Inventory Management Problems and Solutions: EOQ, Safety Stock, and Newsvendor Model
Inventory Management Problems and Solutions
Question 1: Economic Order Quantity (EOQ) at PBK Coffee Store
PBK is a campus coffee store that uses EOQ to manage its inventory of whole bean coffee. The demand is known to be a constant 1,440 pounds/year. PBK’s fixed cost associated with placing and receiving an order from its supplier is $18/order. Its purchase cost is $1/pound, and its estimated annual holding cost is $3.60/pound/year.
a. Optimal Order Quantity with Accurate Holding Cost
Assuming that PBK’s estimate of its holding cost is accurate, what is PBK’s optimal order quantity?
Annual Demand = 1,440 pounds/year
Holding Cost = $3.60/pound/year
Setup/Order Cost = $18/order
EOQ = √(2 * 1,440 * 18 / 3.6) = 120 pounds
b. Optimal Order Quantity with Inaccurate Holding Cost
Suppose that PBK’s holding cost estimate is grossly inaccurate. In particular, suppose that PBK’s actual annual holding cost is $6.40/pound/year. Given that PBK’s annual holding cost is really $6.40/pound/year, what should be PBK’s optimal order quantity?
Annual Demand = 1,440 pounds/year
Holding Cost = $6.40/pound/year
Setup/Order Cost = $18/order
EOQ = √(2 * 1,440 * 18 / 6.4) = 90 pounds
c. Cost of Estimation Error
Given that PBK’s annual holding cost is really $6.40/pound/year, what is the cost per year of PBK’s estimation error? (In other words, given that PBK’s annual holding cost per year is really $6.40/pound/year, how much does it cost PBK per year because it uses the incorrect order quantity from Part (a) instead of the correct order quantity from Part (b)?)
Case 1: Using Incorrect Order Quantity (120 pounds)
Average Annual Cost = Annual Holding Cost + Annual Order Cost
= (Q/2) * H + (R/Q) * K = (120/2) * 6.4 + (1,440/120) * 18 = $600/year
Case 2: Using Correct Order Quantity (90 pounds)
Average Annual Cost = Annual Holding Cost + Annual Order Cost
= (Q/2) * H + (R/Q) * K = (90/2) * 6.4 + (1,440/90) * 18 = $576/year
Extra Cost = $600 – $576 = $24/year
Question 2: Inventory Management Alternatives
The daily demand for a product is distributed normally with a mean of 100 units and a standard deviation of 30 units. The supply lead time is 4 days. The company is considering the following two alternatives for its inventory management system:
- Alternative 1: Place an order every 6 days
- Alternative 2: Order 600 units every time the inventory level drops to 430 units
a. Safety Stock for Alternative 1
Compute the safety stock for Alternative 1, assuming that the target fill rate for the alternative is 95%.
P = 6 days, L = 4 days
Expected Demand = 100 * (6 + 4) = 1,000 units
Standard Deviation of Demand during P+L = √(P+L) * σ = √(10) * 30
0.95 = 1 – (Expected Shortage per Replenishment Cycle / Order Quantity)
Expected Shortage per Replenishment Cycle = L(z) * σP+L
L(z) = (1 – Service Level) * Q / σP+L = (1 – 0.95) * 600 / (30 * √10) = 0.3162
From the Loss function table, z = 0.18
Safety Stock = z * σP+L = 0.18 * 30 * √10 = 17.08 units ≈ 18 units
b. Cycle Service Level for Alternative 2
Compute the cycle service level for Alternative 2.
Reorder Point = Expected Demand during Lead Time + Safety Stock
430 = 100 * 4 + Safety Stock
Safety Stock = 30
Safety Stock = z * σL
30 = z * 30 * √4
z = 0.5
From the z-table, the service level = 69.15%
c. Order Frequency Comparison
Which inventory management system (Alternative 1 or Alternative 2) places orders more frequently?
Both order at the same frequency, i.e., every 6 days.
d. Average Pipeline Inventory Comparison
Which inventory management system (Alternative 1 or Alternative 2) has a higher average pipeline inventory?
Average Pipeline Inventory = Demand over the supply lead time = 400 units
It is the same for both alternatives.
Question 3: Newsvendor Model at Sam’s Club
Sam’s Club sells a particular model of air conditioner that it purchases from Kool King. Most sales are made in June and July. Sam’s makes a one-time purchase of air conditioners in May at a cost of $200 each and sells each air conditioner for $400. Any air conditioners unsold at the end of July are marked down to $175 and sold in a special sale. All marked-down air conditioners are sold. The total (June + July) customer demand distribution for air conditioners is given below:
Demand: 6,000, 7,000, 8,000, 9,000, 10,000, 11,000, 12,000, 13,000, 14,000
Probability: 0.04, 0.08, 0.12, 0.16, 0.2, 0.16, 0.12, 0.08, 0.04
a. Optimal Order Quantity
What is Sam’s optimal order quantity?
Cu = $400 – $200 = $200
Co = $200 – $175 = $25
Target Service Level = Cu / (Cu + Co) = 200 / (200 + 25) = 0.8889
Looking at the cumulative probability, the optimal order quantity is 13,000 units because P(Demand ≤ 13,000) = 0.96, which is the smallest quantity that meets or exceeds the target service level.
b. Expected Profit
What is the expected profit for the order quantity in part a?
Expected Sale = (6,000 * 0.04) + (7,000 * 0.08) + (8,000 * 0.12) + (9,000 * 0.16) + (10,000 * 0.2) + (11,000 * 0.16) + (12,000 * 0.12) + (13,000 * 0.08) + (13,000 * 0.04) = 9,960 units
Expected Leftover = 13,000 – 9,960 = 3,040 units
Expected Profit = (9,960 * $400) + (3,040 * $175) – (13,000 * $200) = $1,916,000
c. Mid-Season Inventory Adjustment
Suppose that one month of summer has passed, and it is now July 1st. Sam’s has 7,000 air conditioners in inventory. Management now believes that customer demand during July will be:
Demand: 3,000, 3,500, 4,000, 4,500, 5,000, 5,500, 6,000, 6,500, 7,000
Probability: 0.04, 0.08, 0.12, 0.16, 0.2, 0.16, 0.12, 0.08, 0.04
Due to an unusually hot summer in Europe, a European distributor has offered to purchase any number of air conditioners that Sam’s is willing to provide (on July 1) for $325/unit. Should Sam’s sell the distributor any of its inventory? If so, how many units?
Let Q denote the quantity to keep for local sales. Therefore, 7,000 – Q is the quantity to sell to the European distributor. The costs associated with the decision are:
Co = $325 – $175 = $150
Cu = $400 – $325 = $75
Target Service Level for Local Sales = Cu / (Cu + Co) = 75 / (75 + 150) = 0.333
Since P(Demand ≤ 4,500) = 0.4, which is greater than 0.333, the optimal quantity to keep for local sales (Q) is 4,500.
Therefore, Sam should sell 7,000 – 4,500 = 2,500 units to the European distributor.
Note: The initial purchase price of $200 per unit is a sunk cost and should not factor into the current decision.
Question 4: Production Planning at Teddy Bower, Inc.
Teddy Bower, Inc. is a boutique manufacturer of women’s parkas. Bower produces and sells two different parkas: Parka A and Parka B. The variable production cost for Parka A is CA = $110/unit, and the variable production cost for Parka B is CB = $95/unit. Bower has a salvage market for all parkas that do not sell to its retailers during its single selling season: The salvage value for Parka A is sA = $40/unit and the salvage value for Parka B is sB = $25/unit. The demand distribution for each parka is given below:
Parka A
Demand: 5,000, 6,000, 7,000, 8,000, 9,000, 10,000
Probability: 0.10, 0.20, 0.15, 0.25, 0.10, 0.20
Parka B
Demand: 8,000, 9,000, 10,000, 11,000, 12,000
Probability: 0.20, 0.25, 0.20, 0.25, 0.10
Bower has two production runs before the start of the selling season. The second production run comes after its retailers place their orders (which means that Bower will know exactly how many units of each parka to produce with its second production run). The first production run comes before its retailers place their orders (which means that Bower only knows the above demand distributions at the time of its first production run). Due to capacity issues with its second production run, Bower knows that it must produce a total of 18,000 parkas with its first production run. Bower’s challenge is how to allocate those 18,000 units between its two parka products.
a. Probability of No Salvage for Parka A
If Bower used its first production run to produce 6,000 units of Parka A and 12,000 units of Parka B, then what would be the probability that no units of Parka A will have to be salvaged?
For Bower to have leftovers of Parka A, demand must be less than 6,000.
P(Demand < 6,000) = P(Demand = 5,000) = 0.1
Hence, the probability of no leftovers = 1 – 0.1 = 0.9
b. Expected Size of Second Production Run
If Bower used its first production run to produce 6,000 units of Parka A and 12,000 units of Parka B, then what would be the expected size of the second production run (in terms of how many units of Parka A and how many units of Parka B)?
The actual number of parkas produced in the second run would be determined by observing the demand. If the number of parkas produced in the first production run is greater than the actual demand, then no further parkas should be produced. If the number is less, then the shortage [Actual Demand – Quantity produced in the first run] should be produced in the second run. The expected size of the second run is simply the expected shortage.
Parka A: (0 * 0.1) + (0 * 0.2) + (7,000 – 6,000) * 0.15 + (8,000 – 6,000) * 0.25 + (9,000 – 6,000) * 0.1 + (10,000 – 6,000) * 0.2 = 1,750 units
Parka B: 0 units (Demand is never greater than 12,000)
c. Minimizing Expected Mismatch Cost
To minimize the expected mismatch cost, how many units of each parka should Bower produce with its first run of 18,000 total units?
The cost of mismatch in this example is the sum of the cost of understocking and the cost of overstocking. Since in this case, you will always meet the demand (through the second production run), there will be no understocking cost. Hence, in order to minimize the mismatch cost, the overstocking cost should be minimized.
Co(A) = $110 – $40 = $70
Co(B) = $95 – $25 = $70
Cost of Mismatch = Co(A) * Expected Leftover of A + Co(B) * Expected Leftover of B
= 70 * [Expected Leftover of A + Expected Leftover of B]
Let QA be the quantity for A and QB for B. It is easy to see that if there is a solution so that the probability of leftover for the unit number QA is equal to the probability of leftover for the unit number QB, then the solution is optimal. With this logic, optimal QA = 8,000 and optimal QB = 10,000. Note: Prob(D < QA) = Prob(D < QB) = 0.45.
Another way to get the same answer is by finding the expected leftovers of all five combinations; and these are:
(6,000, 12,000): [1,000 * 0.1] + [4,000 * 0.2 + 3,000 * 0.25 + 2,000 * 0.05 + 1,000 * 0.35] = 2,100
(7,000, 11,000): [2,000 * 0.1 + 1,000 * 0.2] + [3,000 * 0.2 + 2,000 * 0.25 + 1,000 * 0.05] = 1,550
(8,000, 10,000): [3,000 * 0.1 + 2,000 * 0.2 + 1,000 * 0.15] + [2,000 * 0.2 + 1,000 * 0.25] = 1,500
(9,000, 9,000): 1,750
(10,000, 8,000): 2,350
Question 5: Emergency Room Process Analysis
An emergency room (ER) is currently organized so that all patients register through an initial check-in process. At his or her turn, each patient is seen by a doctor and then exits the process, either with a prescription or with admission to the hospital. Currently, 55 patients per hour arrive to the ER on average, 10% of whom are admitted to the hospital. On average, 7 people are waiting to be registered and 34 are registered and waiting to see a doctor. The registration process takes, on average, 2 minutes per patient. Among patients who receive prescriptions, the average time spent with a doctor is 5 minutes. Among those admitted to the hospital, the average time is 30 minutes.
a. Average Time in Wait Area 1
On average, how long does a patient spend in Wait area 1?
Little’s Law: Iq = Tq / a or Tq = Iq * a
Iq = 7, a = 60/55 minutes between customer arrivals
Therefore, Tq = 7 * (60/55) hours = 7.63 minutes
b. Average Time for Prescription Patients
On average, how long does a prescription patient (or, a patient who leaves the ER with a prescription) spend in the ER?
Total Time = Wait time in area 1 + Registration + Wait time in area 2 + Time with Doctor
= [7 * (60/55)] + 2 + [34 * (60/55)] + 5 = 51.7 minutes
Question 6: Short Questions
SQ1: Stocking Quantities for Discontinued vs. Ongoing Products
Consider two products, A and B, that have identical cost, retail price, and demand parameters and the same short selling season (the summer months from May through August). The newsvendor model is used to manage inventory for both products. Product A is to be discontinued at the end of the season this year, and the leftovers will be salvaged at 75% of the cost. Product B will be re-offered next summer, so any leftovers this year can be carried over and sold next year while incurring a holding cost on each carried-over unit equal to 20% of the product’s cost. How do the stocking quantities for these products compare?
a) Stocking quantity of product A is higher.
b) Stocking quantity of product B is higher.
c) Stocking quantities are equal.
d) The answer cannot be determined from the data provided.
Justification: Cu is the same for both products. Co for product A = 25% * Cost, Co for product B equals 20% * Cost. Hence, the critical fractile (critical ratio) for B is higher, the optimal service level is higher, and the resulting ordering quantity for product B is higher.
SQ2: Impact of Demand Uncertainty on Holding Cost
In the periodic review model, assume that the mean of demand in a period remains the same and the target inventory position is kept at a constant level. If the demand uncertainty (the standard deviation of demand in each period) increases, what happens to the annual holding cost?
a) The annual holding cost would increase.
b) The annual holding cost would decrease.
c) The annual holding cost would remain unchanged.
d) The annual holding cost may go up or down.
Justification: If the target inventory position does not change while the mean demand increases, the safety stock does not change either (only the service level changes). Hence, the holding cost remains unchanged.
SQ4: Order Quantity Adjustment After Shortage
In the continuous review inventory model, if you are short in a cycle, then you need to increase the order quantity in the next cycle to make up for the shortage. Do you agree or disagree?
Disagree
Justification: In the continuous review policy, the order size is fixed.
SQ5: Cycle Service Level with Order Quantity Equal to Expected Demand
For a newsvendor item, the costs and demand are such that the optimal order quantity is equal to the expected demand. The demand has a normal distribution. What is the cycle service level?
If Q* equals the mean demand, z = 0, and the service level equals 50%.