Optimal Prices in the Towards a Precise . . . Towards a Precise . . - PowerPoint PPT Presentation

Practical Situation: . . . How to Describe . . . Formulation of the . . . Optimal Prices in the Towards a Precise . . . Towards a Precise . . . Presence of Discounts: Precise Optimization . . . A New Economic Main result Proof: Part I Application of Proof: Part II – Main Idea Title Page Choquet Integrals ◭◭ ◮◮ Hung T. Nguyen 1 , 2 and Vladik Kreinovich 3 ◭ ◮ Page 1 of 12 1 Mathematics, New Mexico State University, USA 2 Economics, Chaing Mai University, Thailand Go Back 3 Computer Science, University of Texas at El Paso Full Screen hunguyen@nmsu.edu, vladik@utep.edu Close Quit

Practical Situation: . . . 1. Practical Situation: Discounts How to Describe . . . Formulation of the . . . • In many real-life situations, there is a “package deal”: Towards a Precise . . . customers who buy two or more items get a discount. Towards a Precise . . . • When planning a trip, a deal including airfare and hotel Precise Optimization . . . costs less than the airfare and hotel on their own. Main result • There are often also special deals when we combine Proof: Part I airfare with car rental. Proof: Part II – Main Idea Title Page • There are deals that cover all three items plus tickets to local attractions, etc. ◭◭ ◮◮ ◭ ◮ • When a customer buys a house, there are often package deals to also buy furniture and/or car. Page 2 of 12 • In fast food restaurants, such group discounts are a Go Back norm. Full Screen • Usually, there are several such deals with different com- Close binations and different discounts. Quit

Practical Situation: . . . 2. How to Describe Discounts in Precise Terms How to Describe . . . Formulation of the . . . • Let us denote, by n , the total number of different items, Towards a Precise . . . and let us denote these items by x 1 , . . . , x n . Towards a Precise . . . • Then, the set of all the items is X = { x 1 , . . . , x n } . Precise Optimization . . . • Let v ( x i ) denote the price of each individual item. Main result • A discount means that for some sets of items S ⊆ X , Proof: Part I the overall price v ( S ) is v ( S ) < � v ( x i ). Proof: Part II – Main Idea x i ∈ S Title Page • For example, if x 1 is the airfare and x 2 is the hotel, then ◭◭ ◮◮ a package deal means that v ( { x 1 , x 2 } ) < v ( x 1 ) + v ( x 2 ). ◭ ◮ • Not all combinations lead to package deals. Page 3 of 12 • For some combinations S , the cheapest price v ( S ) is Go Back exactly the sum � v ( x i ) of individual prices. x i ∈ S Full Screen • We assume that for every set S ⊆ X , we know the Close cheapest price v ( S ) that we have to pay for this set S . Quit

Practical Situation: . . . 3. Natural Assumption: the Larger the Group, the How to Describe . . . Better the Discount Formulation of the . . . Towards a Precise . . . • Natural assumption: the larger the group, the better Towards a Precise . . . the discount. Precise Optimization . . . • Situation: we want to buy items from two sets S and S ′ . Main result • Option: buy them separately and pay the sum Proof: Part I Proof: Part II – Main Idea v ( S ) + v ( S ′ ) . Title Page • Cheaper option: ◭◭ ◮◮ – to buy the whole group S ∪ S ′ – and then ◭ ◮ – buy additionally all the duplicate items S ∩ S ′ . Page 4 of 12 • Formal description: for all sets S and S ′ , we have Go Back Full Screen v ( S ) + v ( S ′ ) ≥ v ( S ∪ S ′ ) + v ( S ∩ S ′ ) . Close Quit

Practical Situation: . . . 4. Formulation of the Practical Problem How to Describe . . . Formulation of the . . . • What if we want to buy several items of each type, i.e., Towards a Precise . . . we want to buy: Towards a Precise . . . • d 1 items of type x 1 , Precise Optimization . . . • d 2 items of type x 2 , etc. Main result • Example: we want to plan a group tour in which: Proof: Part I Proof: Part II – Main Idea • some tourists want to rent car, Title Page • some want to visit certain attractions, etc. ◭◭ ◮◮ • Problem: What is the best way to use all available ◭ ◮ discounts? Page 5 of 12 • Comment: Go Back • prices v ( S ) correspond to sets ( d i = 0 or d i = 1); Full Screen • we want to extend to bags ( multisets ), when Close d i = 0 , 1 , 2 , . . . Quit

Practical Situation: . . . 5. Towards a Precise Formulation: What Is Given How to Describe . . . Formulation of the . . . • Each way to use the discounts consists of: Towards a Precise . . . • selecting discounts – i.e., sets S 1 , . . . , S m – and Towards a Precise . . . • selecting how many times t 1 , . . . , t m we use each of Precise Optimization . . . the discounts. Main result Proof: Part I • Totally, we should get exactly d 1 objects of type x 1 , Proof: Part II – Main Idea d 2 objects of type x 2 , etc. Title Page • Each set S i ⊆ X can be identified with its characteris- ◭◭ ◮◮ tic function, for which: ◭ ◮ • χ S j ( x i ) = 1 if the item x i is in the set S j and Page 6 of 12 • χ S j ( x i ) = 0 otherwise. Go Back • For each selection of sets S i and times t i , the overall Full Screen price is equal to t 1 · v ( S 1 ) + . . . + t m · v ( S m ) . Close Quit

Practical Situation: . . . 6. Towards a Precise Formulation: What We Want How to Describe . . . Formulation of the . . . Let X = { x 1 , . . . , x n } be a finite set. Towards a Precise . . . • By a discount function , we mean v : 2 X → R + 0 that Towards a Precise . . . maps subsets S ⊆ X into non-negative real numbers Precise Optimization . . . s.t. Main result v ( S ) + v ( S ′ ) ≥ v ( S ∪ S ′ ) + v ( S ∩ S ′ ) . Proof: Part I • A task is a tuple d = ( d 1 , . . . , d n ) of natural numbers. Proof: Part II – Main Idea Title Page • A purchasing plan is a pair P = � ( S 1 , . . . , S m ) , ( t 1 , . . . , t m ) � , with S j ⊆ X and t j ∈ Z . ◭◭ ◮◮ • A plan P satisfies the task d if for every x i , we have ◭ ◮ m Page 7 of 12 � d i = t j · χ S j ( x i ) . Go Back j =1 Full Screen • A price v ( P ) of the plan is def Close v ( P ) = t 1 · v ( S 1 ) + . . . + t m · v ( S m ) . Quit

Practical Situation: . . . 7. Precise Optimization Formulation of the Practical How to Describe . . . Problem Formulation of the . . . • Given: a discount function v : 2 X → R + Towards a Precise . . . 0 and a task Towards a Precise . . . d = ( d 1 , . . . , d n ), Precise Optimization . . . • Among: all the purchasing plans Main result P = � ( S 1 , . . . , S m ) , ( t 1 , . . . , t m ) � Proof: Part I Proof: Part II – Main Idea which are consistent with the task d , i.e., for which Title Page m ◭◭ ◮◮ � d i = t j · χ S j ( x i ) , ◭ ◮ j =1 Page 8 of 12 • Find: the purchasing plan P with the smallest price v ( d ) = min Pv ( P ), where Go Back Full Screen v ( P ) = t 1 · v ( S 1 ) + . . . + t m · v ( S m ) . Close Quit

Practical Situation: . . . 8. Main result How to Describe . . . Formulation of the . . . Theorem. For every discount function v and for every Towards a Precise . . . task d , if we order the values d i in the increasing order Towards a Precise . . . d (1) ≤ d (2) ≤ . . . ≤ d ( n ) Precise Optimization . . . and order the items accordingly, then Main result v ( d ) = d (1) · v ( { x (1) , x (2) . . . , x ( n ) } )+ Proof: Part I Proof: Part II – Main Idea ( d (2) − d (1) ) · v ( { x (2) , . . . , x ( n ) } ) + . . . + Title Page ( d ( i ) − d ( i − 1) ) · v ( { x ( i ) , x ( i +1) , . . . , x ( n ) } ) + . . . + ◭◭ ◮◮ ( d ( n ) − d ( n − 1) ) · v ( { x ( n ) } ) . ◭ ◮ Comments. Page 9 of 12 • The above expression is exactly (discrete) Choquet in- Go Back tegral. Thus, the optimal price is the Choquet integral. Full Screen • There are other economic applications of Choquet in- Close tegral – to decision making. Quit

Practical Situation: . . . 9. Proof: Part I How to Describe . . . Formulation of the . . . v ( d ) = d (1) · v ( { x (1) , x (2) . . . , x ( n ) } )+ Towards a Precise . . . ( d (2) − d (1) ) · v ( { x (2) , . . . , x ( n ) } ) + . . . + Towards a Precise . . . ( d ( i ) − d ( i − 1) ) · v ( { x ( i ) , x ( i +1) , . . . , x ( n ) } ) + . . . + Precise Optimization . . . Main result ( d ( n ) − d ( n − 1) ) · v ( { x ( n ) } ) . Proof: Part I This value is attained when we take the following purchas- Proof: Part II – Main Idea ing plan P 0 : Title Page • d (1) copies of the set { x (1) , x (2) . . . , x ( n ) } , ◭◭ ◮◮ • d (2) − d (1) copies of the set { x (2) , . . . , x ( n ) } , ◭ ◮ • . . . , Page 10 of 12 • d ( i ) − d ( i − 1) copies of the set { x ( i ) , x ( i +1) , . . . , x ( n ) } , Go Back • . . . , and Full Screen Close • d ( n ) − d ( n − 1) copies of the set { x ( n ) } . Quit