Lindo Systems
MODEL:
! Product bundle pricing (PriceProdSimp);
! Producer wants to choose a price for each of a number of
products, based on knowledge of the "willingness
to pay" (Reservation price) of various market segments
for each of the products. The producer wants to choose
prices so as to maximize revenue minus costs.
You can think of a product as a bundle of features.
If the price of a product/bundle is > than a customer's
reservation price, then the customer will not buy that product.
Each customer, i, will buy the one product/bundle, j,
that maximizes his consumer surplus
( = reservation_price(i,j) minus price i must pay for j.)
Each customer/market segment is described by:
Its size,
Reservation price for each product;
! Keywords: Bundling, Consumer Choice, Equilibrium, Pricing,
Product Management, Marketing, Uncertainty;
SETS:
CUST:
SIZE, ! Each cust/market has a size;
DISC, ! Discount off list price willing to give to I;
DISD, ! Discount given to dealer(who sells full price);
FM, ! Fixed cost of developing market I;
YM, ! = 1 if we develop market I, else 0;
SRP; ! Consumer surplus achieved by customer I;
BUNDLE:
COST, ! Each product/bundle has a cost/unit to producer;
FP, ! Fixed cost of developing product J;
YP, ! = 1 if we develop product J, else 0;
PRICE, ! List price of product J;
PMAX; ! Max price that might be charged;
CXB( CUST, BUNDLE):
RP, ! Reservation price of customer I for product J;
EFP, ! Effective price I pays for J, = 0 if not bought;
X; ! = 1 if I buys J, else 0;
ENDSETS
DATA:
CUST = HOME STUD LEGL BUSN; ! Customer/market segments;
SIZE = 5900 3000 2000 3100; ! Customer/market segment sizes;
! Fixed market development costs;
FM = 0 0 0 0;
! Discount off list price to each customer, 0 <= DISC < 1.
Assumes grey marketers will not buy in cheaper market and
sell in more expensive market;
DISC = 0 0 0 0;
! Discount/tax off list to each dealer, 0 <= DISD < 1;
DISD = 0 0 0 0;
! We sell three basic products: Spreadsheet, Wordprocessor, Presenter software,
as well as all combinations of the three features;
BUNDLE = BS BW BP BSW BSP BWP BSWP; ! Products/bundles of products;
! Reservation/max_willing_to_pay prices of markets for products;
RP = 190 90 50 280 240 140 330 ! Home;
160 180 70 340 230 250 410 ! Student;
50 250 85 300 135 335 385 ! Legal;
300 100 120 400 420 220 520; ! General Business;
! Variable costs of each product;
COST = 100 20 30 120 130 50 150;
! Fixed product development costs;
FP = 0 0 0 0 0 0 0;
ENDDATA
!--------------------------------------------------------;
SUBMODEL BUNPRICE:
! Variables:
X(i,j) = 1 if market segment i buys bundle j.
YM( i) = 1 if market segment i buys a bundle (we market to segment i);
! The seller wants to maximize the profit contribution;
[PROFIT] MAX = OBJ;
OBJ =
@SUM( CXB( I, J):
SIZE( I) * EFP( I, J) ! +Revenue;
- COST( J)* SIZE( I) * X( I, J) ! -Variable cost;
- EFP( I, J) * SIZE( I) * DISD( I))! -Discount to dealers;
- @SUM( BUNDLE: FP * YP) ! -development cost;
- @SUM( CUST: FM * YM); ! -Market development cost;
! Each customer can buy at most 1 bundle;
@FOR( CUST( I):
@SUM( BUNDLE( J) : X( I, J)) <= YM( I);
@BIN( YM( I)); ! Must be 0 or 1;
);
! Force development costs for j to be incurred
if we sell it to any customer i;
@FOR( CXB( I, J):
X( I, J) <= YP( J); ! for product J;
! The X's are binary, yes/no, 1/0 variables;
@BIN( X( I, J));
);
! Compute consumer surplus for customer I;
@FOR( CUST( I):
SRP( I) = @SUM( BUNDLE( J): RP( I, J) * X( I, J) - EFP( I, J));
! Customer chooses maximum consumer surplus;
@FOR( BUNDLE( J):
SRP( I) >= RP( I, J) - ( 1 - DISC( I)) * PRICE( J)
);
);
! Force effective price to take on proper value;
@FOR( CXB( I, J):
! zero if I does not buy J, at most reservation price;
EFP( I, J) <= X( I, J) * RP( I, J);
! cannot be greater than published price less discount;
EFP( I, J) <= ( 1 - DISC( I)) * PRICE( J);
! cannot be less than published price if bought less discount;
EFP( I, J) >= ( 1 - DISC( I))* PRICE( J)
- ( 1 - X( I, J))* PMAX( J);
);
! Compute upper bounds on prices;
@FOR( BUNDLE( J):
PMAX( J) = @MAX( CUST( I): RP( I, J)/(1 - DISC( I)));
);
ENDSUBMODEL
CALC:
@SOLVE( BUNPRICE);
@WRITE(@NEWLINE(1),' Max profit for vendor= ', OBJ, @NEWLINE(1));
@WRITE(@NEWLINE(1),' Profit maximizing price for each bundle:',@NEWLINE(1));
@WRITE(' ');
@FOR( BUNDLE( j):
@WRITE( @FORMAT( BUNDLE( j),'6s'));
);
@WRITE(@NEWLINE(1));
@WRITE(' ');
@FOR( BUNDLE( j):
@WRITE( @FORMAT( PRICE( j),'6.0f'));
);
@WRITE(@NEWLINE(1));
@WRITE(@NEWLINE(1),' Purchase matrix:',@NEWLINE(1));
@WRITE(' ');
@FOR( BUNDLE( j):
@WRITE( @FORMAT( BUNDLE( j),'6s'));
);
@WRITE(@NEWLINE(1));
@FOR( CUST(i):
@WRITE( @FORMAT(CUST(i),'6s'));
@FOR( BUNDLE( j):
@IFC( X(i,j) #GT# 0.5:
@WRITE(' 1 ');
@ELSE
@WRITE(' ');
);
);
@WRITE(@NEWLINE(1));
);
@WRITE(@NEWLINE(1));
@WRITE(@NEWLINE(1),' Consumer surplus, Market segment vs. Product/Bundle:',@NEWLINE(1));
@WRITE(' ');
@FOR( BUNDLE( j):
@WRITE( @FORMAT( BUNDLE( j),'6s'));
);
@WRITE(@NEWLINE(1));
@FOR( CUST(i):
@WRITE( @FORMAT(CUST(i),'6s'));
@FOR( BUNDLE( j):
@WRITE( @FORMAT( RP(i,j) - PRICE( j),'6.0f'));
);
@WRITE(@NEWLINE(1));
);
ENDCALC
END