M d Example: Room Control Optimization 16888 E5077 Want: -temperature in ideal range 68-72F humidity above 56% is undesirable assume: T=c'x temperature Formulate as a molP undesirable H=c'x humidity mind,+d min s t Using 68/72'TIFI deviational ideal cx+d,≥68 variables cx-d+<72 H undesirable cx-d≤56 56H[%] Ax≤b,x,d1,d1,d≥0 ideal o Massachusetts Institute of Technology -Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics
11 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Example: Room Control Optimization Example: Room Control Optimization Want: - temperature in ideal range 68-72 ºF - humidity above 56% is undesirable Assume: temperature humidity T H = = 1 2 c x c x U T 68 72 T [ºF ] undesirable ideal UH 56 H [%] undesirable ideal Formulate as a MOLP { } { } 1 1 2 1 1 2 2 112 min min s.t. 68 72 56 , , , , 0 d d d d d d ddd − + + − + + −−+ + + ≥ − ≤ − ≤ ≤ ≥ 1 1 c x c x c x Ax b x Using deviational variables
M Aggregated Utility 16888 E5077 The total utility becomes the weighted sum of partial utilities sometimes called multi-attribute utility analysIS MAUa E.g. two utilities combined: U(JI,J2)=Kk,kU(1)U(2)+k,U(1)+k,U(2) For 2 objectives: K=(-K-k2)/k,k2 Combine single utilities into overall utility function kis determined during interviews Steps: MAUA K is dependent scaling facto 1. Identify Critical Objectives/Attrib 2. Develop Interview Questionnaire 3. Administer Questionnaire 1.0 customer 14. Develop Agg. Utility Function customer 2 5. Analyze Results customer 3 Caution: "Utility"is a surrogate for value" but while " value interviews Attribute j;(performance j) has units of [$], utility is o Massachusetts Institute of Technology unitless Engineering Systems Division and Dept of Aeronautics and Astronautics
12 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Aggregated Utility Aggregated Utility The total utility becomes the weighted sum of partial utilities: Attribute Ji customer 1 customer 2 customer 3 Caution: “Utility” is a surrogate for “value”, but while “value” has units of [$], utility is unitless. interviews Combine single utilities into overall utility function: Steps: MAUA 1. Identify Critical Objectives/Attrib. 2. Develop Interview Questionnaire 3. Administer Questionnaire 4. Develop Agg. Utility Function 5. Analyze Results (performance i) Ui(Ji ) ki’s determined during interviews K is dependent scaling factor … sometimes called multi-attribute utility analysis (MAUA) 1.0 E.g. two utilities combined: ( ) 1 2 12 1 2 1 1 2 2 U J J Kk k U J U J k U J k U J , ()( ) () ( ) = ++ For 2 objectives: 1 2 12 K =−− (1 ) / k k kk
Mlesd Notes about Utility Maximization 50 7 tility maximization is very common and well accepted Usually U is a non -linear combination of objectives J Physical meaning of aggregate objective is lost (no units) Need to obtain a mathematical representation for U(Jy for all to include all components of utility Utility function can vary drastically depending on decision maker .e.g. in U.s. Govt change every 3-4 years o Massachusetts Institute of Technology -Prof de Weck and Prof. Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics
13 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Notes about Utility Maximization Notes about Utility Maximization • Utility maximization is very common and well accepted • Usually U is a non-linear combination of objectives J • Physical meaning of aggregate objective is lost (no units) • Need to obtain a mathematical representation for U(Ji ) for all I to include all components of utility • Utility function can vary drastically depending on decision maker …e.g. in U.S. Govt change every 3-4 years