> with(plots, display, textplot);
> restart;
> Digits := 6;
                                      6
Formulation of the model

> a := .15;
                                    0.15
> n := 1;
                                      1
> P := proc (r) options operator, arrow; n*(1+a)^((1/100)*r)*(1500-r) end proc;
              (1/100 r)           
r -> n (1 + a)          (1500 - r)
> plot(P(r), r = 0 .. 1500, labels = ["Rebate amount", "Profit"], labeldirections = [horizontal, vertical], labelfont = [TIMES, BOLD, 20]);

> H := proc (r) options operator, arrow; 1500 end proc;
r -> 1500
> plot([P(r), H(r)], r = 0 .. 1500, labels = ["Rebate amount", "Profit"], labeldirections = [horizontal, vertical], labelfont = [TIMES, BOLD, 20]);

Computation of the optimal rebate amount

> diff(P(r), r);
                            (1/100 r)                  (1/100 r)
             0.00139762 1.15          (1500 - r) - 1.15         
> dPdr := diff(P(r), r);
                            (1/100 r)                  (1/100 r)
             0.00139762 1.15          (1500 - r) - 1.15         
> rmax := solve(dPdr = 0, r);
                                   784.498
> P(rmax);
                                   2141.83
Sensitivity Analysis


> restart;
> alpha := alpha;
                                    alpha
> P := (1+alpha)^((1/100)*r)*(1500-r);
                                  (1/100 r)           
                       (1 + alpha)          (1500 - r)
> rstar := solve(diff(P, r) = 0, r);
                         100 (15 ln(1 + alpha) - 1)
                         --------------------------
                               ln(1 + alpha)       
> plot(rstar, alpha = 0.1e-1 .. .5, labels = ["alpha", "optimal rebate"], labeldirections = [horizontal, vertical], labelfont = [TIMES, BOLD, 20]);

> drdalpha := diff(rstar, alpha);
                     1500              100 (15 ln(1 + alpha) - 1)
           ------------------------- - --------------------------
           (1 + alpha) ln(1 + alpha)                2            
                                       ln(1 + alpha)  (1 + alpha)
> drdalpha*alpha/rstar(alpha);
                        1                 //          1500           
        --------------------------------- ||-------------------------
        100 (15 ln(1 + alpha)(alpha) - 1) ||(1 + alpha) ln(1 + alpha)
                                          \\                         

             100 (15 ln(1 + alpha) - 1)\                           \
           - --------------------------| alpha ln(1 + alpha)(alpha)|
                          2            |                           |
             ln(1 + alpha)  (1 + alpha)/                           /
> simplify(%);
                         alpha ln(1 + alpha)(alpha)               
          --------------------------------------------------------
                       2                                          
          ln(1 + alpha)  (1 + alpha) (15 ln(1 + alpha)(alpha) - 1)
> S := simplify(drdalpha*alpha/rstar(alpha));
                         alpha ln(1 + alpha)(alpha)               
          --------------------------------------------------------
                       2                                          
          ln(1 + alpha)  (1 + alpha) (15 ln(1 + alpha)(alpha) - 1)
> plot(S, alpha = 0.1e-1 .. .2, -10 .. 10, labels = ["alpha", "sensitivity"], labeldirections = [horizontal, vertical], labelfont = [TIMES, BOLD, 20]);

> alpha := .15;
                                    0.15
> S;
                                0.8511849245
> rstar(.15);
                                 784.4976373
> alpha := 0.7e-1;
                                    0.07
> S;
                                 64.98243499
> rstar(0.7e-1);
                                 21.99235033
> alpha := .10;
                                    0.10
> S;
                                 2.219987269
> rstar(.10);
                                 450.7941312
> alpha := .12;
                                    0.12
> rstar(.12);
                                 617.6108707
> 
>