topic Help with constraining in Mathematical Optimization, Discrete-Event Simulation, and OR

Help with constraining

Audron — Wed, 10 Jul 2019 00:02:02 GMT

Hello,

I'm trying to solve an elasticity problem in which I'm maximizing revenue such as the example presented here:

http://support.sas.com/documentation/cdl/en/ormpex/66104/HTML/default/viewer.htm#ormpex_ex21_sect009.htm

However, I'm trying to add a constraint so that the price of product A has to be the same price as the price in product B in the output. Which you'll see my attempt in the rack_rate_con constraint on the code.

Here is my current SAS Code:

/*Previous Prievious Price and Demand Data*/
data product_data;
   input product $  prev_demand prev_price;
   datalines;
Land      60  120
Trans     30  120
;
run;


/*Elasticity Data*/
data elasticity_data;
   input i $ j $ elasticity;
   datalines;
Land    Land   -0.85
Trans   Trans   -0.27
;
run;



proc optmodel;
/*   set <str> RAWS;*/
/*   num supply {RAWS};*/
/*   read data raw_material_data into RAWS=[raw] supply;*/

   set <str> PRODUCTS;
   num prev_demand {PRODUCTS};
   num prev_price {PRODUCTS};
/*   num percent {PRODUCTS, RAWS};*/
   read data product_data into PRODUCTS=[product] prev_demand prev_price;
      
   num elasticity {PRODUCTS, PRODUCTS} init 0;
   read data elasticity_data into [i j] elasticity;


 var Price {PRODUCTS} >= 0;

   var Demand {PRODUCTS} >= 0;

   
   max TotalRevenue 
      = sum {product in PRODUCTS} Price[product] * Demand[product];

   con Demand_con {i in PRODUCTS}:
      (Demand[i] - prev_demand[i]) / prev_demand[i] 
    = sum {j in PRODUCTS} elasticity[i,j] * (Price[j] - prev_price[j]) / 
      prev_price[j];


   con Price_index_con:
      sum {product in PRODUCTS} prev_demand[product] * Price[product]
   <= sum {product in PRODUCTS} prev_demand[product] * prev_price[product];

/* Attempting to add constraint to make sure the price of the two recommendations is the same*/ 
   con rack_rate_con {i in PRODUCTS}: 
     sum {PRODUCTS} Price[i] 
   = sum {j in PRODUCTS} Price[j]; 

solve with NLP / algorithm=activeset MULTISTART SEED=7888422;

print Price Demand TotalRevenue; 
print Price_index_con.dual; 
print rack_rate_con.dual; 
run;

I do not error out, but it's not suggesting any changes. However, If I add the FEASTOL=2 option, it starts suggesting rates that are close enough to each other, but the calculated demand is way off when enabling this option.

If I remove the rack_rate_con constraint altogether, the calculated demand looks reasonable.

Any help is appreciated.

Re: Help with constraining

RobPratt — Wed, 10 Jul 2019 01:27:43 GMT

The EXPAND statement shows the following model:

Var Price[Land] >= 0                                                                           
Var Price[Trans] >= 0                                                                          
Var Demand[Land] >= 0                                                                          
Var Demand[Trans] >= 0                                                                         
Maximize TotalRevenue=Price[Land]*Demand[Land] + Price[Trans]*Demand[Trans]                    
Constraint Demand_con[Land]: 0.0166666667*Demand[Land] + 0.0070833333*Price[Land] = 1.85       
Constraint Demand_con[Trans]: 0.0333333333*Demand[Trans] + 0.00225*Price[Trans] = 1.27         
Constraint Price_index_con: 60*Price[Land] + 30*Price[Trans] <= 10800                          
Constraint rack_rate_con[Land]: Price[Land] - Price[Trans] = 0                                 
Constraint rack_rate_con[Trans]: Price[Trans] - Price[Land] = 0

This does seem to do what you want, but notice that the last two constraints are duplicates. A simpler approach to ensure that both products have the same price is to not index the Price variable by product. For example, the variable declaration would be:

   var Price >= 0;

Both approaches yield optimal objective value 10800.

Re: Help with constraining

Audron — Wed, 10 Jul 2019 07:05:38 GMT

I think I got it to work, and here is my new code:

/*Previous Prievious Price and Demand Data*/
data product_data;
   input product $  prev_demand prev_price;
   datalines;
Land      60  120
Trans     30  120
;
run;


/*Elasticity Data*/
data elasticity_data;
   input i $ elasticity;
   datalines;
Land    -0.85
Trans    -0.27
;
run;



proc optmodel;
   set <str> PRODUCTS;
 
   num prev_demand {PRODUCTS};
   num prev_price {PRODUCTS};
   read data product_data into PRODUCTS=[product] prev_demand prev_price;
   num elasticity {PRODUCTS};
   read data elasticity_data into [i] elasticity;

   var Price >= 0;
   var Demand {PRODUCTS} >= 0;


   max TotalRevenue = sum {product in PRODUCTS} Price * Demand[product];

   con Demand_con {i in PRODUCTS}:
      (Demand[i] - prev_demand[i]) / prev_demand[i]  = sum {PRODUCTS} elasticity[i] * (Price - prev_price[i]) / prev_price[i];



   solve with NLP / algorithm=activeset MULTISTART SEED=7888422;
   print Price Demand TotalRevenue; 
   print Demand_con.dual;
   expand;
RUN;