Quiz on Deterministic part of course October 22, 2002

Transcription

1 Engneerng ystems Analyss for Desgn Quz on Determnstc part of course October 22, 2002 Ths s a closed book exercse. You may use calculators Grade Tables There are 90 ponts possble for the regular test, or 1 pont per mnute. Ponts assocated wth each queston correspond to the estmated tme t mght take to answer them. There are also a possble 10 ponts of extra credt. Fnal scores wll be based on 90 possble ponts. Item Max Your Name (provded we can read t) 1 Concepts 6 What's the best desgn? 20 Money, Money, Money 1 hp t out! 20 Extra Credt 10 Total 100 Percentage grade on bass of 90 maxmum: core Yours I have completed ths test farly, wthout copyng from others or from a textbook Please sgn your name legbly 1

2 Concepts (6 ponts -- 2 ponts per part) Wrte a short defnton or descrpton explanng the followng: Producton Functon A physcal functon represents the techncally effcent transformaton of physcal resources nto products. Techncal Effcency Techncal effcency represents the maxmum product that can be obtaned from any gven set of resources. Economc Effcency Economc effcency represents the best desgn economcally. Not the same as Techncal Effcency. Margnal Product A margnal product s the change n output due to a unt change n a specfc nput. Isoquant An soquant s a locus on the producton functon of all equal specfc levels of product. Returns to cale Returns to scale s the rato of the rate of change n output due to a proportonal change n all nputs smultaneously. Economes of cale Economes of scale exst for a producton process when t s cheaper to produce n quantty. How are the prevous two related (or not)? Both refer to the dea of somehow gettng proportonately more as the scale of producton ncreases. Ther prncpal dfference arses from the fact that the noton of economes of scale ncorporates nformaton about the nput cost functon. hadow Prces A shadow prce s the rate of change of the obectve functon wth respect to a partcular constrant. 2

3 Opportunty Costs An opportunty cost s the rate of degradaton of the optmum per unt use of a non-optmal varable n the desgn. Lagrangean Multplers For the problem: optmze ( ) g subect to h ( ) = b and 0, Lagrangean s L = g( ) λ [ h ( ) b ]. The parameters of λ are known as Lagrangean multplers, and actually they are the shadow prces. Optmalty Crtera n Margnal Analyss MP MC = 1 λ for all Complementary lackness λ = 0 for all so that ether lambda or s s equal to zero s Expanson Path The expanson path s the locus of all the optmum desgns for every level of output Y. Cost Functon A cost functon descrbes the optmal, the least cost of producng any level of product Y. Actvtes An actvty s a specfc way of combnng basc materals or resources to acheve some obectve or output. Fxed Charge Problem A fxed charge problem s a specfc amount, typcally a cost, assocated wth any level of a decson varable. For example, cost of = c 0 + c. It cannot ordnarly be handled by LP software. Data Tables An Excel tool to execute senstvty analyss automatcally by examnng the consequences of varyng one, two, and sometmes three varables at once.

4 What's the best desgn? (20 ponts) You are gven a producton functon: And the cost of the resources as: 0.6 R 0.8 6R + 10 a) What can you say by mmedately, by nspecton, about the returns to scale? About the economes of scale? Justfy your answer. ( ponts) Returns to scale are decreasng because < 1. Economes of scale can not be told by nspecton. b) What s the optmal relatonshp between the resources R and? (10 ponts) MP R = 1.8R MP = 0.9R (oluton usng Y n the expresson s good also = (0.6/R ) Y ; = (/) Y) MC R =.8R MC = 12 Applyng Optmalty Crtera n Margnal Analyss: MP MC R R MP = MC R.8R R = = R 0.8 c) What s the assocated cost functon? (6 ponts) Z = R 0.6 C = 6 5 = ( ) = 0 = 5 so the cost functon s 0 C = Z. 99Z and has no economes of scale. (5 )

5 Money, Money, Money (1 ponts) What s Net present value? ( Ponts) Net Present Value = Present Value Revenues Present Value Costs Present value s the dscounted value of future sums of money usng the approprate dscount rate. What are the maor advantages and dsadvantages of the Beneft/Cost rato as a crteron of evaluaton? (6 Ponts) Advantages: It compares proects on a common scale It provdes a drect ndcaton of whether a proect s worthwhle (the rato exceeds 1) It provdes an easy means to rank proects n order of relatve mert Dsadvantages: It requres all benefts to be assgned a monetary value Ambguty of the treatment of recurrng costs Bas n favor of captal-ntensve proects Relatve rank of proects can depend on dscount rate used Why mght the rank order of proects change when you calculate ther beneft-cost ratos usng dfferent dscount rates? ( Ponts) Lower dscount rates favor proects wth longer-term benefts. Thus, proects wth longer-term benefts can appear better than proects wth benefts accrung sooner f a lower rate s used n the analyss. 5

6 hp t out! (20 ponts) A plant manager wants to mnmze the cost of shpments from plants A and B (capacty of 1000 and 500) to markets K, L, M (requrements of 00, 800, 200, respectvely). The shppng costs are as n the table From To K L M A B a) et up the Lnear Program. You may use a vector notaton f convenent. (7 ponts) Let be the shpment from plant to market. Mnmze ubect to ,, + + = 00 = 800 = 200, ,, 0 b) uppose the results gave the followng results: (9 ponts) Requrement at hadow Prce Range A 0 (0. 800) B 0 (00, 600) K 20 (200, 00) L a M 0 (250, 50) What s the meanng of the shadow prce on producton at B? The rate of cost reducton f capacty of B ncreases. What can you say about the shadow prce on requrements at K f these rse to 50 unts? Greater than 20 because as a constrant tghtened, the shadow prce ncreases beyond the range What can you say about the shadow prce at L? a > 0 because f I change requrement that wll change amount shpped and ths has a cost 6

7 c) The manager thnks t mght be a good dea to set up a faclty between B and M at a cost of throughput. The obect would be to reduce the shppng costs on ths route. How does the LP handle such an extenson to the basc formulaton? ( ponts) Ths s a fxed charge problem. We can solve two problems: - one wthout the new faclty - one wth the new faclty, but drop off the fxed charge (20) off when solvng the LP, and add the fxed charge (20) back to get the fnal result Comparng the optmal solutons of the above two problems, whchever lower s the fnal soluton. The above s vald only for trval problems. The better answer s that standard LP does not know how to deal wth ths problem. For 10 ponts extra credt: d) uppose that the actual cost of shppng between B and M s not but 80 ( volume _ on _ route), a ( volume _ on _ route). Under what condtons could you ncorporate ths feature nto the LP? ( ponts) If a 1, then the feasble regon s convex (the feasble regon s above the curve because we can always spend more) and we can ncorporate t nto the LP. If a=1, we can solve LP drectly; f a>1, we need pecewse approxmaton of the curve. If a<1, we cannot deal wth ths problem usng LP. How would ncludng ths feature change the LP? how specfc equatons. (6 ponts) Ths s for the case that a > 1. If a = 1, we can ncorporate the feature drectly nto the LP. Let denote volume_on_route. We have the formulaton related to ths feature: 1, Defne breakponts on that defne the new varables ', ", "', etc. and then '< V 1, ",< V 2, etc for as many segments as relevant, also > 0. a ' '' ''' 2, Defne ( volume _ on _ route) as c 1 + c2 + c wth c determned by the breakponts selected, and change the obectve functon to ' '' ''' ( + + ), Insert the new defntons of n the constrants for shpments out of B and nto M. 7