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# Inventory Control & Forecasting HELP! watch

1. A store sells shampoos which are ordered from a manufacturer. Annual demand for the shampoos is 5000. The store incurs an annual holding cost of 20 percent of purchase price and incurs a fixed order placement, transportation and receiving cost of £49 each time an order for the shampoos is placed, regardless of the order quantity. The price charged by the manufacturer varies according to the following discount pricing:

Order quantity Unit Price
0 – 999 £5.00
1000 – 2499 £4.85
2500 and over £4.75

Determine the quantity of shampoos that the supermarket should order each time it makes an order

2. A shoe company is having a spring-summer promotion of the latest model of a men’s shoe. Because the shoe is designed for spring and summer months, it cannot be expected to sell in the autumn. The company can buy the shoes for a reasonable price of £30 per pair and sell them for £40 per pair. The company plans to hold a special August clearance sale in an attempt to sell all shoes not sold by July 31. The sale price in August will be £15 per pair; however, there will be an additional associated cost of £1 per unsold pair of shoes.
As there is no statistical data, the company estimates that they can sell any number between 30 and 50 pairs of shoes, inclusive, with equal probabilities.

How many pairs of shoes should the Company order to maximize their profit?

Hi Guys, I am doing a mathematics module in my business degree and those are the questions I need some help with, each question is worth 30 marks each, can somebody shine some light onto this please?
2. FORMULAE FOR INVENTORY CONTROL

Deterministic inventory control models

Notation
D – number of units demanded per year,
h – cost of holding one unit in inventory for one unit of time,
K – setup or ordering cost,
q – order quantity

Basic EOQ model

Total cost = ordering cost + purchasing cost + holding cost =
=

D/q* orders are placed each unit of time

Quantity discount EOQ model

If q < b1, each item costs p1 pounds
If b1 £ q < b2, each item costs p2 pounds

If bk-2 £ q < bk-1, each item costs pk-1 pounds
If bk-1 £ q < bk = ¥, each item costs pk pounds

Beginning with the lowest price, determine for each price, qi* that minimises TC(q) for bi-1 £ q < bi

Determine qk*, qk-1*,…, until qi* is admissible
qi* = EOQi
The optimal order quantity will be the member of {qk*, qk-1*, …, qi*} with the smallest TC(q)

Probabilistic inventory control models

News vendor problem

q - the number of units to order
D - random variable representing demand
p(d) – probability of demand taking value d
c0 – overstocking cost d £ q
cu – understocking cost d ³ q + 1

F(q) = P(D £ q)

If D is a discrete random variable

If D is a continuous random variable

Notation
K - ordering cost
J – cost of reviewing inventory level
h - holding cost/unit/year
L - lead time for each order (deterministic)
q - order quantity
D - random continuous variable representing annual demand
mean E(D), variance var D, standard deviation σD =
X - random variable representing lead time demand
mean E(X) = L x E(D) var X = L x (var D) σX = x σD
DL+R – random demand during a time interval of length L + R,
mean E(DL+R) = (L+R) x E(D), standard deviation = x σD
If L - random variable representing lead time
mean E(X) = E(L) x E(D) var X = E(L)x(var D) + E(D)2x(var L)
cB - cost incurred for each unit short, if shortages are backlogged
cLS – cost incurred for each lost sale if a shortage results in a lost sale
r - inventory level at which order is placed (reorder point)

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