## Advanced Math

Advanced Math

1) Each week Brian visits the fish market in Seattle and procures quantities of a variety of fish that he will take to sell in Yakima. He has capacity to store only 415 pounds of fish in his truck. Suppose he concentrates on two types of fish that appear to be the best choices today. Details of these two types are

tabulated below:

Haddock Salmon

Cost per pound $4.00 $4.50

Selling price per pound $10 $12

Expected demand, in pounds 300 100

Standard deviation of demand, in pounds 30 20

How much of each type of fish should be purchased to maximize expected profit?

Solve this problem using a Lagrangian method. The standard normal cumulative and complementary cumulative distribution is tabulated as an attachment for a range of useful values. For each fish type, you may express your solution as ?=

?? + ? where ? is the number of pounds of that type of fish to buy, ? is one of the listed values in the standard normal table and ? and ? are the mean and standard deviation, respectively, of fish demand of that type. That is, you do not need to

interpolate between values in the table.

2) A certain type of equipment has two parts subject to random failures. The two

parts have lead time demand distributions with low means and low coefficients

of variation (CoV):

Part 1 Part 2

Mean 4.95 2.52

CoV 0.3 0.5

We choose to approximate these distributions with binomial distributions with the

following parameters:

Part 1 Part 2

n 9 7

p 0.55 0.36

Binomial probabilities can be computed recursively using:

?(? – 1? = (?

? ? ???? = ??

1 – ?? ??-? + 1

? ?

for ? = 1,2,…,?. The attached spreadsheet printout calculates the following

quantities for each part:

Column Heading Formula

I ? = 0,1,…,?.

p(i) ??

P(i) ???

?

???

1-P(i) ? ??

?

?????

B(i) ? (? – ?)??

?

?????

= ?(1 – ?(?))

?

???

B(i)-B(i-1) ? (? – ?)??

?

?????

??(1-?(? – ? +

?

???

Other parameters which may (or may not) be relevant are:

Part 1 Part 2 System

Annual demand rate 50 25

Cost per unit 10 20

Number of pieces of

equipment

100

Budget 1

The system follows an (? – 1,?-(type continuous review ordering policy for each part. We are tasked with determining appropriate target stock levels, ?? and ??, for the two parts. Consider the following optimization problem:

(a) Describe an algorithm to solve this problem efficiently (an algorithm which explicitly considers all possible combinations of stocking levels for the two parts is not efficient).

(b) Find the optimal combination of stock levels for the two parts,

(c) Report the optimal objective value, and

(d) Verify that the constraint is satisfied.

3) A certain item has an annual demand rate of 1200 units. Assume 240 days per year. The coefficient of variation of daily demand is 1.5. The lead time is 15 days.

Demand over the resupply lead time can be approximated by a Laplace distribution. The fixed cost of placing an order is $400. The holding cost per unit per year is $20. The per incident backorder cost is $12 and the backorder cost per

backorder per year is $200.

(a) Find a system of equations (two equations in two unknowns: ? and ? ;or three equations in three unknowns: ?, ? ,and ?) that will permit you to find the optimal combination of ? and ? for this item.

(b) If possible, find a closed form solution for the optimal value of ?.

(c) For the data provided, find optimal values of ? and ? .You do not need to report an integer solution.

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