Difference: SurfboardProduction (26 vs. 27)

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He has consulted a market research firm about the effect of the price he charges on the demand for the surfboards. They have provided him with the following information. If the price of a Malibu board is and the price of a Super Thruster board is , then the demands for the two types of boards ( and respectively) are given by:

The manufacturer would like to know how much he should charge for his boards and which boards to make in order to maximize his return (the money he makes) from board sales. | |*FORM FIELD ProblemFormulation*|ProblemFormulation|To formulate this problem we will use the 4 steps for formulating a mathematical programme.

1. Identify the Decision Variables

The manufacturer has two decisions to make:

1. the price to charge for the surfboards;
2. how many surfboards to make.

Define the set of types of surfboards to make and then define the decision variables:

2. Formulate the Objective Function

The manufacturer wants to maximise his return from selling the surfboards. He does not want to have any left over as that is a waste of resin, so he will not make enough boards to satisfy demand. Thus, he assumes he will sell all the boards he makes. His return will be the number of boards of each type he makes multiplied by the price he chooses for that type of board:

3. Formulate the Constraints

The manufacturer only has 72 kilograms of resin to make his boards. Define a set of raw materials and as the upper limit of each raw material available (for this example and . Also, define as the amount of raw material required (in kg) to make a surfboard of type . The manufacturing constraints are:

For this example

for resin.

Furthermore, he does not believe he will sell more than the estimated demand, i.e., , where is defined by the market research equations. Thus, he won't make more than those estimates.

Finally, he cannot sell a negative number of boards, i.e., and he can only sell an integer number of boards, i.e., .

The complete formulation is:

where and are the coefficient and exponent of the demand expression for surfboard respectively. | |*FORM FIELD ComputationalModel*|ComputationalModel|To build our mathematical programme in AMPL we first define the sets for surfboards and raw materials in surfboard.mod:

set SURFBOARDS;
set MATERIALS;

Next, we add the parameters for surfboards to surfboard.mod:

param Coefficient {SURFBOARDS};
param Exponent {SURFBOARDS};

As well as the parameters for raw materials:

param Supply {MATERIALS};

param Recipe {MATERIALS, SURFBOARDS};

Our final step, before implementing our formulation in surfboard.mod is to define the decision variables:

var Price {SURFBOARDS};
var Production {SURFBOARDS} >= 0, integer;

Now the formulation is straightforward:

maximize TotalProfit:
  sum {s in SURFBOARDS} Price[s] * Production[s];

subject to MaterialSupply {m in MATERIALS} :
  sum {s in SURFBOARDS} Recipe[m, s] * Production[s] <= Supply[m];

subject to SurfboardDemand {s in SURFBOARDS} :
  Production[s] <= Coefficient[s] * (Price[s] ^ Exponent[s]);

Note Both the objective and the SurfboardDemand are nonlinear expressions.

The formulation data is specified in surfboard.dat. The sets are straightforward (note the use of ' to define the whitespace in Super Thruster):

set SURFBOARDS := Malibu 'Super Thruster' ;
set MATERIALS  := Resin ;

param:           Coefficient Exponent :=
 Malibu                12000     -1.5
'Super Thruster'       12000     -1.8 ;

Using a list to define the Supply parameter and a table to define the Recipe parameter completes surfboard.dat:

param Supply :=
Resin     72 ;

param Recipe :
      Malibu 'Super Thruster' :=
Resin      3               4 ;

Since our mathematical programme is a nonlinear programme we cannot use CPLEX to solve our problem. In (the Student Edition) of AMPL we can use MINOS for nonlinear programming. The script file for solving this problem would typically be similar to:

reset;

model surfboard.mod;
data surfboard.dat;

option solver minos;
solve;

display Price, Production;

However, if you run this script file you will observe the following error:

This error occurs because the initial values for Price (and Production) are set to 0. Both demand estimation functions have negative exponents, so are undefined for a value of 0, i.e.,

both of which are undefined.

We can set the initial values of the Price variables to be non-zero:

let Price['Malibu'] := 1;
let Price['Super Thruster'] := 1;

Note that MINOS does not actually solve for integer variables:

MINOS 5.5: ignoring integrality of 2 variables

MINOS also terminated before determining optimality

MINOS 5.5: too many major iterations.

By default, the number of major iterations allowed by MINOS is 50. If you increase the number of major iterations by changing the MINOS options:

option minos_options 'major_iterations=100';

|

NEOS Optimization Server

The nonlinearity in this problem is introduced by the price variables. If these variables are fixed we can use CPLEX to find the optimal production levels. By looping over possible pairs of values for the prices and solving for the production levels we can build up a picture of the objective function.

param Objective {1..150, 1..150};

fix Price;

for {p in 1..150 by 1, q in 1..150 by 1} {
  let Price['Malibu'] := p;
  let Price['Super Thruster'] := q;
  
  option solver cplex;
  solve;

  let Objective[p, q] := TotalProfit;
}

Using this picture we can try to discern if the optimal solution we found using MINLP (shown below) is the global optimum.

:                   Price   Production    :=
Malibu             100           12
'Super Thruster'    58.1427       8

Hint This for loop may take a long time to finish. Also, you may want to print the TotalProfit parameter to a file using AMPL. | |*FORM FIELD StudentTasks*|StudentTasks|

1. Try different initial values for both Price and Production. Do you believe that the MINLP solution is the global optimum?

   What to hand in Hand in a summary of your different experiments along with an answer to the global optimality question. Be sure to justify your answer.

2. Experts Only Use AMPL and CPLEX to create a picture of the objective function. Does the solution we found using MINLP look like the global optimum?

   What to hand in Your picture of the objective function. Your answer to the global optimality questions. Be sure to justify your answer.