Acknowledgments
Lemma 1: Properties of K-convex functions
6.4 Case Study
AMS is a growing fashion house. It started as a small family business in selling novelty T-shirts a couple of decades ago. Nowadays, it is a recognized forerunner in the global casual apparel industry. Its products are divided into two categories: novel and basic. Novel products are designed to be put in the market for one season only, while basic products are offered for at least two seasons. Unlike the basic products that might have inventory left over from previous years, all the excess novel products are salvaged at the end of their selling season. Furthermore, all the novel products are produced before their selling season.
Five new novelty T-shirts are designed for the next season. The cost of each T-shirt is $3. Traditionally, the cost is 30% of the selling price and the quantity of production is the average of the modes of the forecast.
According to a $10 selling price, the expert forecasts of their independent demand are shown in Table 6.1.
Steven James is a product manager just hired to work under the Director of Novel Products and asked to report on the sales plans of these T-shirts. As a top graduate from an industrial engineering department who has a keen interest in inventory and pricing models, Steven is very enthusiastic in his new job and is confident that he will contribute significantly to AMS. After checking the current sales plans for the new novelty T-shirts, he wants to improve the current production plan and also try to convince his boss that a better pricing scheme should be implemented. In order to achieve these objectives, he needs to answer the following questions in his report:
What is the expected profit for the current sales plan?
• What is the optimal production plan for the current pricing scheme? What is the
correspond-• ing expected profit?
What is the potential increase in expected profit in deploying a different pricing scheme?
•
Discussion with the sales department reveals that excess novelty T-shirts have a salvage value of $.50.
Furthermore, for a selling price from $5 to $15, each independent demand can be approximated by an additive model with a 10% drop in the $10 low-demand estimate per dollar increase in selling price. That is, demand = a − br + ε for 5 < r < 15, with a, b and ε as shown in Table 6.2.
6.4.1 Exercises
1. Suppose that the demand for a product is 20 units per month and the items are withdrawn at a constant rate. The setup cost each time a production run is undertaken to replenish inventory is $10.
The production cost is $1 per item, and inventory holding cost is $0.20 per item per month. Assuming
TABLE 6�1 Forecasts of Low, Medium, and High Demand for the Novelty T-Shirts
T-Shirts Demand (Probability)
Swirl 10,000 (.2) 40,000 (.5) 80,000 (.3)
Strip 5000 (.25) 10,000 (.25) 50,000 (.5)
Sea 4000 (.1) 7000 (.5) 15,000 (.4)
Stone 3000 (.3) 9000 (.4) 20,000 (.3)
Star 8000 (.4) 10,000 (.4) 12,000 (.2)
shortages are not allowed, determine the optimal production quantity in a production run. What are the corresponding time between consecutive production runs and average cost per month?
2. Consider a situation in which a particular product is produced and placed in in-process inventory until it is needed in a subsequent production process. The number of units required in each of the next three months, the setup cost, and the production cost that would be incurred in each month are shown in Table 6.3. There is no inventory of the product, but 1 unit of inventory is needed at the end of the three months. The holding cost is $200 per unit for each extra month the product is stored. Use dynamic programming to determine how many units should be produced in each month to minimize the total cost.
3. In Example 2 (Section 6.3.2.1), if the demand−price relationship were (100r-3)ε, what would the optimal price and procurement level be?
4. Consider the hot dog stand example (Example 1 in Section 6.3.1). Now suppose that we would like to determine the optimal procurement policy over the next week (assume four games a week and we are only concerned about the game days). Each order costs the vendor $10.00 for gas and park-ing. Assume that any hot dog left at the end of the day is stored for the next game day and is not sold at the entertainment district. Each excess hot dog costs us $0.50 for handling and proper refrigera-tion. Also, let us assume that there are other vendors next door. In case of a shortage, extra hot dogs can be purchased from the neighboring hot dog vendors for $2.50 each and, hence, no demand is lost. Find the optimal procurement policy for the vendor over the next four sales periods.
5. Following the outline given in Section 6.3.2, prove that Jt for t = 0, 1, . . . , N − 1 is K-convex when order setup cost k is positive.
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TABLE 6�2 Parameters for the Additive Demand-Price Models
T-Shirts a b ε (Probability)
Swirl 20,000 1000 0 (.2) 30,000 (.5) 70,000 (.3)
Strip 10,000 500 0 (.25) 5000 (.25) 45,000 (.5)
Sea 8000 400 0 (.1) 3000 (.5) 11,000 (.4)
Stone 6000 300 0 (.3) 6000 (.4) 17,000 (.3)
Star 16,000 800 0 (.4) 2000 (.4) 4000 (.2)
TABLE 6�3 Requirement and Production Information
Month Requirement Setup Cost ($) Production Cost ($)
1 2 500 800
2 4 700 900
3 3 400 900
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