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Assignment 2: A manufacturing firm produces and sells a product to a single independent retailer. The product is sold during a single selling season. The manufacturer has production cost is $6.00 per unit and the retailer sells the product to the consumers at a price of $28.00 per unit. Unsold units at the end of the selling season have a salvage value of $2.00. If the retailer cannot meet the demand during the selling season then she has a goodwill penalty cost of $3.00 and the manufacturer has a goodwill penalty cost of $2.5. To procure the product the retailer has a cost of $1.00 per unit. The retailer places an order with the manufacturer at the start of the selling season and the delivery is instantancous The consumer demand x follows a Gamma distribution in the interval (0,co) with shape parameter a - 1 and scale parameter ß - 100. Hint: The Gamma distribution has the probability density function 1 f(x) T(a) 8° Where l(a) is called the Gamma function defined as r(n) - (n - 1)!. The expected value of the Gamma distribution is EX aß In Excel the Gamma distribution can be found by the formula GAMMADIST (English) and the inverse of the Gamma distribution can be found by the formula GAMMA.INV (English). For students who used Excel in Danish or other languages, please use the Excel help or google to find the corresponding formulas in Excel. You have to solve the exercises A-F below and they are weighted evenly. A) With the parameters given above you have to 1 1. Show that the probability density function has the formf(x) = 100 100 2. Show that the distribution function has the form F(x)-1-e- - 3. Show that the expected sale has the form S(q) - 100(1-e - 4. Construct a diagram showing the expected dernand, the expected sales, the expected leftover inventory and expected lost sales as functions of the retailer's order quantity. B) Suppose that the supply chain members make joint decisions 1. Which order quantity is the optimal for the supply chain? 2. What is the optimal supply chain profit?

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