Chapter 1



ISyE 3104: Introduction to Supply Chain Modeling:

Manufacturing and Warehousing

Instructor : Spyros Reveliotis

Summer 2006

Solutions for Homework #2

Chapter 4

1. The questions that inventory control addresses are when to order and how much to

order.

2. Too much:

a) Money that is tied up in inventories could be invested elsewhere.

b) Costs of supporting inventory could be high, including cost of storage, taxes, insurances, pilferage, etc.

c) Production/distribution inefficiencies could arise.

d) Stock can become obsolete.

Too little:

a) Insufficient stock to meet customer demand, i.e., opportunity cost of lost sales

b) Loss of customer goodwill

c) Expediting costs for filling potential backorders

d) Production disruptions in case of raw material or work-in-process (WIP) inventory; for example, in a sequential production process, a lack of sufficient buffer inventory could cause production to come to a halt.

5. Total cost in 5 years = 12,500 + 5(2,000) = $22,500. Let s = yearly savings as fraction of sales. The s solves (80,000)(s)(5) = $22,500

s = 0.05625

10. a. The optimal order quantity

[pic]

b. The time between placement of orders

[pic]months

c. The average annual cost of holding and setup

[pic]

d. The reorder level r = λγ = 280(3/52) = 16.15

r = 16 units

12. λ = 1250(0.18) = 225

c = $18.50

I = 0.25

h = Ic =0.25 (18.50) = 4.625

K = 28

a. The optimal order quantity [pic]

The time between placement of orders [pic] weeks.

b. The reorder point r = λγ = 255 (6/52) = 25.96 ≈ 26 units

c. If Q = 225, the average inventory level is Q/2 = 225/2 = 112.5. The annual holding cost is (112.5)(4.625) = $520.31. At the optimal solution, the annual holding cost is (52/2)(4.625) = $120.25. The excess holding cost is $400.06 annually.

The annual holding and set-up cost incurred by this policy is $520.31 + 28 = $548.31 since there is only one set-up annually.

The average annual holding and set-up cost at the optimal policy is

[pic]

Therefore, the annual difference = $306.91.

13. If the set up cost is $15, the true optimal [pic]

The actual annual holding plus setup cost, when ordering at Q=229, is equal to [pic]

Hence, the resulting error is equal to 109.98-73.3=36.5$.

17. a. Notice that an annual demand of 0.6 million pounds with 250 working days per year is equivalent to 2400 pounds per day. Hence,

[pic]lbs

b. Up time = Q/P = 44,612 / 10,000 = 4.46 days

Cycle time = Q/λ = 44,612 / 2,400 = 18.59 days

Proportion of uptime in a cycle = 4.46/18.59 = 0.24

Proportion of downtime in a cycle = 1-0.24 = 0.76

c. Annual holding and set-up cost

[pic]

If the compound sells for $3.9 per lb, then the annual profit is

(3.9-3.5)(600,000) – 40347.5 = $199,652.5

22. λ = 20,000

K = 100

I = 0.20

c0= $2.50

c1= $2.40

c2= $2.30

[pic]

[pic]

[pic]

Only Q(0) is realizable.

Hence, the minimum achievable cost for source C is at Q = 4000 and it is equal to:

[pic]

The minimum achievable cost for source B is at Q = 3000 and it is equal to:

[pic]

Finally, the minimum achievable cost for source A is at Q = Q(0) = 2828 and it is equal to:

[pic]

It follows that the optimal order size is Q = 4,000.

b. Holding cost and set up cost = [pic]

c. T = Q/λ = 4000/20000 = 0.2 years = 2.4 months. Hence γ >T. This implies that we need to place the replenishment order more than one cycle ahead from the cycle in which it will be consumed. More specifically, setting the reorder point at r = λ (γ−Τ) = 20,000 [(3-2.4)/12] = 1,000, we shall receiving replenishment orders according to the pattern indicated below:

24. Since Q(2) is realizable, it must be optimal. This can be argued as follows: Every other point on the “total annual cost (TAC)” curve corresponding to price c2 will be above to the point of this curve corresponding to Q(2) (since this is the corresponding EOQ value). Furthermore, for any given Q, the corresponding points of the TAC curves obtained for prices c1 and c0 will be above the point of the TAC curve obtained for price c2 (we showed that in class). Hence, it is not possible to have a better total annual cost than that obtained with Q(2).

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