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Chapter17[1] - Inventory Production Supply Chain Mgt...

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Inventory, Production & Supply Chain Mgt. Chapter 17 493 17 Inventory, Production, and Supply Chain Management 17.1 Introduction One carries inventory for a variety of reasons: a) protect against uncertainty in demand, b) avoid high overhead costs associated with ordering or producing small quantities frequently, c) supply does not occur when demand occurs, even though both are predictable (e.g., seasonal products such as agricultural products, or anti-freeze) d) protect against uncertainty in supply, e) unavoidable “pipeline” inventories resulting from long transportation times (e.g., shipment of oil by pipeline, or grain by barge) f) for speculative reasons because of an expected price rise. We will illustrate models useful for choosing appropriate inventory levels for situations (a), (b), (c) and (d). 17.2 One Period News Vendor Problem For highly seasonal products, such as ski parkas, the catalog merchant, L. L. Bean makes an estimate for the upcoming season, of the mean and standard deviation of the demand for each type of parka. Because of the short length of the season, L.L. Bean has to make the decision of how much to produce of each parka type before it sees any of the demand. There are many other products for which essentially the same decision process applies, for example, newspapers, Christmas trees, anti-freeze, and road salt. This kind of problem is sometimes known as the one-period newsvendor problem. To analyze the problem, we need the following data: c = purchase cost/unit. v = revenue per unit sold. h = holding cost/unit purchased, but not sold. It may be negative if leftovers have a positive salvage value. p = explicit penalty per unit of unsatisfied demand, beyond the lost revenue.
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494 Chapter 17 Inventory, Production & Supply Chain Mgt. In addition, we need some information about the demand distribution (e.g., its mean and standard deviation). For the general case, we will presume for any value x : F ( x ) = probability demand ( D ) is less-than-or-equal-to x . 17.2.1 Analysis of the Decision We want to choose: S = the stock-up-to level (i.e., the amount to stock in anticipation of demand). We can determine the best value for S by marginal analysis as follows. Suppose we are about to produce S units, but we ask, “What is the expected marginal value of producing one more unit?” It is: - c + ( v + p ) * Prob{ demand > S } – h * Prob{ demand d S } = - c + ( v + p ) * ( 1 – F ( S )) – h * F ( S ) = - c + v + p – ( v + p + h ) * F ( S ). If this expression is positive, then it is worthwhile to increase S by at least one unit. In general, if this expression is zero, then the current value of S is optimal. Thus, we are interested in the value of S for which: -c + v + p – ( v + p + h ) * F ( S ) = 0 or re-arranging: F ( S ) = ( v + p – c ) / ( v + p + h ) = ( v + p – c ) / [( v + p – c ) + ( c + h )]. Rephrasing the last line in words: Probability of not stocking out should = (opportunity shortage cost)/[(opportunity shortage cost) + ( opportunity holding cost)].
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