IT代写 TCU00(y) = 2DK > 0 y3

Math269: Operational Research
Active Learning Session 1 (Inventory control)
Dr Jessica Banks

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Attendance

Inventory control – traditional model
Shelves are re-stocked from the warehouse as space becomes available

Inventory control – just in time model Advantages:
I fresher produce
I wider variety
I lower storage costs
Disadvantages:
I vulnerable to variation in supply and demand
I little space available to stockpile goods

The chicken crossed the road, just not to our restaurants. . .
We’re sorry.
A chicken restaurant without any chicken. It’s not ideal. Huge apologies to our customers, especially those who travelled out of their way to find we were closed.
In February 2018, distribution problems meant KFC had to close the majority of their UK stores until they were able to deliver more supplies.

Inventory Control Policy Questions:
I How much should I order? I How often?
0@ Total 1A ✓Purchasing◆ ✓Setup◆ ✓Holding◆ ✓Shortage◆ Inventory = Cost + Cost + Cost + Cost
Holding cost:
How much it costs to be in possession of an item (usually storage costs)
Shortage cost:
Losses as a result of not having an item available when a customer wants it (e.g. immediate loss of revenue, loss of goodwill)

Question 1a
Give examples of items with di↵erent retail properties: (they do not all need to be di↵erent)
I with a high holding cost;
I with a low holding cost;
I with a high shortage cost;
I with a low shortage cost;
I that you would prefer to hold;
I that you would prefer to short;
I that you would prefer to backlog;
I that you would prefer not to backlog.
Ideas: ice cream; toilet paper; sand; chewing gum; magazine subscriptions; wedding rings; bread; sofas; watch batteries; school ties; original paintings

high holding cost: low holding cost: high shortage cost: low shortage cost: prefer to hold:
prefer to short:
prefer to backlog: prefer not to backlog:
Backlogging: If an item is not in
you when it is in stock.
No backlogging: If an item is not in stock, the customer doesn’t buy the item, or buys from a competitor.
toilet paper, wedding rings chewing gum, watch batteries bread
wedding rings
chewing gum sofas
bread, toilet paper
stock, the customer gets it from

Question 1b
Give examples of items with di↵erent retail properties: (they do not all need to be di↵erent)
I with constant demand;
I with seasonal demand;
I with deterministic demand;
I with probabilistic demand;
I where a fractional size order makes sense;
I where any integer size order makes sense (but not a fractional one);
I where you might expect to have to order in certain multiples. Does it make sense for the demand D to not be an integer?

constant demand: seasonal demand: deterministic demand: probabilistic demand: fractional order size: integer order size:
fixed multiples order size:
toilet paper, watch batteries school ties, ice cream magazine subscriptions sofas
original paintings toilet paper
Can D be fractional?
If 2310 shopping bags per year are required, what is the demand per month?
Would it make sense to talk about the average demand for the , the Millennium Dome, or the Three Gorges Dam?

Single-item Static Continuous Review Model
0 Total 1 ✓Setup◆ ✓Holding◆ @InventoryA = Cost + Cost
D: constant demand, in units per unit time y: replenishment amount (i.e. order size) K: set-up cost
h: holding cost, per unit per unit time

Question 2a
I Explain, in your own words, why the total cost per unit time in the single-item static continuous review model is given by
TCU(y) = DK + hy . y2
I Calculate the minimum point of this function of y, and verify that it is a minimum.
I Assume that, for a particular company, D = 11, K = 5 and
h = 0.07. What is the integer value of y that gives the lowest total cost per unit time?

Inventory Level
Average inventory level: y/2 Average holding cost: hy/2
Holding cost per cycle:
Total cost per unit time:
K + hy2 2D
= DK + hy y2

Find the minimum:
TCU(y) = DK + hy y2
TCU0(y) = DK + h y2 2
DK +h=0 ) y?=r2DK (y?)2 2 h
Check it is a minimum:
TCU00(y) = 2DK > 0 y3

D = 11, K = 5, h = 0.07
)y? =r2DK =r2⇥11⇥5 ⇡39.64
h 0.07 Check integer values either side:
TCU(39) = 11 ⇥ 5 + 0.07 ⇥ 39 ⇡p2.7753 39 2
⇤ 11 ⇥ 5 0.07 ⇥ 11000/7 TCU(y ) = p11000/7 + 2
TCU(40) = 11 ⇥ 5 + 0.07 ⇥ 40 = 2.775 40 2

Sensitivity analysis
I I know it’s unlikely the model and reality match perfectly. I How much does it matter?
i.e. How much will inaccuracies change the conclusion?
Practical applications:
I Would a di↵erent choice be safer?
I Do I need to spend more money on market analysis?

Question 2b
I If the company can only place integer-sized orders, and wants to keep the total cost per unit time within 1% of the optimal value, what is the range of possible order sizes they can make?
I Suppose that the company wants to keep the total cost per unit time within 10% of the optimal value. For what integer values of the demand will using the estimate D = 11 give a suitable outcome?

I want costs within 1% of optimal value TCU(y)  1.01
I can only order integer amounts
1.01=1✓y +y?◆=1✓ y +39.64◆
TCU(y?) 2 y? y 2 39.64 y
y2 80.07y + 1571.33 = 0 ) y ⇡ 34.43, 45.64
For integer size orders keeping costs within 1% of optimal, we need y 2 [35, 45] ✓ [34.43, 45.64].
Extension: why don’t we use 0.99 in place of 1.01?

I want costs within 10% of optimal value TCU(y)  1.1
I estimated demand is D = 11
TCU(y?) 2 11 D
TCU(y?) 1.1=TCU(y)=1 rD+r11!
d = r D ) d2 2.2d + 1 = 0 11
d ⇡ 0.642, 1.558
D = 11d2 ⇡ 4.53,26.7
This estimate is suitable for integer values of demand D 2 [5, 26] ✓ [4.53, 26.7].

Single-item Static Continuous Review Model with Planned Shortages
0 Total 1 ✓Setup◆ ✓Holding◆ ✓Shortage◆ @InventoryA = Cost + Cost + Cost
D: constant demand, in units per unit time y: replenishment amount (i.e. order size) K: set-up cost
h: holding cost, per unit per unit time
w: (maximum) shortage amount
p: shortage cost, per unit per unit time

Inventory Level
costpercycle =K+
K + h(yw)2 + pw2
h(y w)2 2D
TCU = 2D 2D y/D
DK h(y w)2 pw2 =y+ 2y +2y
(y w) Dt

Optimal values:
y? =r2DKsp+h hp
w ? = sr 2 D K sr h
? ? 2DK p yw=rhsp+h
2K p+h Dh p
t? = y⇤/D =

Question 3a
A fast-food restaurant chain specialising in one particular type of food has faced significant shortages.
You are hired as an Operational Research consultant to improve the inventory control model.
Before o↵ering advice, what questions would you ask the company? And why?
Sample questions:
I What are your assumed costs? What is the evidence for these?
I What is your current inventory control policy?
I What is your current inventory control model, and what models have you tried in the past?
I What factors might be influencing your shortages? Have there been any changes in practice?

Question 3b
A high budget Hollywood film is due to be released on Blu-Ray next month.
A particular retail chain is planning to use a single-item static demand continuous review model with planned shortages.
Estimate the relative holding and shortage costs.
Comment on whether this model is appropriate, and what considerations may need to be taken into account.

Holding and shortage costs:
I Blu-Ray disks are small but high-value. Storage space will be cheap, but security will add expense (e.g. sta↵ to hold disks behind a counter, security tags). Consideration also needs to be taken of price reductions as the film gets older.
I As Blu-Ray disks are widely available, customers are likely to shop elsewhere if you have a shortage, resulting in loss of goodwill. New films typically have a high profit margin, so this could significantly increase the shortage costs.
Suitability of model:
I Demand is assumed to be static but is more likely to be seasonal, as there is often greater immediate demand for a new film.
I Similarly, shortage costs may change as the film gets older. I ’Single-item’ may not be suitable for a large retail chain.

Question 3c
The ‘Eggceptional Chocolate Company’ stocks high quality Easter Eggs all year round, using a single-item static demand continuous review model with planned shortages as its inventory model.
Discuss the assumptions of this model in this context and suggest alternatives.
Discuss the nature of the holding and shortage costs.

Assumptions:
I Demand is more likely to be seasonal than static, as people are more likely to eat Easter eggs at Easter.
I Backlogged shortages probably make sense if this is a particularly high-quality item. However, it is unlikely that someone would pre-order an Easter egg to arrive after Easter.
Holding and shortage costs:
I Holding costs probably include salvage costs (discounts and waste), since the item is perishable.
I Shortage costs are likely to be standard (immediate loss of revenue, loss of goodwill, cost of running a backlogging system); but shortage costs might be greater at Easter.

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