Application of quasi-integer programming to the solution of menu planning problems with variable portion size

This paper presents the application of a modified mixed-integer programming algorithm to plan menus in which the portion size of the menu items can vary over a specified positive range. Previous mathematical programming formulations of menu planning problems have either required the variables representing menu items to be bivalent, or have formulated the problem with food groups as decision variables and no integer requirements at all. The former gives rise to a zero-one programming problem and the latter to a "feed-mix" problem. In many instances, a more realistic formulation would require that if a menu item is offered, its portion size must be between a specified upper and lower bound. Although this paper addresses itself chiefly to menu planning, it is readily seen that problems in capital budgeting may be tractable with a similar formulation. It is shown how a branch-and-bound algorithm for mixed-integer programming of the type proposed by Beale and Tomlin can be modified to solve the quasi-integer programming problem resulting from a variable portion size formulation of menu planning problems.