A Business Manager for a Grain Distributor Is Asked to Decide

question 13

Essay

A business manager for a grain distributor is asked to decide how many containers of each of two grains to purchase to fill its 1,600-pound capacity warehouse.The table below summarizes the container size,availability,and expected profit per container upon distribution.​ ​
a.Formulate as a linear program with the decision variables representing the number of containers purchased of each grain.Solve for the optimal solution.
b.What would be the optimal solution if you were not allowed to purchase fractional containers?
c.There are three possible results from rounding an LP solution to obtain an integer solution:
(1)The rounded optimal LP solution will be the optimal IP solution.
(2)The rounded optimal LP solution gives a feasible,but not optimal IP solution.
(3)The rounded optimal LP solution is an infeasible IP solution.
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For this problem,(i)round down all fractions; (ii)round up all fractions; and (iii)round off (to the nearest integer)all fractions (Note: Two of these are equivalent.)Which result above (1,2,or 3)occurred under each rounding method?

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Definitions:

Marginal Rate of Substitution (MRS)

The marginal rate of substitution is the rate at which a consumer is willing to substitute one good for another while keeping the utility level constant.

Cobb-Douglas Utility Functions

A mathematical representation of consumer preferences that shows how utility depends on the consumption of different goods, characterized by constant elasticity of substitution.

Pareto Optimal

A state of allocation of resources from which it is impossible to reallocate to make any one individual or preference criterion better off without making at least one individual or preference criterion worse off.

Pure Exchange Economy

An economic model where agents trade existing goods without the production of new goods, focusing on the allocation and distribution of resources.