Difference between profit and contribution in an objective function: Profit is equal to total revenue minus total variable cost minus total fixed cost, which can be modified as total contribution less of total fixed cost. Contribution is equal to total revenue minus total variable cost, which can be modified as total contribution. Difference between profit and contribution in an objective function is inclusion (non-inclusion) of fixed cost.

Objective function:
Profit = Total revenue - total variable cost - total fixed cost
= Unit selling price of product1*qty sold of product 1+...+ Unit selling price of product n*qty sold of product n - Unit variable cost of product1*qty sold of product 1 - ...- Unit variable cost of product n*qty sold of product n - total fixed cost
= contribution margin of product 1*qty sold of product 1 + ...+ contribution margin of product n*qty sold of product n - total fixed cost

Contribution = contribution margin of product 1*qty sold of product 1 + ...+ contribution margin of product n*qty sold of product n

Why is it important for the decision maker to know which of these the objective function?
Above highlighted difference clearly shows the importance of knowing what objective function is as it indicates the scope of the problem. Typically, profit is considered as objective function when investment cost is important (fixed cost) or when various investment alternatives are considered for example make or buy decisions, investment in new facilities/equipments etc. Profit must be an objective function when fixed cost is also a critical factor.

Contribution is the objective function when manager needs to determine what the best is in current situation. Investments are already done and now best product mix needs to be determined. In this case fixed costs are sunk costs and make no difference in the objective function. Fixed costs do not change with change in decisions made unlike profit as an objective function.

Therefore it is important for manager to know what objective function is. In fact, the choice of objective function is determined by nature and scope of the problem under consideration.