# Answered Essay: A product can be produced on four different machines. Each Machine has a production cost per unit processed and a

 A product can be produced on four different machines. Each Machine has a production cost per unit processed and a production capacity (Cap). Using excel solver and LP formulate the model and find the least cost schedule on the machines for the production of 3129 units/day. Be sure to have solver produce the sensitivity report. Then discuss the reduced cost for each decision variable and the shadow price for each constraint. Cost/unit Cap/day Machine1 30 1064 Machine2 30 1341 Machine3 24 1646 Machine4 19 929

LP formulation is as follows

Decision variables: Let X1, X2, X3, X4 be the numbe of units produced on Machine 1,2,3,4 respectively

Objective: Min Z = 30X1 + 30X2 + 24X3 + 19X4

s.t.

X1 + X2 + X3 + X4 = 3129

X1 <= 1064

X2 <= 1341

X3 <= 1646

X4 <= 929

X1, X2, X3, X4 >= 0

Solution by Excel Solver (using Excel QM) and Sensitivity report are following

Reduced cost of each decision variable is 0

Shadow price of first two constraints is 0, this is because these are non-binding constraints. So increasing the RHS of these constraints does not have any effect on the objective value.

Shadow prices of constraint 3 and 4 are negative, which means for each unit increase in RHS, the value of objective function decreases by the corresponding amount. For example if RHS of constraint 3 is increased by 1 unit, then objective value decreases by \$ 6

Shadow price of fifth constraint is 30, which means if RHS of this constraint is increased by 1 unit, then objective value increases by \$ 30. So, this is adverse effect on the objective value.

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