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Operations Management

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**1. **Operations Management Session 23: Newsvendor Model

**2. **Uncertain Demand Uncertain Demand: What are the relevant trade-offs?
Overstock
Demand is lower than the available inventory
Inventory holding cost
Understock
Shortage- Demand is higher than the available inventory
Why do we have shortages?
What is the effect of shortages?

**3. **The Magnitude of Shortages (Out of Stock)

**4. **What are the Reasons?

**5. **Consumer Reaction

**6. **What can be done to minimize shortages? Better forecast
Produce to order and not to stock
Is it always feasible?
Have large inventory levels
Order the right quantity
What do we mean by the right quantity?

**7. **Uncertain Demand What is the objective?
Minimize the expected cost (Maximize the expected profits).
What are the decision variables?
The optimal purchasing quantity, or the optimal inventory level.

**8. **Optimal Service Level: The Newsvendor Problem

**9. **News Vendor Model Assumptions
Demand is random
Distribution of demand is known
No initial inventory
Set-up cost is equal to zero
Single period
Zero lead time
Linear costs:
Purchasing (production)
Salvage value
Revenue
Goodwill

**10. **Optimal Service Level: The Newsvendor Problem

**11. **Optimal Service Level: The Newsvendor Problem

**12. **Optimal Service Level: The Newsvendor Problem

**13. **Compute the Average Demand

**16. **Cumulative Probabilities

**17. **Number of Units Sold, Salvages

**18. **Total Revenue for Different Ordering Policies

**19. **Example 2: Denim Wholesaler The demand for denim is:
1000 with probability 0.1
2000 with probability 0.15
3000 with probability 0.15
4000 with probability 0.2
5000 with probability 0.15
6000 with probability 0.15
7000 with probability 0.1

**20. **Example 2: Marginal Analysis Marginal analysis: What is the value of an additional unit?
Suppose the wholesaler purchases 1000 units
What is the value of the 1001st unit?

**21. **Example 2: Marginal Analysis Wholesaler purchases an additional unit
Case 1: Demand is smaller than 1001 (Probability 0.1)
The retailer must salvage the additional unit and losses $5 (10 – 5)
Case 2: Demand is larger than 1001 (Probability 0.9)
The retailer makes and extra profit of $20 (30 – 10)
Expected value = -(0.1*5) + (0.9*20) = 17.5

**22. **Example 2: Marginal Analysis What does it mean that the marginal value is positive?
By purchasing an additional unit, the expected profit increases by $17.5
The dealer should purchase at least 1,001 units.
Should he purchase 1,002 units?

**23. **Example 2: Marginal Analysis Wholesaler purchases an additional unit
Case 1: Demand is smaller than 1002 (Probability 0.1)
The retailer must salvage the additional unit and losses $5 (10 – 5)
Case 2: Demand is larger than 1002 (Probability 0.9)
The retailer makes and extra profit of $20 (30 – 10)
Expected value = -(0.1*5) + (0.9*20) = 17.5

**24. **Example 2: Marginal Analysis Assuming that the initial purchasing quantity is between 1000 and 2000, then by purchasing an additional unit exactly the same savings will be achieved.
Conclusion: Wholesaler should purchase at least 2000 units.

**25. **Example 2: Marginal Analysis Wholesaler purchases an additional unit
Case 1: Demand is smaller than 2001 (Probability 0.25)
The retailer must salvage the additional unit and losses $5 (10 – 5)
Case 2: Demand is larger than 2001 (Probability 0.75)
The retailer makes and extra profit of $20 (30 – 10)
Expected value = -(0.25*5) + (0.75*20) = 13.75

**26. **Example 2: Marginal Analysis Why does the marginal value of an additional unit decrease, as the purchasing quantity increases?
Expected cost of an additional unit increases
Expected savings of an additional unit decreases

**27. **Example 2: Marginal Analysis We could continue calculating the marginal values

**28. **Example 2: Marginal Analysis What is the optimal purchasing quantity?
Answer: Choose the quantity that makes marginal value: zero

**29. **Analytical Solution for the Optimal Service Level

**30. **Analytical Solution for the Optimal Service Level

**31. **Marginal Value: The General Formula P(D = Q*) = Cu / (Co+Cu)
Cu / (Co+Cu) = (30-10)/[(10-5)+(30-10)] = 20/25 = 0.8
Order until P(D = Q*) = 0.8
P(D = 5000) = = 0.75 not > 0.8 still order
P(D = 6000) = = 0.9 > 0.8 Stop
Order 6000 units

**32. **Analytical Solution for the Optimal Service Level

**33. **Marginal Value: Uniform distribution Suppose instead of a discreet demand of

**34. **Marginal Value: Uniform distribution

**35. **Type-1 Service Level What is the meaning of the number 0.80?
F(Q) = (30 – 10) / (30 – 5) = 0.8
Pr {demand is smaller than Q} =
Pr {No shortage} =
Pr {All the demand is satisfied from stock} = 0.80
It is optimal to ensure that 80% of the time all the demand is satisfied.

**36. **Marginal Value: Normal Distribution Suppose the demand is normally distributed with a mean of 4000 and a standard deviation of 1000.
What is the optimal order quantity?
Notice: F(Q) = 0.80 is correct for all distributions.
We only need to find the right value of Q assuming the normal distribution.

**37. **Marginal Value: Normal Distribution

**38. **Type-1 Service Level Recall that:
F(Q) = Cu / (Co + Cu) = Type-1 service level

**39. **Type-1 Service Level Is it correct to set the service level to 0.8?
Shouldn’t we aim to provide 100% serviceability?

**40. **Type-1 Service Level What is the optimal purchasing quantity?

**41. **Type-1 Service Level How do you determine the service level?
For normal distribution, it is always optimal to have:
Mean + z*Standard deviation
µ + zs
The service level determines the value of Z
zs is the level of safety stock
m +zs is the base stock (order-up-to level)

**42. **Type-1 Service Level Given a service level, how do we calculate z?
From our normal table or
From Excel
Normsinv(service level)

**43. **Example 3 A key component has a cost c = 10, holding cost h (for the period) = 1, salvage value v = 10, and sales price p = 19.
What is the optimal target inventory level at each WH?
What is the total inventory?

**44. **Central Warehouse What is the optimal target inventory level at CWH?

**45. **Today The News Vendor Problem
Trade-off
Write down the objective
Maximize profit
Minimize cost
Optimal order quantity
Marginal analysis
Continuous demand distribution
P( Demand = Q*) = F(Q*) = Cu / (Cu + Co)
Discrete demand distribution
P( Demand = Q*) = F(Q*) = Cu / (Cu + Co)

**46. **Next Lecture Real-life Inventory Systems
Important for the second game
Inventory Performance Measure
Inventory turns/turnover
Briefing of the second run of the simulation game

**47. **Additional Example Your store is selling calendars, which cost you $6.00 and sell for $12.00 You cannot predict demand for the calendars with certainty. Data from previous years suggest that demand is well described by a normal distribution with mean value 60 and standard deviation 10. Calendars which remain unsold after January are returned to the publisher for a $2.00 "salvage" credit. There is only one opportunity to order the calendars. What is the right number of calendars to order?

**48. **Additional Example - Solution MC= Overage Cost = Co = Unit Cost – Salvage = 6 – 2 = 4
MB= Underage Cost = Cu = Selling Price – Unit Cost = 12 – 6 = 6

**49. **Additional Example - Solution Suppose the supplier would like to decrease the unit cost in order to have you increase your order quantity by 20%. What is the minimum decrease (in $) that the supplier has to offer.
Qnew = 1.2 * 63 = 75.6 ~ 76 units

**50. **Additional Example On consecutive Sundays, Mac, the owner of your local newsstand, purchases a number of copies of “The Computer Journal”. He pays 25 cents for each copy and sells each for 75 cents. Copies he has not sold during the week can be returned to his supplier for 10 cents each. The supplier is able to salvage the paper for printing future issues. Mac has kept careful records of the demand each week for the journal. The observed demand during the past weeks has the following distribution:
What is the optimum order quantity for Mac to minimize his cost?

**51. **Session 23 Operations Management 51 Additional Example - Solution Overage Cost = Co = Unit Cost – Salvage = 0.25 – 0.1 = 0.15
Underage Cost = Cu = Selling Price – Unit Cost = 0.75 – 0.25 = 0.50