Exercise Sheet 1
Q1. A business sells storage units at a constant rate of 7 per day. It costs £38 to place an order, and 40p per day to store each unit. It takes two days for a shipment to arrive after ordering.
(a) First assume no shortages are allowed.
(i) What are the parameters D, K and h for the Single-Item Static Continuous Review Model, in pounds per day?
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(ii) Calculate the optimal values y⋆ and t⋆.
(iii) What are the lead time and the reorder point?
(b) Now assume that shortages are allowed.
(i) Calculate the optimal values y⋆, w⋆, y⋆ − w⋆ and t⋆ when p takes the values 0.01, 0.1, 0.4, 1, 10, 100.
(ii) How does the optimal strategy change as the ratio between h and p varies? Does this fit with what you would expect to happen?
(iii) What happens as p → ∞? What happens as p → 0?
Q2. A 24-hour fruit shop buys individual apples for 15p each, and sells them for 35p each. The shop sells 50 apples a day. Their delivery has a fixed cost of £20, independently of how many apples they order, a delivery takes 10 days to arrive, and the storage space costs 1p per day for each apple. Assume that no shortages are permitted.
(i) Determine the optimal number of apples to order, and the reorder point (i.e. the level that will trigger a reorder).
(ii) If the shop may only order apples in multiples of 10, how many apples should be ordered to minimise the increased cost relative to the cost associated with the economic order quantity.
(iii) If the shop may only order apples in multiples of 10, state an interval such that if the number of apples ordered is contained in this interval, the increased cost relative to the cost associated with the economic order quantity is at most 5%.
(iv) The shop implemented the inventory policy derived in part (ii) and, after a week, realised that the policy was not suitable. Why is this model not suitable for this problem?
Q3. A large frozen food shop sells popular frozen ready meals at a rate of 72 per day. Each delivery costs £64. It costs 16p per day to hold each ready meal, but only 9p per day for each ready meal that the shop is short. Assume that shortages are backlogged.
(i) State the economic order quantity, the corresponding optimal maximal shortage, and the cycle length.
(ii) Are any of our assumptions unrealistic?
Q4. Using the EOQ model, assume that the demand in a particular situation has been estimated as D′ = 100. Find the range of integer values D such that if D is the real demand then modelling using the estimate D′ increases costs by at most 10%. Is it safer to under-estimate or over-estimate the demand?
Q1. (a) (i) (ii)
(iii) (b) (i)
WehaveD=7,K=38,h=0.4. y⋆ = 2×7×38 ≈ 36.469
t⋆ = 2×38 ≈ 5.210
The lead time is two days. Since 14 units will be sold within two days, this is the reorder point.
p y⋆ w⋆ y⋆−w⋆ t⋆ 0.01 233.517 228.821 5.700 33.360
0.1 81.548 65.238 0.4 51.575 25.788 1 43.151 12.329 10 37.191 1.430
100 36.542 0.146
16.310 25.788 30.822 35.761 36.396
11.650 7.368 6.164 5.313 5.220
When the shortage cost is very low relative to the holding cost, the preferred strategy is to build up a big backlog of orders, then get a lot of stock delivered that can quickly be passed on to the customers. On the other hand, if shortages are much more expensive than holding stock, the preferred strategy is to order more frequently to avoid running out of stock too quickly. When p = h we find that the two sides are balanced.
Hint: How long will a customer wait from the point they order an item to when they receive it?
that the purchase and sale cost of the apples is irrelevant as without quantity discounts this (i) We have,
Q2. Note first
does not affect the inventory cost.
2DK 2×50×20 h = 0.01
⋆ y⋆ 447.2 t=D= 50 =8.9.
Since the lead time 10 is larger than the cycle, we should reorder during the previous cycle. I.e. the effective lead time is L = 10 − 8.9 = 1.1. Hence the reorder point occurs when the inventory level is LD = 1.1 × 50 = 55.
(ii) We use the formula,
to compute
TCU(y) 1 y y⋆
TCU(y⋆)=2 y⋆ + y (1)
TCU(440) = 1.00013, TCU(y⋆)
TCU(450) = 1.00001. TCU(y⋆)
So it makes very little difference, but an order of 450 apples makes for a marginally cheaper inventory cost.
(iii) We set Equation 1 to 1.05 and solve, to get
y = 326.4, 612.8.
Hence, remembering that we may only order multiples of 10, we should order an amount in the interval [330, 610].
(iv) Hint: What is the longest time between when an apple is delivered and when it is sold? (i) These are,
2DKp+h 2×72×640.09+0.16
y⋆ = h p = 0.16 0.09 = 400,
2DK h 2×72×64 0.16
w⋆ = p p+h = 0.09 0.09+0.16 =256,
t = D = 72 = 5.6.
(ii) Hint: How far in advance do you order a ready meal? We set
1 D 100
2 100 + D = 1.1.
This gives us the quadratic equation
We know that D ≥ 0. Write d =
D, so the equation becomes d + 10 = 2.2.
which has solutions d = 11 ±
21. From this we find
D ≈ 242.8 and D ≈ 41.2.
d2 +100−22d=0,
Therefore the integer values of D that mean costs are increased by at most 10% are D ∈ [42, 242].
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