Problem 1: You buy seafood from a distributor for $10/pound and sell it in a local market at $20/pound. Daily demand is Normally distributed with mean µ = 100 and standard deviation σ = 20. You replenish your seafood every morning (paying no fixed costs) to a base stock level of S. Cold storage at this market is expensive, costing $1/pound/day on the maximum space you need (i.e. on the base stock level S), but ensures that your seafood never perishes. On the other hand, any unfulfilled demand is lost to the other vendors at the market. a) Calculate the base stock level S ∗ that maximizes your profit. b) Calculate the Average Profit that you expect to make daily by stocking to this optimal level S ∗ . Don't forget to include both the product costs and storage costs in your calculation. Note: You can check your answer by verifying on your spreadsheet that setting base stock level S to S ∗ indeed maximizes your profit. c) You are exploring the option of selling your seafood in the same market without cold storage. You would no longer pay storage costs, but now any unsold seafood at the end of each day would perish. Under this model, what is the optimal stocking level? d) Calculate your Average Profit under part c). Is it higher or lower than the optimal daily profit from part b) using cold storage?
Problem 2: Instant Pot sells its signature pressure cooker online, shipping directly to customers from a single distribution center. Daily demand is Normally distributed with mean µ = 100 and standard deviation σ = 20, and independent over time. Due to Instant Pot's unique product, all demand is backlogged, with customers willing to wait if the distribution center does not have inventory to ship their order right away. Nonetheless, Instant Pot strives to maintain a Service Level of SL = 0.95, which means that on 95% of replenishment cycles, inventory does not run out. Therefore, Instant Pot sets z = NORMSINV(0.95). The distribution center is replenished from a supplier with a delay of 2 weeks between ordering and receiving. The supplier produces product for Instant Pot in a fixed batch size of Q = 800 units. Instant Pot continuously reviews the inventory in its distribution center and applies an (Q, R) policy with Q = 800. (a) Calculate the re-order point R desired, given the desired Service Level of 0.95. (b) Calculate the average on-hand inventory (not including pipeline inventory). (c) Suppose that the holding cost for on-hand inventory is $0.01 per unit per day. What is the annual holding cost paid on average over 365 days? (d) Suppose that the supplier now restricts Instant Pot to ordering replenishments once a week, which are due Mondays at 8:00 AM. However, now the supplier allows Instant Pot to order any quantity (at the same unit costs), removing the constraint to order in batches of exactly 800. How should Instant Pot change its replenishment policy, with the goal of still maintaining the Service Level of 0.95? Give a precise description of the new replenishment policy. (e) What is the longest that a customer might have to wait for an Instant Pot to ship to them, before and after the change in part (d)? (You can assume the worst-case demand and inventory patterns. However, assume that the supplier never goes down.)
Problem 3: You designed a new meal replacement drink and turned it into a small business, selling it online in non-perishable plastic bottles. You own the brand but outsource all production to a contract packager ("copacker"). Each time you make a purchase order from the copacker, you pay a fixed cost of K = $400 (mostly for the cost of driving a truck) and variable cost of c = $2 per bottle produced. You offload each truck shipment into your empty garage and hence face no storage or processing costs. However, you value your capital at the interest rate of 0.5% per month. a) Since you sell your drinks using a subscription service, you know that you will face a steady demand of λ = 5000 bottles per month, with no variability in the demand (the existing customers have already committed to their subscriptions, and you are not taking new customers for now). What quantity Q should you order from the copacker each time? Assume that your garage has infinite space and that you can fit up to 60000 bottles into a truck. b) You collect p = $3 for each bottle sold (after accounting for outbound shipping). How much profit can you make each month?