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The four possible points which could be the optimal solution are labeled from one to four. The solutions to these are then given in Table 1, along with the profits which would result from these combinations. The values of each of these points were calculated by solving the simultaneous equations where the lines crossed. It can be seen from Table 1 that the maximum profit would be reached by producing 20 beef dinners and 40 fish dinners each day.
Figure 1: Feasible region for the linear programming problem
Table 1: Resultant profits from each of the critical points
Value of X1
Value of X2
Now Excel may also be used to solve this problem. The solution which is given is shown in Figure 2. From this it may be confirmed that the optimal solution for the restaurant is to prepare 20 beef meals and 40 fish meals each night. The sensitivity report is also given for this model in Figure 3.
Figure 2: Computer Solution to the Linear Programming Problem
Figure 3: Sensitivity Output from the Excel Model
It is also possible that these solutions may be used to answer further questions related to the problems. For example, if the profit from the fish dinners was raised to be equal to the profit from the beef dinners this would then change the optimal solution to one in which any point on the line between points 3 and 4 would produce the same profit. The profit overall would also be increased subject to this increase in price not affecting the number of customers overall. Also, if it was determined that at least 20% of customers would want beef dinners, this also would not change the optimal meal preparation, as this would move one of the lines, would but not alter which of the points gave the optimal solution, where more than 20% are beef dinners anyway.
It is also possible to answer the problems presented in the second case study from the model presented. It would not be worth them paying for an advertisement to increase the total number of meals. This is because it may be seen from Figure 1 and Figure 2 that the labor constraint is binding at the present time. This means that they would not have the labor available to increase their meal output in a way which would create more profit.
Also, as the labor is a binding factor on the current optimal solution to the programming problem, this means that a decrease in labor would reduce the profit available. In fact, this would change the parameters of the optimal solution to 10 beef dinners and 40 fish dinners, with only $640 profit rather than $800. Finally, if the profit obtained from the fish meals was to be changed, then the actual solution to the problem would not change, all that would happen is that the profit would be raised to $880. Therefore, Pierre should have no problems with this!
Conclusions and Recommendations
It may be seen from the solutions presented above that the best strategy for the restaurant based on this would be to produce 20 beef dinners and 40 fish dinners at the present time. From the sensitivity analysis it would however seem that the number of dishes which was produced could be reduced by 2 without altering the optimal solution, so it would possibly benefit them under this model to instead produce only 58 meals. Then if there was a greater demand it would serve them well to re-assess the entire model. For example if they found that they had a greater demand for meals then they would in fact need to add to the labor hours before they would be able to increase their profitability.
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