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Overview - Lecture 1 Overview of optimization and review...

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Lecture 1: Overview of optimization and review Contents 1 Overview of optimization We will illustrate the whole story of optimization using the following simple example. Example 1 MM operates a diamond business. Several times each year, MM travels to Antwerp to replenish his diamond supply. The details of his business are as follows: wholesale price = $700/ct minimum wholesale order q min = 100cts retail price = $900/ct (i.e. profit p = $200/ct) trip: cost f = $2000/trip & time l = 1 wk demand d = 55 cts/wk and no back-order allowed (?) inventory holding cost h = $3 . 50/ct/wk What can he do to maximize profit ? The first step in the optimization story is to translate this English description of the problem into a mathematical description. Such a description is called a mathematical model . A mathematical model consists of (a) decision variables : These are the variables that define the problem. These variables should be chosen in such a way that one is able to write the constraints and the objective in terms of these variables. (b) constraints : These define relation between the various decision variables and also limits on the choice of decision variables. (c) objectives : This the function defined in terms of the decision variables that the decision maker wants to minimize or maximize i.e. optimize . These definitions will not make too much sense right away. They will become clearer as we work through the example so bear with me for the moment. The moment we start using the word model we implicitly mean that we are moving away from reality (whatever that might be!). One of the big tensions in mathematical modeling is the one between (a) validity : how close is the model to reality ? (b) tractability : how easy it is to work with model ? how easily can one solve it ? More often than not, these two requirements pull us in opposite directions. So how does one resolve this issue ? The typical strategy is to start with the simplest (most tractable) model, solve it, and then test the quality of the solution. If it acceptable, stop; otherwise refine the model by adding aspects of reality that were not included the last time around.
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Overview of Optimization 2 r q/ 55 1 wk slope = 55/wk Inventory Figure 1: Inventory Problem 1.1 Mathematical Model Before we get a little too ahead of ourselves with definitions and philosophising about optimization, let us start building our first model. We will make the following assumption: Assumption 1 The demand is deterministic with rate d = 55 cts/wk This is break from reality but it make the problem tractable. Given this assumption, the decision variables reduce to (i) re-order quantity q : the amount to order per trip (ii) re-order level r : re-order when inventory falls below r Since the demand is time-independent, we will also assume that the quantities q and r are fixed and not functions of time. We might, of course, have to revisit this topic in case our model does not give us reasonable solutions.
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