Los factores que deben tenerse en cuenta para un tipo de zapatas

The paper by B e ale , F o rre s t and Taylor presents a rad ical new way in which to handle a multi-commodity sto ch a stic production/inventory problem by approximate techniques. Since th e ir work provided a basis fo r the development o f a more general sto ch astic model and approximate so lutio n procedure in Chapters 5 and 6 an exposition o f i t has been presented here. This chapter has provided a summary and discussion of th e ir work which explains the c ru c ia l steps th a t they took without d e ta ilin g the technical c a lc u la tio n s . E s s e n t ia lly ,t h e y derive a d e te rm in istic non-linear program from the o rig in a l sto ch astic problem involving only the expected values of the random v a ria b le s , but without ignoring t h e ir v a r ia b ilit y . T h is is considered to be encapsulated in a random v a ria b le representing the excess of supply over demand.

Estim ation o f the v a r ia b ilit y o f th is random va ria b le is done it e r a t iv e ly by so lving a sequence o f no n -lin ear programs. The f i r s t is solved with minimal estim ates of i t s v a r i a b i l i t y . This provides the f i r s t approximate so lu tio n , which the authors term th e ir " f i r s t p ass", and i t enables th e ir procedure which estimates the v a r ia b ilit y of supply over demand to be run over the la s t time period and a non-linear program modelling the la s t time period only to be formulated and so lve d . This provides information about the process enabling b etter estim ates of the v a r ia b ilit y of supply over demand in the l a s t time period to be made. These revised estimates are used in the form ulation of a non-linear program modelling the la s t two time p eriods, and so on u n til a non-linear program modelling a l l the time periods in the model is solved again. T his the authors term th e ir " f u ll method".

They t e s t t h e ir approach on four simple examples. The f i r s t two are s u f f ic ie n t ly simple to allow a dynamic programming solution to be obtained. This they do by r e s t r ic t in g the demand d is trib u tio n to d is c re te v a lu e s . They demonstrate th at the so lu tio n thus obtained agrees well w ith that yield ed by th e ir approximate methods. They also used the examples to demonstrate the s e n s it iv it y o f the i n i t i a l production d e cisio n s to the va lu a tio n of the c lo s in g inventory.

T h e ir method has been fu rth e r tested s t a t i s t i c a l l y along with other algorithm s by re p lic a t iv e sim u la tio n . To obtain s u f fic ie n t accuracy by t h is method fo r a reasonable s iz e of experiment control s t a t is t ic s were used. The theory behind them is developed in Chapters 7 and 8, and the re s u lts o f the experiment are given in Chapter 9. These r e s u lt s show that th e ir method performs w ell in p ra ctice fo r the simple examples and y ie ld s an expected revenue very clo se to that given by uie dynamic program ing so lu tio n . There was l i t t l e d iffe re n ce in the performance of th e ir " f i r s t pass" and " f u l l " methods, but the examples had only four time periods in order to keep the computer time requirements of the sim ulation experiments reasonable. I f there were more time periods in the model, th e ir " f u l l " method would out-perform t h e ir " f i r s t pass" method. However, from the s im ila r it y between the expected revenues accrued from using algorithm s g ivin g s lig h t ly d iffe re n t production d e cisio n s and the dynamic programming method i t may be inferred th at f i r s t order deviations from the optimal production ra te s produce only second order changes to the o b je ctive fu n c tio n . This shows that answers to s to c h a stic problems obtained by approximate methods may be r e lia b le in p ra c tic e .

CHAPTER 4

DYNAMIC PROGRAMMING APPROACHES TO PRODUCTION PLANNING

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