Class_7_-_08

Class_7_-_08 - EMSE 154-254 Fall 2008 Class 7 EMSE 154-254...

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EMSE 154-254 Fall 2008 Class 7 Instructor: Hernan Abeledo Sources: Optimization Modeling with LINGO and Applications of Optimization with Xpress-MP 1 EMSE 154-254 Applied Optimization Modeling Today Multiperiod Planning Models Introduction to the Simplex Algorithm for Linear Programming

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EMSE 154-254 Fall 2008 Class 7 Instructor: Hernan Abeledo Sources: Optimization Modeling with LINGO and Applications of Optimization with Xpress-MP 2 Multi-period Planning Problems Most of the problems we have considered thus far have been essentially one-period problems. However, an important use of optimization is in multi-period planning. In fact, most large linear programs encountered in practice are multi-period models (also known as “dynamic” models). Models for planning over time represent the real world by partitioning time into a number of periods. The portion of the model corresponding to a single period might be some combination of product mix, blending, and other models. These single-period or static models are linked by: 1. A linking or inventory variable for each commodity and period. The linking variable represents the amount of commodity transferred from one period to the next. 2. A “material balance” or “sources = uses” constraint for each commodity and period. The simplest form of this constraint is: “beginning inventory + production = ending inventory + goods sold”. For example, suppose there is a single explicit decision to make each period: how much to produce of a single product. Let Pj be this decision variable for period j . Further, suppose we have contracts to sell known amounts dj of this product in each period j . Define the decision variable Ij as the amount of inventory left over at the end of period j . By this convention, the beginning inventory in period j is I j-1 .
EMSE 154-254 Fall 2008 Class 7 Instructor: Hernan Abeledo Sources: Optimization Modeling with LINGO and Applications of Optimization with Xpress-MP 3 The LP formulation has one “sources of product = uses of product” constraint for each period. For example, if d 2 = 60 and d 3 = 40, then the constraint for period 2 is: I 1 + P 2 = 60 + I 2 or, equivalently, I 1 + P 2 - I 2 = 60. The constraint for period 3 is: I 2 + P 3 - I 3 = 40. Notice how I 2 “links” periods 2 and 3. (i.e., appears in both the constraints for periods 2 and 3). In some models, net outflow may be not equal to the net inflow into the next period: For example, if the product is cash, then one of the linking variables may be short-term borrowing or lending. For each dollar carried over from period 2 by lending, we will enter period 3 with \$1.05 if the interest rate is 5% per period. On the other hand, if the product concerns bananas at a grocery store and there is a predictable spoil rate of 10% per period (say, per day), then the above two constraints would be modified to: .90 I 1 + P 2 - I 2 = 60 .90 I 2 + P 3 - I 3 = 40.

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Class_7_-_08 - EMSE 154-254 Fall 2008 Class 7 EMSE 154-254...

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