The Optimal Wine. A Study in Design Optimization. April 26, Erin MacDonald Alexis Lubensky Bryon Sohns

Transcription

1 The Optmal ne A Study n Desgn Optmzaton Aprl 6, 004 Ern MacDonald Alexs Lubensky Bryon Sohns Unversty of Mchgan ME 555 Desgn Optmzaton Professor Panos Y. Papalambros

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3 Unversty of Mchgan Table of ontents ABSTRAT... NTRODUTON... 4 NOMENLATURE DTONARY... 6 SUB-SYSTEM OPTMZATON NE FORMULATON... 8 DESRPTON... 8 Pctoral Representaton of the Process... 8 MATHEMATAL MODEL... 8 Nomenclature... 9 Obectve Functon and onstrants... 0 SPEFYNG THE EQUATONS... Purchase of Grapes... Processng the Grapes nto ne... Fndng Revenues... 4 SUMMARY MODEL... 4 OPTMZNG THE NE FORMULATON... 7 A Soluton to the Optmal ne Formulaton... 7 Analyzng the Soluton... 8 SUB-SYSTEM OPTMZATON BOTTLE DESGN... 0 DESRPTON... 0 Obectve... 0 Selecton of Adectves... 0 Prevous ork... onsumer Survey... MATHEMATAL MODEL... 4 Nomenclature... 4 onstrants... 5 Desgn Varables and Parameters... 6 SUMMARY MODEL... 8 Feasblty of Model... 8 MODEL ANALYSS...8 hoce of Optmzaton Package... 8 hangng parameters to varables... 8 Monotoncty and ell boundedness... 9 Redundant onstrants... 0 Smplfcaton...0 Senstvty of Parameters...

4 The Optmal ne A Study n Desgn Optmzaton RESULTS AND ONLUSONS... onclusons... SUB-SYSTEM OPTMZATON ESTABLSHNG A NE PORTFOLO...4 DESRPTON...4 MATHEMATAL MODEL...4 Notaton...4 Obectve Functon and onstrants...5 SPEFYNG THE EQUATONS...5 SUMMARY MODEL...8 RESULTS...8 TOTAL SYSTEM OPTMZATON DEVELOPNG A BUSNESS PLAN...40 DESRPTON...40 MATHEMATAL MODEL...40 Nomenclature...40 Obectve Functon and onstrants...4 Specfyng the Euatons...4 DSUSSON OF RESULTS...46 ONLUSON AND AREAS FOR FUTURE RESEARH...5 APPENDX DEVELOPNG THE NE FORMULATON MODEL...5 PROESSNG GRAPES NTO NE...5 Fermentaton...5 Malolactc Fermentaton...55 Oak Barrel Agng...56 Steel and Bottle Agng...57 QUALTY DETERMNES REVENUE...58 TESTNG THE MODEL TH REASONABLE NPUTS...59 Values for Parameters and onstants...59 Screen Shots of the Model n Excel...6 RUNNNG THE OPTMZATON...6 neformulaton.m...6 neformulatonob.m...64 APPENDX BOTTLE DESGN SURVEY AND MODEL...65 SAMPLE SURVEY PAGE...65 BOTTLE SHAPES AND PROPORTONALTES USED N SURVEY...66 BBLOGRAPHY...68 EB RESOURES...68 BOOKS, PAPERS AND OTHER RESOURES...69

5 Unversty of Mchgan Abstract Proect Motvaton The wne maker s often consdered an artst who carefully selects the grapes, chooses how to process them, and blends them to extract the desred tastes, colors, and aromas. entures of practce have taught wnemakers how to extract the subtle flavors from the frut whch can be mxed and enhanced wth the wne maker s sklls. An empty bottle s a canvas for a new creaton where the aromas of almond, black current, strawberry, or butter, substtute the colors used by a panter. hat role, then, can the rgor of mathematcs have n wne makng? an we desgn such a thng as an optmal wne? e venture n ths study to apply an engneerng methodology to the world of ndustral wne makng. Proect Scope For wne makng, optmalty s n the eye and palette of the beholder. e wll defne a few characterstcs that may be of nterest to a wnery when establshng what wnes to produce. lear trade-offs exst between producng the least expensve (cost optmal wne and producng the optmal tastng wne based on customers preferences. Our study strves to balance these trade-offs by optmzng the overall expected proft of a portfolo of wnes. e wll lnk three dfferent decson areas of the wne-makng process n a system-wde optmzaton. Many decsons go nto determnng the expected proft of a wnery. e have decded to model the followng aspects of the busness ne Formulaton The obectve of ths sub-system optmzaton s to maxmze the proft of a partcular wne formulaton gven grape costs and ualtes as well as the manufacturng steps contrbutng product costs and features. The resultng ualty of the desgned wne wll determne the revenues whch can be extracted from a formulaton. The costs of the grapes and the steps ncurred n manufacturng wll be subtracted from the revenues to obtan an expected proft. The optmal formulaton wll gve the proporton of each grape to be used as well as the amount of processng reured from each of a preestablshed set of steps n the manufacturng process. ne Portfolo An optmal portfolo for partcular wne varetals means a partcular number of brands wth dfferent prce ponts that optmzes the proft for the wnery. The selecton of dfferent types of wne, each havng a partcular characterstc, allows a wnery to capture ts consumers. A combnaton of wne type and prce pont can lead to the selecton of the most proftable product portfolo. The selecton of products balances the use of manufacturng resources, the avalablty of captal, and the proft cannbalzaton that occurs by ntroducng smlar products to the market. ne Bottle n some nstances, the desgn of the wne bottle plays a large role n entcng a consumer to purchase a partcular wne. The semantc role of the bottle shape as a marketng tool wll be nvestgated and ntegrated nto overall wne ualty. Gven a wne formulaton and consumer preferences for semantc bottle characterstcs, we wll fnd optmal bottles for dfferently-flavored wnes wthn a portfolo, wth respect to overall expected proft.

6 The Optmal ne A Study n Desgn Optmzaton ntroducton Ths optmzaton model lnks decsons from manufacturng, desgn, and busness n order to maxmze proft for a portfolo of wne brands, or labels, for a md-szed wne producer. The overall model conssts of three subsystem models that can be optmzed ndvdually or ntegrated and optmzed as a whole. ndvdually, the three subsystems behave as descrbed n the black-box fgure below. Black-box representaton of stand-alone optmzaton problems nputs (Parameters Outputs Grape propertes Processng effects osts Preferences Busness nformaton ne Formulaton Maxmum proft Optmal hardonnay recpe Grape purchase plan Semantc preferences Physcal bottle measurements Adectves Bottle Desgn Maxmum semantc conveyance Optmal bottle shape to convey gven adectve Manufacturng nformaton Market nformaton Brand Portfolo Maxmum proft Optmal - Number of brands - Prce per brand - Quantty per brand - Qualty of brand hen evaluated separately, the wne formulaton model and the brand portfolo model both am to maxmze proft. The bottle desgn model ams to maxmze the conveyance of semantc meanng through the bottle desgn. ndvdually, the three models wll contrbute valuable nformaton to a wnery. The ntegraton of the three models allows a wnery to mprove ts product offerngs, wthn the chardonnay category, as a whole. The ntegraton of the models occurs as descrbed n the fgure below. The brand portfolo model wll send ualty reuests to the wne formulaton model. Based on these reuests, the wne formulaton model wll send flavor adectves for each wne n the portfolo to the bottle desgn model. The bottle desgn model wll then send aggregated desrabltes for each wne to the brand portfolo model. The wne formulaton model wll send costs to the brand portfolo model. Based on ths nformaton, f the wne costs need to be adusted, the brand portfolo model wll then modfy ts ualty reuests and re-send them to wne formulaton. Ths process wll terate untl maxmum proft s acheved for a determned number of brands at a specfc prce ponts wth specfc ualtes. Each brand wll receve a desgned bottle shape, grape purchase plan, and chardonnay recpe. 4

7 Unversty of Mchgan ntegraton of Models All Parameters Maxmum proft Optmal - Number of wne brands - Prce per brand - Quantty per brand - Qualty of brand - Bottle shape per brand - hardonnay recpe - Grape Purchase Plan 5

8 The Optmal ne A Study n Desgn Optmzaton Nomenclature Dctonary The followng notaton s used n the three models descrbed n ths paper. Varables b uantty of label produced n bottles ( =... n m amount of manufacturng step used n number of labels to produce (dscrete p sellng prce of label ( =... n B ualty of bottle for label ( =... n ualty of wne for label ( =... n u s proporton of grape (u s for "uva" bottle style for label ( =...n (dscrete ( x, y a Set of ponts that descrbe actual bottle shape for gven flavor adectve (=... Functons A A B D ( D H K L P P Q R V X M B G M x T amount sold Surface area of B ( p, D(, ( D X cost cost of cost of, Desreablty of Heght of characterstc % of volume lost on process proft ( p, ~ ost per bottle of ~ ~ ualty of revenue, b, n, ~ Volume of X B B purchasng a grape of type manufacturng process Desrablty of the bottle (n characterstc (formed by a partcular manufacturng process Maxmum radus of wne exposed to ar (n Maxmum demand for label ( =... n a wne ualty characterstc x of agregated grape purchase Total Proft label ( =... n nteror of bottle (n producng label ( =... n the bottle (n 6

9 Unversty of Mchgan Parameters A M Maxmum surface area of wne that can be exposed to ar (n B producton capablty of wnery n bottles F K, G x, fxed cost to produce a label of wne Matrx of coeffcents that adusts the desrablty of a wne wth respect to bottle shape (=...7 characterstc x of grape K Exponental Decay onstant for anabalzaton Functon D mportance of ualty characterstc n relaton to the other characterstcs Target ntensty of ualty characterstc (expressed as a % G market prce of grape H M Maxmum heght of the bottle (n M cost of one unt processed on manufacturng step ( S, S =,,... Set of ponts that descrbe desred bottle shape for gven flavor adectve x y a T Thckness of glass on sdes of bottle (n V V V V V Volume of wne n a standard bottle (n A Volume of mnmum amount of ar that must be present n a standard bottle (n B Volume of maxmum amount of ar that must be present n a standard bottle (n D L M Volume of dmplen bottom of bottle (n % of volume lost per unt of manufacturng on process Maxmum outer radus of the bottle (n nner radus necessary for proper cork fttng (n G χ maxmum proporton of grape χ M maxmum amount of manufacturng step allowed or avalable χ maxmum cost allowed onstants ς ς R Manufacturng costs not explctly modeled Revenue scalng factor 7

10 The Optmal ne A Study n Desgn Optmzaton Sub-system optmzaton ne Formulaton Descrpton The obectve of ths sub-system optmzaton s to maxmze the proft of a partcular wne formulaton gven grape costs and ualtes as well as the manufacturng steps contrbutng product costs and features. The resultng ualty of the desgned wne wll determne the revenues whch can be extracted from a formulaton. The costs of the grapes and the steps ncurred n manufacturng wll be subtracted from the revenues to obtan an expected proft. The optmal formulaton wll gve the proporton of each grape to be used as well as the amount of processng reured from each of a preestablshed set of steps n the manufacturng process. PTORAL REPRESENTATON OF THE PROESS Mathematcal Model Turnng grapes nto wne nvolves an ntrcate set of processes whch start n the vneyards and end when the wne s fnally consumed. Ths model wll focus on some of the key processes startng wth 8

11 Unversty of Mchgan the purchase of the grapes are endng wth the agng of the wne n ts bottle. The frst secton descrbes the notaton whch wll be utlzed n ths model. The followng secton descrbes an optmzaton formulaton that allows for a varety of grape and process characterstcs. The next secton, Specfyng the Euatons, determnes the specfc set of characterstcs and processes selected to model realty n a way that we can obtan meanngful suggestons. NOMENLATURE Varables u proporton of grape purchased (u s for "uva" m amount Parameters K G x, D L G cost % of of manufacturng step used (measured n days characterstc x of grape mportance of ualty characterstc n relaton to the other characterstcs Target ntensty of ualty characterstc (expressed as a % G market prce of grape M V of one unt processed on manufacturng step volume lost per unt of χ maxmum proporton of grape manufacturng on process χ M maxmum amount of manufacturng step allowed or avalable χ maxmum cost allowed onstants ς Manufacturng costs not explctly modeled R ς Revenue scalng factor euvalent to the revenue obtaned from an expensve bottle of wne 9

12 The Optmal ne A Study n Desgn Optmzaton Functons A amount sold cost G cost of purchasng a grape of type D M cost of manufacturng process Desreablty of a wne ualty characterstc K L x characterstc x of agregated grape purchase % of volume lost on process P proft Q ualty of characterstc (formed by a partcular manufacturng process R revenue OBJETVE FUNTON AND ONSTRANTS MAX P Maxmze proft here P A D Q G M ( R G ( u * K x, = = K c = ς + = G * u = M R = R( D = D L L = V * m, D * m G = Q m, K * A V L (, Q ~ + M Proft s the revenue mnus the cost multpled by the amount sold. Amount sold s eual to the volume of lud collected from the grapes mnus the amount loss n makng the wne. ost s the sum of the cost of purchasng the grapes and manufacturng them. Grape cost s gven by the purchase prce tmes by the proporton bought. Manufacturng cost s determned by the cost of the process multpled by the amount of that process used. Revenue s a functon of the desrablty of the ualty characterstcs and ther relatve mportance. The desrablty of a partcular ualty characterstc s determned by the ualty attaned and the target ualty. Qualty characterstcs are a functon of the relevant process and the characterstcs of the grapes. The characterstcs of the grapes to be processed are determned by the proportons of the grapes purchased. The volume lost n a process s determned by the amount 0

13 Unversty of Mchgan S.T. h u = of that process utlzed. The sum of the proportons purchased of each grape must sum to 00%. g u 0 The proporton purchased of any grape can not be less than 0%. g g G u χ - m 0 0 The amount grape purchased must not exceed a prespecfed amount. Negatve manufacturng s not permtted. g g 4 5 m L χ M 0 0 The amount of processng used can not exceed a prespecfed amount. Loss s a percentage and can not exceed 00%. g 6 L 0 Loss a percentage and can not be less than 0%. g 7 χ 0 The cost of producng the wne must not be hgher than a pre-specfed amount. g m 0 The wne must age for at least half a year A varety of specfc euatons can be used n the model above. Specfcally the way revenue, desrablty, and ualty characterstcs come together depend on the functons selected. The next secton wll descrbe the functons n greater detal. Specfyng the Euatons PURHASE OF GRAPES neres purchase grapes f they do not own vneyards or f they need to complment ther own producton. The grape prce s determned by a varety of characterstcs. For a gven varetal (.e. hardonnay, Pnot Nor, Shraz, etc. prce s most nfluenced by the sugar content and the ualty of the grape. Modern technology allows wneres to chemcally analyze the grape for varous ualty measures before makng a purchase but tradtonally the color of the grape was used as a proxy for the desred ualty. Many small wneres stll rely on color to negotate the prce to be pad to the grape grower. n our model, color wll be a subectve measure that reflects the ualty whch n turn nfluences the prce of the grape and the agng characterstcs of the wne. The followng notaton descrbes the characterstcs of the grapes G x K, represents a characterstc x of grape. The model wll use three types of grapes ( ranges from to. There are three characterstcs ( x ranges from to where s the volume of lud as a % of the total volume, s the amount of sugar measured n the Brx scale, and s the color measured on a relatve scale where 00 s deal and each ncrement or decrease of one unt represents an undesred devaton from the deal color by one percent. The color scale s arbtrary and specfc to the wne beng created.

14 The Optmal ne A Study n Desgn Optmzaton n addton to the grape characterstcs, the costs of the grapes are also fed nto the model as parameters descrbed by the followng notaton G represents the market prce of grape. Ths partcular model does not deal wth the uantty of wne beng produced thus the purchasng decson s not descrbed as a uantty of each type of grape but rather asks what s the proporton of the total purchase that should be taken by each type of grape u represents the percentage of the total grape purchase by volume made up of grape. PROESSNG THE GRAPES NTO NE Many more processes are reured to make wne than what are ncorporated n ths model. The wnemaker has flexblty n changng many characterstcs of appearance, aroma, flavor, texture, and aftertaste. To choose what processes to model we followed the followng crtera The process had to contrbute to a uantfable wne characterstc that would nfluence the prce of the wne n the market. The resultng change n wne characterstcs as a result of the process served to better serve a target segment for the wne rather than target a dfferent segment altogether. More smply, the process must not change the type of wne completely. The process s controllable and adustable based on the wnemaker s decsons. The manufacturng steps selected for ths model wth ther respectve varables are shown below m Fermentaton m Malolactc Fermentaton m Oak Barrel Agng m 4 Steel and Bottle Agng Qualty haracterstcs Below are the euatons that wll be used n the optmzaton model. For a descrpton on how the model was developed and what each euaton means, see Appendx Developng the Model. Parameters K K K = Volume of agregate grape purchase = grape sugar content n Brx Scale = grape color/ualty characterstc Defntons The nomenclature used n ths secton serves only to smplfy the notaton of the euatons shown on the next page. The symbols x, z, f, g, α, w, ϖ, andη are ntermedate calculatons for ths secton only.

15 Unversty of Mchgan x = 4 ( K * K z = ( m f ( x = ( x + (5. 67* 0 g( x,z = zf ( x = alcohol level α = ( z f ( x h( x, z =. h( α = (6. 0* 0 ϖ = (( K -00/ 5 00 x -(. * x α + (. 09* 0 + (4. * 0 5 α -6 x ( 6. * 0 α + ( 5. * 0-8 α 4 η = m m ϖ Qualty haracterstc Model Euatons The ualty characterstcs are functons of the number of days that the n a partcular manufacturng process (expressed by the varables. The number of days the grape uce s allowed to ferment m affects the sweetness and the alcohol level. Malolactc fermentaton determnes acdty and smoothness. Oak barrel agng affects the oak flavor of the wne. Frutness and nut/wood flavors are nfluenced by the total age of the wne. The characterstcs of the grape blend purchased also play an mportant role n the ualty of the resultng wne. Sweeter grapes lead to a potentally sweeter or more alcoholc wne. Hgher ualty grapes provde the potental for fruter and more complex wnes wth better nut and wood flavors. The euatons for these relatonshps s gven below. For a more detaled dscusson on how these euatons were derved as well as ther graphcal representatons, see Appendx Developng the ne Formulaton Model. Sweetness Q = Q ( m, K h( x, z = Alcohol ntensty Q = Q ( m, K = g( x, z Acdty Q = Q ( m = ( m / Smoothness Q 4 0. = Q4 ( m = ( m /

16 The Optmal ne A Study n Desgn Optmzaton 4 ( ( ( ( ( + + = = = = + = = , ( , ( 0 0 ( m m. e e K m Q Q m - m -. K m Q Q *m. - m Q Q η η ϖ Nut/ood Flavor + e Frutness Oak m FNDNG REVENUES Revenues are a functon of the desrablty of each characterstc of the wne. The desrablty of a partcular characterstc and the resultng revenue are gven by the followng euatons ( ( = + = R D D R Q Q Q D * *, ( 0.98( 6(0.5 4(, ( ς Summary Model Below s the mathematcal model used to obtan the optmal wne formulaton. The prevous secton descrbes these euatons n greater detal. Appendx descrbes how the model was developed. MAX P Proft here ( A R P * = (Revenue ost*amount = L K A Amount s determned by the ntal lud tmes the percent lost durng manufacturng. + + = M G c ς 4 ost s the sum of the cost of the grape purchase, the cost of manufacturng, and a constant. G u G * = 5 ost of purchasng grapes. M m M * = 6 ost of manufacturng.

17 Unversty of Mchgan D ( D D R 7 R ( D, = ς * * Revenue s the weghted average of the desrabltes tmes a constant. 8 9 D Q ( = Q, Q m = 4( Q, K ~ = G ( u K + 6(0.5 ( 0.98( Q The desrablty of the attaned ualty s based on the target ntensty. The euatons for the ualty characterstcs Q to Q are gven below. ( 7 0 K The characterstcs of the x * x, grape blend are determned by the characterstcs of the grapes purchased. L L = V * m Amount lost durng manufacturng ntermedate alculatons for Q. ( x, z, f, g, α, w, andϖ apply to ths secton only x = 4 ( K * K z = ( m f ( x = ( x + (5. 67* 0 g( x,z = zf ( x = alcohol level α = ( z f ( x h( x, z =. h( α = (6. 0* 0 ϖ = (( K -00/ 5 00 x -(. * x α + (. 09* 0 + (4. * x 4 α - ( 6. * 0-6 α + ( 5. * 0-8 α 4 η = m m ϖ 5

18 The Optmal ne A Study n Desgn Optmzaton Euatons for Qualty haracterstcs Sweetness Q ( m, K h( x, z = Alcohol ntensty Q ( m, K = g( x, z Acdty Q = ( m 0. / Smoothness Q 4 = ( m 0. / Oak Q 5 = *m + Frutness Q 6 =. - m m ϖ + e ( 0.( m -70 Nut/ood Flavor Q 7 = η + e η ( ( ( m m e 8 65 S.T. h u = Sum of the percentages purchased of each type of grape must eual 00%. g u 0 The proporton purchased of any partcular grape must G g u χ 0 be between 0% and a specfed maxmum. g - m 0 The number of days the grape s processed n each M g4 m χ 0 step must be greater than 0 and less than a specfed maxmum. g5 L 0 The amount of wne lost must be between 0% and g6 L 0 00%. 6

19 Unversty of Mchgan g m 0 The wne must age at least half a year before beng sold. g 8 χ 0 The wne cost can not be hgher than a gven amount. (Ths constrant s not used when obtanng the optmal wne. Optmzng the ne Formulaton Matlab and fmncon was used to obtan the optmal soluton. A soluton was obtaned n less than a second on a moderately powerful P. As wll be shown below, ths rapd soluton s due to the nature of the euatons n the regon or reasonable solutons. More nformaton on the specfc parameters selected can be found n Appendx. A SOLUTON TO THE OPTMAL NE FORMULATON Qualty Measures Target ntensty Relatve mportance Attaned ntensty Resultng Desrablty Sugar Alcohol Acd Smooth Oak Frut Nut Tannn Grape Purchase Grape ( u Grape ( u Grape ( u Manufacturng Days Fermentaton ML Fermentaton Oak Barrel Agng Steel Barrel Agng ( m ( m ( m ( m ost and Revenue Proft Revenue Total ost Purchase ost Manufacturng ost Amount

20 The Optmal ne A Study n Desgn Optmzaton ANALYZNG THE SOLUTON Ths functon s acheves a sngle maxmum pont f the ualty attaned matches the target ualty. As the actual ualty devates from the target, the obectve functon s penalzed. Specfcally, the revenue obtaned from a partcular wne formulaton ncreased as the actual ualty measures approach the target. The costs of buyng the grapes and the costs of manufacturng provde the counterweght n the obectve functon makng t so that achevng the targets for each ualty characterstc not necessarly the most proftable soluton. Gven the nature of the desrablty functon, hardly any constrants are actve at the optmum. Two constrant where actve wth the parameters selected the constrant that specfes that the sum of the proportons of grapes purchased must eual 00% ( and the constrant that specfes that we can not sell grapes ( g. The frst actve constrant s an eualty so t s actve by defnton. As long as there s one grape type that s less desrable than the other two, g wll be actve. Other constrants could become actve wth other parameters but do not become actve f the parameters chosen are normal n the eyes of a wnemaker. The problem can be dvded nto parts wth can be smpler to solve and provde nsghts nto how the model works. The varable m representng the number of days n ML fermentaton determnes smoothness and acdty whle havng ust a very mnor effect on frutness and nut flavor. Snce smoothness and acdty are only determned by ths one varable, a sub problem could be formed that balances the costs and benefts of achevng a partcular level of each of these characterstcs. f we temporarly gnore the effects on frutness and nut flavor, the value of m at the optmal would be gven by solvng the followng euaton D smoothness smoothness smoothness acdty acdty acdty smoothness + h ( m M ( (, Q + D (, Q = M m As long as the daly cost of storng wne n steel barrels or n ther bottles s less than the cost of processng the wne through malolactc fermentaton, the model wll avod ncreasng m to ncrease frutness or nut flavor. Thus, for reasonable cost parameters, the euaton above provdes the optmal value for the number of days of ML fermentaton. These varables can then be elmnated from the larger optmzaton problem and solved separately. Smlar analyses can be used to obtan the optmal values for the remanng acdty varables; however, there are now nteractons wth the characterstcs of the grapes purchased. For example, as m ncreases, the level of alcohol also ncreases, and the level of sugar drops. However, both the level of sugar n the wne and alcohol s hgher as the level of sugar n purchased grapes ncreases. Solvng for these varables ndependently s no easer than solvng the whole optmzaton. n the case of sugar and alcohol, t s useful to look at the monotonctes of the functons. Assumng prces stay fxed, proft ncreases monotoncally wth the sweetness of the grapes. For a gven grape sweetness (n the range of grape sweetness attanable n the real world, the desrablty of the sweetness decreases monotoncally wth the number of days n fermentaton whle the desrablty of the alcohol level ncreases monotoncally wth ths varable. Ths relatonshp holds because the target sweetness s hgher than what s attanable wthout producng a wne wth an unacceptably low alcohol level. f we temporarly gnore the costs ncurred wth fermentaton, the optmal m would be that that sets the contrbuton to revenue from the level of alcohol eual to the contrbuton to revenue obtaned from the wne s sweetness. Snce the cost to ferment the wne s not zero, both the alcohol level and the sweetness wll be under the target. The optmal soluton presented prevously support ths analyss. Steel and bottle agng s the least expensve process when measured by day. Snce the frutness and nut flavors gan complexty over tme, m 4 wll be the least expensve way of agng the wne after t acheves the optmal oak flavor. After the bottle s sold t wll contnue to age untl t s purchased and m 8

21 Unversty of Mchgan opened by a consumer. The model could easly be extended to ncorporate the amount of tme the bottle should be kept before drnkng. Ths could be done by addng another varable that represents storage wth no cost (as seen by the wnery. Dependng on the preference of frut over nut flavors, the bottle may be stored from a few weeks to a few years. n s ute ntutve that better grapes make better wne but come at a hgher prce. However, snce there are three characterstcs to each grape, s not ntutve why one grape s selected over another. The table below descrbes the grapes avalable for purchase n ths scenaro Grape haracterstcs Grape Type Grape Type Grape Type Volume 85.% 96.0% 98.7% Sugar (Brx olor/qualty Prce (per uantty to make bottle $4. $.40 $.09 Grapes of type are the most expensve due to ther exceptonal ualty. However, they are not as sweet or ucy as grapes of type. Grapes of type are nexpensve and very ucy but are nether very sweet or of a hgh ualty. The model determned that the optmal wne would be made up of a blend of 8% of type and 8% of type. Type was completely reected, actvatng the mnmum constrant specfyng that no less than a 0% purchase s allowable. So what would t take for type to be used n an optmal formulaton? The table below provdes the change n the grapes characterstcs, and the resultng change n the blend. hange n Grape Type % Type % Type % Type Orgnal Answer 8.4% 8.57% 0% ncrease ualty to % 7.9% 7.% ncrease sugar level to 7.7.5% 70.% 7.7% Decrease prce to 0 No hange The results above show that the purchase of grape type s most dependent on ts ualty. Gven the current characterstcs, the grape would not be purchased at any prce. The sugar content would have to be ncreased way above the norm for chardonnay grapes to select these grapes. f the ualty were ncreased to 66, a number below that of both other types of grapes, t would stll represent a worth whle purchase. 9

22 The Optmal ne A Study n Desgn Optmzaton Sub-system optmzaton Bottle Desgn Descrpton hen a consumer buys a spaghett sauce, they brng to ths purchase a set of preferences for sauces that help them make a purchasng decson. There are many brands of sauces wth smlar prces, so t may seem dffcult for a consumer to choose among the dfferent optons avalable. Fortunately, spaghett sauce packagng offers the consumers many clues that help them fnd a sauce they lke. The consumer can see the sauce clearly through the glass contaner to determne f t s chunky or smooth. They can read the name and ngredents of the sauce,.e. Trple Mushroom, to determne f the sauce contans flavors that they prefer. hen a consumer begns the selecton process for a bottle of whte wne, they have certan preferences that they are lookng to satsfy n ther wne purchase. For example, they may want a wne that goes well wth fsh or they may be buyng a bottle of wne for a barbeue. They may know that they lke sweet wnes and not dry wnes. However, unlke spaghett sauce, whte wne bottles have lttle or no vsual ndcaton of the flavors and texture the consumer wll experence upon drnkng the wne. Typcally the consumer s presented wth an artfully desgned bottle and label that merely state the type and orgn of the grape used n the wne and the year and locaton of manufacture of the wne. n a gven wne prce range, the consumer may have as many choces as they do wth spaghett sauce, but lttle to no gudance n selectng a varety that wll sut ther preferences. onsderng the average wne purchaser may spend only seconds selectng a bottle of wne, there s not much tme to convey flavors and textures to the consumer. th added selecton tme and assstance from a sales employee, a consumer would probably be able to fnd a number of bottles that satsfy ther ntal preferences. However, n a less-than-mnute tme frame, t s necessary to send an mmedate message to a consumer n order to convnce them to purchase a partcular wne. f the message that the wnery attempts to convey n ther packagng s not the same as the message the consumer s nterpretng, the result can be a negatve mpact on not ust a partcular bottle of wne, but the entre wnery. For example, a consumer uckly purchases a bottle of whte wne at the supermarket because they thnk t looks sweet. Upon drnkng the wne, the consumer dscovers the wne s actually very dry. Now that the consumer has plenty of tme to study the wne name and packagng, t s possble that the consumer would remember the brand of wne and wnery negatvely and avod t altogether next tme they make a wne purchase. OBJETVE The purpose of ths subsystem desgn s to utlze the shape of a wne bottle to convey flavor adectves to the consumer. The optmzaton problem begns wth a survey that explores the consumer s nterpretaton of flavors assocated wth pre-defned bottle shapes and proportons. The flavor adectves selected to be explored are Sweet and Fruty, Dry and rsp, and Nutty and Oaky. From the nformaton gathered n ths survey, the model wll produce three bottle shapes that are optmzed to semantcally convey each of the above three flavors, whle satsfyng wne bottle sze constrants. Through ths study, t s hoped that the wne bottles wthn the overall portfolo desgn wll each gve an accurate portrayal of the predomnate flavor of the wne wthn each bottle, addng to consumer satsfacton and adng n the wne selecton process. SELETON OF ADJETVES The selecton of adectves s based on research, prevous knowledge of chardonnays, and a dscusson wth a wne retaler. The fnal three adectval pars were selected from a larger lst of adectves because Sweet and Fruty, Dry and rsp, and Nutty and Oaky are all pars of words that could suggest shapes n the mnd. 0

23 Unversty of Mchgan PREVOUS ORK There are two types of knowledge about obects knowledge stored n the head and knowledge stored n the world. Knowledge n the world s accessble and retrevable, whle knowledge n the head s not readly retrevable and can reure learnng and memorzaton n order to be stored. However, knowledge n the world reures nterpretaton and can be unaesthetc n desgn (Norman, 988. nes have very lttle worded nformaton on the bottles, and use artwork to convey a stylstc message about the wne to the consumer (n-world knowledge. For nformaton about a wne that reures less nterpretaton, t s useful to research wne before headng to the store, or ask an experenced wne shop employee for assstance n selectng a wne (n-head knowledge. Product semantcs are meanngs conveyed through a product s desgn attrbutes to the customer. hether ntentonal or unntentonal, product semantcs add n-world knowledge to a product and nform the customers desgn preferences. t s possble that the semantc meanng the desgner ntends to convey to the consumer s not the meanng that the consumer actually nterprets (Krppendorff, 995. For example, a popular wne label, Turnng Leaf, has a pcture of an autumn leaf on a dark bottle. Although the desgner and wnery may see the label as artstc, eye-catchng, and expensve-lookng, a consumer may nterpret deeper meanngs, such as a flavor that s crsp and cool, lke autumn. f the wne s, n fact, sweet and fruty, the consumer has been confused by the nterpreted product semantcs. t s therefore mportant that the n-world knowledge conveyed by product attrbutes send a clear and ntentonal message. t follows that n recent years the emphass n product desgn has shfted from functonalsm, where form follows functon, to product semantcs, where form follows meanng (Krppendorff, 995. Semantcs, the study of meanng, has been appled to a large range of engneerng desgns, begnnng wth research n Kanse Engneerng n the 950s. Kanse engneerng, and most offshoots n the realm of semantc desgn research, employ adectve pars that evoke opposte meanngs to gauge the customer s response to product attrbutes over a range of values (Nagamach, 995. Ths study wll employ a smplfed verson of kanse research, askng surveyed customers to match a wne bottle shape to a sngle adectve-descrpton, nstead of usng a gradent rankng. Ths study wll contrbute knowledge to the feld of semantcs and optmzaton by lnkng a purely semantc model wth a busness model, the brand portfolo model, and a manufacturng model, the wne formulaton model. Past studes have utlzed semantcs to convey consumer preferences ntegrated wthn a busness-based model. The potental problem wth ths approach s that semantcs s somewhat dscrete n nature f research ndcates that a certan product attrbute conveys a certan meanng that s not a guarantee that a somewhat smlar, but compromsed attrbute conveys a smlar meanng. hen other consderatons from busness are consdered along wth semantc preferences n defnng an attrbute, t s unclear that the orgnal semantc preferences are mantaned n the compromsed attrbute. n ths optmzaton study, semantcs are defned dscretely for a gven adectve wthout compromses from other realms of the model. Although trade-offs are consdered, they occur at a macroscopc level and do not compromse ndvdual bottle shapes. ONSUMER SURVEY A consumer survey nforms the set of ponts ( S, x S y that represents the deal semantc shape of the wne bottle for each flavor adectve. The survey uses a combnaton of three dfferent bottle shapes combned wth three dfferent proportonaltes. The bottle shapes are based on measurements taken from three wne bottles wth markedly dfferent shapes. The proportonaltes are based on the desgner s ntuton for the range of the user s acceptance for unusual bottle shapes. See the appendx for a pcture of the bottle shapes and proportonaltes. The survey s a dscrete choce survey, n whch beta values are calculated for each of the factors at each level, ndcatng maxmum lkelhood for each factor at every level, and probablty of choce for every combnaton of factors and levels. Three dentcal surveys have been constructed, one for each adectval par Sweet and Fruty, Dry and rsp, and Nutty and Oaky. The surveys are orthogonally balanced wth 8 uestons for two factors (shape and proportonalty at three levels each. The dependent a

24 The Optmal ne A Study n Desgn Optmzaton varable s bottle proporton and the ndependent varable s bottle shape. The survey s a full-factoral survey, whch means that any nteracton between the two varables, or factors, wll be accounted for n the fnal analyss of the survey results and wll not affect the outcome. Each of the eghteen uestons n the three surveys asks the consumer to dentfy whch of three bottles contans the wne that s, for example, the most Sweet and Fruty. For a sample page of the survey, please see the appendx. The three surveys were admnstered va the nternet at zoomerang.com. There were full responses to the Sweet and Fruty Survey, 8 full responses to the Dry and rsp survey and 9 full responses to the Nutty and Oaky survey. t s unclear why fewer people responded to the Nutty and Oaky survey. Possble explanatons nclude that t was more dffcult for respondents to understand the flavor adectves Nutty and Oaky. Perhaps the poor response rate was due to the order n whch the surveys were taken the uestonnare becomes ute repettve by the tme the respondent reaches the thrd survey. The survey reveals some nterestng results the bottle shape consumers dentfed most strongly as Sweet and Fruty s the same bottle that consumers dentfed most strongly as Dry and rsp. Ths s bottle shape A wth a beta value of 0.40 for Sweet and Fruty and a beta value of 0.4 for Dry and rsp. onversely, for Nutty and Oaky, consumers preferred ether bottle shape B or euvalently over A, wth beta values of 0.09 for both B and bottle shapes. Smlarly, the proportonalty that consumers dentfed most strongly as Sweet and Fruty s the same proportonalty that consumers dentfed most strongly as Dry and rsp. Ths s the vertcally stretched proportonalty, wth a beta value 0.8 for Sweet and Fruty and 0.9 for Dry and rsp. For both Sweet and Fruty and Dry and rsp, the stretch preference s almost lnear wth respect to the three eually spaced proportonalty ratos tested (0.9.,,.0.9. Once agan conversely, the proportonalty most preferred for Nutty and Oaky wnes s the horzontally stretched bottle, wth a beta value of The strongest beta value seen n the entre survey s the non-preference for vertcally stretched Nutty and Oaky bottles, wth a beta value of n summary below are the three preferred bottles and, on the followng page, the beta values determned from the consumer survey. Sweet and Fruty deal Bottle Dry and rsp deal Bottle Nutty and Oaky deal Bottle or

25 Unversty of Mchgan Sw eet and Fruty Betas Reveal Preference for Vertcally Stretched Proportonalty Sw eet and Fruty Betas Reveal Preference for Shape A Beta 0-0. Horzontal Stretch Normal Vertcal Stretch Beta 0-0. Shape A Shape B Shape Dry and rsp Betas Reveal Preference for Vertcally Stretched Proportonalty Dry and rsp Betas Reveal Preference for Shape A Beta 0-0. Horzontal Stretch Normal Vertcal Stretch Beta 0-0. Shape A Shape B Shape Nutty and Oaky Betas Reveal Preference for Horzontally Stretched Proportonalty Nutty and Oaky Betas Reveal Preference for Shapes B and Beta 0-0. Horzontal Stretch Normal Vertcal Stretch Beta 0-0. Shape A Shape B Shape

26 The Optmal ne A Study n Desgn Optmzaton onclusons from onsumer Survey The preferred proportonaltes for all three flavor adectval pars rested at an extreme of the tested proportonalty range. Therefore, t would be deal to admnster another consumer survey that recentered the proportonalty ranges n order to fnd the maxmum preferred proportonalty wthn the bounds of the surveyed range. t would be nterestng to nvestgate why both Sweet and Fruty and Dry and rsp ndcated the same bottle shape and proportonalty to consumers. A cluster analyss could tell f one group of consumers found the ndcated bottle shape Sweet and Fruty whle another found t Dry and rsp, or f, nstead, the same consumers found that the bottle expressed both adectval pars. The obvous next step n the nvestgaton would be to test the lnk between the words Sweet and Fruty and the flavor of an actual wne that tastes sweet and fruty. The same should be done for Dry and rsp and Nutty and Oaky. Next, the lnk between the actual wne flavor and the bottle shapes should be nvestgated. The combnaton of these three surveys would most lkely reveal an explanaton as to why the two adectval pars ndcated the same bottle shape and proportonalty. Mathematcal Model NOMENLATURE Varables ( x, y =,,... a Set of ponts that descrbe actual bottle shape for gven flavor adectve Parameters ( S, S V x y a =,,... Set of ponts that descrbe desred bottle shape for gven flavor adectve Volume of wne n a standard bottle (n V V V A B D Volume of mnmum amount of ar that must be present n a standard bottle (n Volume of maxmum amount of ar that must be present n a standard bottle (n Volume of dmple n bottom of bottle (n A M Maxmum surface area of wne that can be exposed to ar (n M Maxmum outer radus of the bottle (n nner radus necessary for proper cork fttng (n H T M Maxmum heght of the bottle Thckness of (n glass on sdes of bottle (n Functons V Volume of nteror of bottle (n A X Surface area of wne exposed to ar (n X Maxmum radus of the wne bottle (n H X Heght of the bottle (n 4

27 Unversty of Mchgan OBJETVE FUNTON For a dscrete adectve a, the obectve functon serves to mnmze the devaton between the output bottle shape, represented as a seres of eleven ponts, and a flavor adectve, represented as a seres of eleven ponts. Ths mnmzaton occurs at each x value and y value separately. mnf = ( x Sx + = = ( y S y ONSTRANTS Volume of nteror The nsde of the bottle must be able to hold 45.8 cubc nches (750 mlllters of wne, V, and a mnmum amount of ar, V A, n order to allow for a bt of breathng and proper corkng wthout spllage. That provdes a lower bound to the nteror volume of the bottle. An upper bound s provded by the fact that too much ar n the bottle allows the propertes of the wne to change as t s stored. Ths maxmum ar volume, V B, s currently set at 0.0 cubc nches, based on measurements from actual bottles of wne. The functon V determnes the nteror volume of the bottle. ( V V + V ( V D + V + V B A V + V D 0 0 Surface area of wne to exposed ar Durng storage, t s deal to have the smallest possble surface area of wne exposed to the ar wthn the bottle. The surface area of the wne exposed to the ar wll be constraned to a maxmum surface area, A M. The functon A X calculates the surface area of the wne exposed to the ar. The optmzaton model calculates the fll heght of the bottle, and then estmates the surface area of the wne at ths heght. t makes the assumpton that n a well optmzed bottle, the fll heght wll occur n the neck regon of the bottle, namely between ponts 0 and. To avod nvolved geometry, t assumes that the radus of the wne at the fll heght s eual to the radus of the pont that defnes the bottom of the concal regon n whch the fll heght occurs. Ths s a conservatve and accurate estmate. A X A M 0 ork regon of bottle To create a sense of unty wthn the wne portfolo and make the corkng process easer, all bottles of wne n the portfolo wll have the same top and cork. t s therefore necessary to have all bottle shapes resolve to the same shape n the cork regon. The y-coordnate of the top pont of bottle x,, wll be constraned to produce a cork regon radus of. y = ( y c Realstc wdth and heght e do not want to upset any store owners wth bottles that they cannot dsplay on ther shelves easly. ne bottles n a local wne store have been measured to determne maxmum heght, H M, and maxmum wdth (dameter, M, parameters. H X X H M M 0 0 5

28 The Optmal ne A Study n Desgn Optmzaton Desred slopes between ponts The purpose of the survey s to determne deal bottle desgn based on shape and proportonalty. Ths deal shape and proportonalty most lkely wll not fall wthn the constrants of the model. The goal s to fnd a realstc, or actual shape that most closely emulates the mportant propertes of the desred shape and also conforms to the constrants of the model. Durng ntal model runs, mnmzng the dstance between deal and actual coordnates that defne the bottle shape was not enough to transfer the shape of the bottle from the realm of the desred to the realm of the realstc. Therefore, an addtonal eualty constrant was added to the model. Ths constrant, or rather, set of ten constrants, states that the slope between two adacent ponts on the actual bottle must eual the slope between two adacent ponts on the desred bottle. y x + + y x = S S y( + x( + S S y x =,,...0 DESGN VARABLES AND PARAMETERS Desgn Varables The desgn varables are expressed as a set of eleven ponts that defne the nteror outlne of the wne bottle. The regon of the bottle beng consdered n ths desgn problem s detaled n the fgure below. The very top of the bottle, where the cork s placed, wll reman fxed n the dfferent desgns to allow the same cork to be used n all bottles and create a contnuous brandng feature through the wne portfolo. ork Regon fxed for all Bottle Regon represented as a set of ponts Desred vs. Actual nches Act ual Desred Example of eleven ponts defnng desred bottle shape (parameter and actual bottle shape (varable, once constrants are appled. 6

29 Unversty of Mchgan ne bottles typcally have domed bottoms. t s ntally assumed that all bottle desgns consdered wll have the same dome-shaped bottom. Ths statement s relaxed n the analyss of the model, where the dome sze s changed from a parameter to a varable. Ths dome s defned n the model as V D, representng the volume that the dome subtracts from the nteror volume of the bottle. V D can also be seen as the volume the dome adds to the volume of the wne tself. Parameters ( S, S =,,... x y a Set of ponts that descrbe desred bottle shape for gven flavor adectvalpar Ths parameter s determned from the consumer survey detaled above. V = 750 ml = 45.8 n Volume of wne n a standard bottle (n V A = 0.5n Volume of mnmum amount of ar that must be present n a standard bottle (n Ths parameter was determned from measurements of exstng wne bottles. V B = 0.0 n Volume of maxmum amount of ar that can be present n a standard bottle (n Ths parameter was determned from measurements of exstng wne bottles. V D = 5.0 n Volume that dome n bottom of bottle subtracts from nteror bottle space (n Ths parameter was determned from measurements of exstng wne bottles. t changes from a parameter to a varable n the model analyss. A M = n Maxmum surface area of wne that can be exposed to ar (n Ths parameter was determned from measurements of exstng wne bottles. M =.5 n Maxmum outer radus of the bottle (n Ths parameter was determned from measurements of exstng wne bottles and s relaxed n the model analyss. = 0.4 n nner radus necessary for proper cork fttng (n The value for ths parameter s based on cork and bottle measurements. M H = n Maxmum heght of the bottle (n Ths parameter was determned from measurements of exstng wne bottles. 7

30 The Optmal ne A Study n Desgn Optmzaton T = 0.5 n Thckness of glass on sdes of bottle Ths parameter was determned from measurements of exstng wne bottles. Summary Model MN mnf = ( x Sx + = = ( y S y (n Mnmze the devaton between bottle shape and the deal expresson of flavor n bottle shape. S.T. h c x = T a The top pont of the wne bottle s fxed to a gven cork regon radus. y + y h x x + S = S y( + x( + S S y x =,,... 0 The slopes between actual adacent ponts must be eual to the slopes between desred adacent ponts. g ( V + V g V ( V X g A A M D + V + V 0 B A V + V D 0 0 The space nsde the bottle must nether too bg nor too small. The surface area of wne exposed to ar cannot be too large. g 4 X M 0 The wdth of the bottle must be approprate for shelvng n wne stores. g 5 H X H M 0 The heght of the bottle must be approprate for shelvng n wne stores. FEASBLTY OF MODEL The model has provded feasble answers to a varety of dfferent desred bottle shapes. One such feasble model output s attached n the appendx, as dsplayed on the excel spreadsheet that performed the calculatons. Graphcal dsplays of fnal feasble results for the two bottle shapes dentfed from the consumer survey are ncluded n the Results secton. Model Analyss HOE OF OPTMZATON PAKAGE Excel Solver was selected as the optmzaton package for ths partcular sub-problem. Excel provdes a graphc nterface that allows for easy assessment of the valdty of solutons n ths vsually-based optmzaton problem. Furthermore, because the desred bottle shape provdes an deal startng pont for the optmzaton, the reduced gradent method works effcently at fndng local optma. As the startng pont s moved further from the desred bottle shape, the algorthm s less effectve. HANGNG PARAMETERS TO VARABLES For very tall and thn bottles desgns, such as the deal bottle shape for the Sweet and Fruty and Dry and rsp flavors, there s a conflct between the constrant on maxmum heght and the constrant on mnmum volume, because t s desrable to keep the wdth of the bottle thn, n order to 8

31 Unversty of Mchgan mantan the semantc message of the shape. For tall and thn desgns, t s nterestng to consder the volume of the dome n the bottom of the bottle as a desgn varable rather than a parameter. For the followng tall and thn bottle desgn, Solver cannot fnd a feasble soluton under the current defnton of the model nfeasble Soluton for Tall Thn Bottle --- Dmple Dome as Parameter Soluton Desred St art ng Pont nches nfeasble Soluton for tall and thn bottle f the volume of the dome n the bottom of the bottle s changed from a parameter to a varable, a feasble soluton s acheved, as shown n the fgure below. The addton of the dome as a varable also reures an addtonal constrant that the dome s volume be greater than or eual to zero. n the feasble soluton, the new dome volume constrant s actve, meanng that the volume of the dome s zero. The maxmum wdth constrant s no longer actve, as would be desrable for a tall and thn bottle. The wne surface area constrant has become actve. Feasble Soluton for Tall Thn Bottle - -- Dmple Dome as Varable Soluton Desred St art ng Pont nches Feasble soluton for tall and thn bottle wth actve dmple volume constrant MONOTONTY AND ELL BOUNDEDNESS By nspecton of results from varous optmzaton runs, the model s closely bounded by the upper and lower lmts on the volume of the nteror and also by the constrant on the maxmum heght of the bottle or the maxmum radus of the bottle, dependng on the bottle shape. ntally, the shape of the desred bottle began at x=0 and the model ncluded constrants that all varables must be greater than or eual to 0, n order to ensure the valdty of volume calculatons. 9