system of control

Consider an (s, Q) system of control for a single item and with normally distributed forecast error. The management
wish to have an expected number of stockout occasions per year equal to N. The following model represents such
control system where the cost function is minimized subject to the constraint given on the number of stockout occasions
per year.

Q AD
TC(k , Q) k L vr
2 Q
subject to
D
pu (k ) N
Q

The two simultaneous equations that Q and K must satisfy if the management wishes to minimize the total expected
costs of replenishment and carrying inventory subject to the expected number of stockout occasions per year being N are

Q
pu (k ) N
D
1
2 AD 2N L
Q 1
vr Dfu (k )

The characteristics of the item under consideration are:

A= $5 r= 0.16/yr v= $2/unit D= 1000 units/year L= 80 units N= 0.5/year

Using Microsoft Excel find the followings

a) Simultaneous best (Q, K) pairs and the associated total cost for this policy.
b) If the policy is changed to be B1-costing at B1 = $100, what are the best (Q, k) pairs and the associated total
cost?

Note that,

1- fu(k) and pu (k) can be found using the Excel function NORM.DIST
2- The inverse cumulative normal distribution function that finds k can be found using the Excel function
NORM.S.INV.
3- Stop iterating at two decimal numbers accurate results

The Excel sheet should be submitted as a hard copy with your solution of this assignment, and a soft copy of the Excel
sheet should be submitted electronically using Electronic Assignment Submission https://fis.encs.concordia.ca/eas/
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Problem 2

Consider a family of four items with the following characteristics

A= $100, r= 0.2/ year

Item i DiVi ($/year) ai ($)
1 100,000 5
2 20,000 5
3 1,000 21.5
4 300 5
a) Using the iterative algorithm given in the class find appropriate values of the cycle time T1 and the integer
multiplier associated with each item i.
b) Suppose that we restricted attention to the case where every item is included in each replenishment of the
family. What does this says about the multipliers? Find the best value of T1.
c) Find the cost difference in the answers to part a and b.

Problem 3

The Steady-Milver Corporation produces ball bearings. It has a family of three items, which run consecutively, do not
take much time for changeovers. The characteristics of the items are as follows:

Item i ID Di Raw Value Value after ai ($)
(units/year) material added production
($/unit) ($/unit) vi ($/unit)
1 BB1 2000 2.50 0.50 3.00 5
2 BB2 1000 2.50 0.50 3.00 2
3 BB3 500 1.60 0.40 2.00 1
The initial setup cost for the family is $30. Management has agreed on an r value of 0.10$/$/yr. Production rates are
substantially larger than demand rates.

a) What are the preferred production quantities of the three items?
b) Raw material for product BB1 is acquired from a supplier distinct from that for the other two products. Suppose
that the BB1 supplier offers an 8 percent discount on all units if an order of 700 or more is placed. Should
Steady-Milver take the discount offer?
N N
A a1 ai / ni T1 ( 1 n)
i i N
i 2 i 2
Note that the total cost is C Di vi
T1 2 i 2

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Problem 4

Consider five products produced in the same machine. Various data are given in the table.

Product 1 2 3 4 5

Demand per week 900 1000 1000 500 1500

Production rate per week 13,000 15000 11000 9000 18,000

Setup time in weeks 0.0071 0.0214 0.0142 0.0142 0.0071

Ordering costs 25 25 5 10 50

Holding cost per unit and week 2 1 5 3 4

a) Check whether the independent solution is feasible or not. Find the total cost, is it a lower bound or the
optimal cost?
b) Use a cyclic schedule with a common cycle. Determine batch quantities and the total cost.
c) Assuming a power of two multipliers policy, use Doll and Whybark iterative algorithm to find a
feasible convergent solution. How far is this solution from the cost obtained in part a?