Industrial & Systems Engineering Department

ISE $303$: Operations Research I

Term $232$

IBL Task $-$ Application on Sensitivity and Post$-$Optimal Analyses

Due Date: $15$ May $2024($Submission on Blackboard$)$

A furniture company manufactures four products: chairs, office tables, computer tables, and book shelves.

The manufacturing process involves three different departments: cutting, painting, and assembly. The

company also needs to adhere to supply contracts with two major distributors. They decided to use a Linear

Programming $($LP$)$ model to optimize their production plan. The problem is formulated below considering

the availability limit for each department and the demand requirements for each distributor with the

objective of maximizing the profit of the company.

LP Formulation:

Decision Variables:

$x_1$: Number of chairs produced.

$x_2$ : Number of office tables produced.

$x_3$: Number of computer tables produced.

$x_4$ : Number of book shelves produced.

Objective Function:

Maximize $z=35x_1+30x_2+55x_3+20x_4$

Constraints:

Subject to:

$4x_1+3x_2+5x_3+2x_{4}1750,($Cutting labor constraint$)$

$3x_1+4x_2+2x_3+x_{4}1000,($Painting labor constraint$)$

$2x_1+2x_2+3x_3+3x_{4}1400,($Assembly labor constraint$)$

$2x_1+3x_2+x_3+2x_{4}650($Distributor A contract$)$

$x_1+x_2+2x_3+x_{4}450,($Distributor B contract$)$

$x_1,x_2,x_3,x_{4}0,($Non$-$negativity constraints$)$

The problem is solved using the Big$-$M Method $(M=100)$ and the optimal tableau is given below:

where $s_1,s_2,$ and $s_3$ are the slacks for the Cutting, Painting, and Assembly labors constraints, respectively,

while $s_4$ and $s_5$ are the surplus variables for the Distributor A and Distributor B contracts, respectively. Based on the given formulation and the given optimal tableau, you are required to prepare a detailed report

to answer the following questions:

a$.$ Is the obtained solution a unique optimal solution $($YES$/$NO$)?$ justify your answer? If your answer

is $"$NO$",$ then find an alternative one.

b$.$ Identify which of the five constraints is binding and which is non$-$binding, showing how you

recognize the constraint state.

c$.$ Find the dual price for each of the five constraints.

d$.$ Find the feasibility range for each dual price.

e$.$ Find the optimality range for each coefficient in the objective function.

f$.$ Write a code to solve the above LP using any optimization software $($Lingo$,$ GAMS, Gurobi,

Python, AMPL, etc.$)$

NOTE: The final answers to the above requirements are given below.

g$.$ Given the following Tableau with $B^{-1}$ provided. Complete the tableau and show your work in

detail.

Dual prices and feasibility range for each constraint are as follows:

Optimality range for each coefficient in the objective function are as follows:

please show all steps