variable production cost is cjQj, the lost sales cost is ljSj, the profit (negative cost) is PQj, and the inventory carrying cost is . This yields the above total cost function on a daily basis.

1.4

Notation:	xj : production in period j, j = 1,,n
dj : demand in period j, j = 1,,n
Ij : inventory at the end of period j, j = 0, 1,,n.

The production scheduling problem is to:

1.6 Let X denote the set of feasible portfolios. The task is to find an x* X such that there does not exist an for which and with at least one inequality strict. One way to find is to solve:

Maximize

for different values of (1, 2) > 0 such that 1 + 2 = 1.

1.10 Let x and p denote the demand and production levels, respectively, and let Z denote a standard normal random variable. Then we need p to be such that P(p < x 5) 0.01, which by the continuity of the normal random variable is equivalent to P(xp + 5) 0.01. Therefore, p must satisfy

where Z is a standard normal random variable. From tables of the standard normal distribution we have P(Z 2.3267) = 0.01. Thus, we want or that the chance constraint is equivalent to p 161.2869.

1.13 We need to find a positive number K that minimizes the expected total cost. The expected total cost is