A jeweler and her apprentice make silver pins and necklaces by hand. Each week they have 80 hours of labor and 36 ounces of silver available. It requires 8 hours of labor and 2 ounces of silver to make a pin, and 10 hours of labor and 6 ounces of silver to make a necklace. Each pin also contains a small gem of some kind. The demand for pins is no more than six per week. A pin earns the jeweler $400 in profit, and a necklace earns $100. The jeweler wants to know how many of each item to make each week in order to maximize profit. a. (2 pts) Formulate an integer programming model for this problem. b. (3 pts) Solve this model using the branch and bound method. Compare this solution with the solution without integer restrictions and indicate if the rounded-down solution would have been optimal.

Suppose that fizzo and pop hop are the only two firms that sell orange soda. the following payoff matrix shows the profit (in millions of dollars) each company will earn depending on whether or not it advertises: pop hopadvertise doesn’t advertisefizzo advertise 10, 10 18, 2doesn’t advertise 2, 18 11, 11for example, the upper right cell shows that if fizzo advertises and pop hop doesn't advertise, fizzo will make a profit of $18 million, and pop hop will make a profit of $2 million. assume this is a simultaneous game and that fizzo and pop hop are both profit-maximizing firms.if fizzo decides to advertise, it will earn a profit if pop hop advertises and a profit if pop hop does not advertise.if fizzo decides not to advertise, it will earn a profit if pop hop advertises and a profit if pop hop does not advertise.if pop hop advertises, fizzo makes a higher profit if it chooses (not to advertise, to .if pop hop doesn't advertise, fizzo makes a higher profit if it chooses (not to advertise, to . suppose that both firms start off not advertising. if the firms act independently, what strategies will they end up choosing? fizzo will choose to advertise and pop hop will choose not to advertise.both firms will choose to advertise.fizzo will choose not to advertise and pop hop will choose to advertise.both firms will choose not to advertise.again, suppose that both firms start off not advertising. if the firms decide to collude, what strategies will they end up choosing? fizzo will choose not to advertise and pop hop will choose to advertise.both firms will choose not to advertise.fizzo will choose to advertise and pop hop will choose not to advertise.both firms will choose to advertise.