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Flair Furniture Company The flair furniture company produces inexpensive tables and chairs. The production process for each is similar in that both require a certain number of hours of carpentry work and a certain number of labor hours in the painting and varnishing department. Each table takes 4 hours of carpentry and 2 hours of painting and varnishing shop. Each chair requires 3 hours of carpentry and 1 hour of painting and varnishing. During the current production period, 240 hours of carpentry time available and 100 hours of painting and varnishing time available. Each table sold with a profit of Rs. 7; each chair produced is sold for a Rs. 5 profit.

Flair Furniture’s problem is to determine the best possible combination of tables and chairs to manufacture in order to reach the maximum profit. The firm would like this production mix situation formulated as an LP problem.

Canonical Form of a Linear Programming Problem.

• The objective function is of maximization type. Maximize z = - minimize (-z).• All the constraints are of the ``’’ type, except for the

non-negative constrains. If the inequality is ``” sign, multiply both side by –1 to

make it ``”. An equation can be replaced by two weak inequalities

in opposite directions. Example : Maximize z = cx subject to Ax b , x 0.

Standard form of a Linear Programming Problem.

• All the constraints are expressed in the form of equation, except for the non-negative restrictions.

• The right hand side of each constraint equation is non-negative.

Example : Optimize z = cx,

subject to Ax = b, x 0.

AssumptionsFor a LP to be an appropriate representation of the real world 4 assumptions must be satisfied

1. Proportionality assumption: Contribution to objective functionContribution to LHS of constraints

2. Additivity assumption: Contribution to objective functionContribution to LHS of constraints

3. Divisibility assumption: Each variable can be allowed to take on fractional values

4. Certainty assumption: Each parameter (objective function coefficient, rhs and technological coefficients) is known with certainty

Proportional to value of decision variable

By any decision variable independent of the values of all other decision variables

If a real-life problem satisfies all of the above assumptions, LP can be used to optimize the problem

Binding & Non-binding constraints

A constraint is binding if the LHS and the RHS of the constraint are equal when the optimal values of the decision variables are substituted into the constraint.

A constraint is non-binding if the LHS and the RHS of the constraint are unequal when the optimal values of the decision variables are substituted into the constraint.

The above imply that an equality constraint (= sign) will always be a binding constraint

Sensitivity Analysis • Sensitivity analysis (or post-optimality analysis) is

used to determine how the optimal solution is affected by changes, within specified ranges, in:– the objective function coefficients– the right-hand side (RHS) values

• Sensitivity analysis is important to the manager who must operate in a dynamic environment with imprecise estimates of the coefficients.

• Sensitivity analysis allows him to ask certain what-if questions about the problem.

The Dual in Linear Programming• In LP the solution for the profit-maximizing

combination of outputs automatically determines the input amounts that must be used in the production process.

• If the optimal output combination, suggested by the LP solution, uses up all available inputs, we say that the capacity constraints are binding.

• Under this condition, any reduction/increase in the use of inputs will reduce/increase the company’s profits.

Right-Hand Sides • Let us consider how a change in the right-hand side for a

constraint might affect the feasible region and perhaps cause a change in the optimal solution.

• The improvement in the value of the optimal solution per unit increase in the right-hand side is called as its Marginal Value.

• Marginal values are referred to as opportunity costs, because a limitation on available resources may result in the loss of opportunity to earn greater profit.

• It also referred to as dual price, shadow price..• The range of feasibility is the range over which the dual

price is applicable.

Dual Price

• Graphically, a dual price is determined by adding +1 to the right hand side value in question and then resolving for the optimal solution in terms of the same two binding constraints.

• The dual price is equal to the difference in the values of the objective functions between the new and original problems.

• The dual price for a nonbinding constraint is 0.• A negative dual price indicates that the objective

function will not improve if the RHS is increased.

The Flair Furniture Company • The primal linear programming problem represents the

perspective of the company’s objective to optimize resource allocation by establishing quantities that will maximize profit.

• The dual linear programming represents a production plan that optimizes resource allocation by ensuring that each product is produced at that quantity such that its marginal opportunity cost equals to marginal return.

• Objective: To deploy all resources optimally so that the aggregate value of increasing any resource (carpentry time or painting time) by one more unit is minimized.

• Constrains: Marginal opportunity cost to produce a table / chair marginal return of a table/chair.

The daily requirement of a patient is 20 and 30 units of vitamins A and B respectively. The food F contains 3 units of A and 4 units of B, the food G contains 2 units of A and 3 units of B. Find the minimum cost of buying the vitamins if the cost per unit of F and G be Rs. 6 and Rs. 5 respectively.

Let x units of F and y units of G be required to get minimum amount of vitamins.

Minimize z = 6x + 5y

Subject to 3x + 2y 20, 4x + 3y 30, x, y 0.

A dealer sells the mentioned vitamins (in the previous slide) A and B separately. His problem is to fix the cost per unit of A and B in such a way that the price of F and G do not exceed the amount of the previous slide problem. His problem is also to get a maximum amount in selling the vitamins.

Let u and v be the price per unit A and B respectively.

Maximize z* = 20u + 30v

Subject to 3u + 4v 6, 2u + 3v 5, u, v 0.

Standard form of primalFor maximization problem constraints is of

``” sign.For Minimization problem constraints is of

``” sign.

The initial problem is Minimize Z = cx subject to Ax b, x 0.The corresponding dual problem is Maximize Z* = b/w subject to A/w c/, w 0.

Rule of Transformation • Co-efficient matrix of the dual is the transpose of

the co-efficient matrix of the primal. • Interchanging of c and b• Changing the direction of the sign of inequality• Minimization (maximization) of the objective

function instead of maximization (minimization).

The initial problem is

Maximize Z = cx subject to Ax b, x 0. The corresponding dual problem is

Minimize Z* = b/w subject to A/w c/, w 0.

Write down the dual..

Maximize z = 3x + 2y

subject to 2x + y 5 x + y 3, x, y 0.

Primal-dual problems

• Symmetric primal-dual problems All constraints of both primal and dual are

inequations and the variables are non-negative.

• Unsymmetric primal-dual problems All constraints of primal are equations and primal

variables are non-negative.

• Mixed type problemSome constraints of primal are equations and some

variables are unrestricted.

Properties

• Dual of dual is primal • If the k-th constraint in a primal is an

equality , then the corresponding dual variable is unrestricted in sign

• If the p-th variable of a primal is unrestricted in sign, then the corresponding p-th constraint of the dual is an equality.

If x* and w* are the feasible solutions to primal and dual, respectively, such that c/x* = b/w*, then x* and w* are optimal solutions respectively.

Fundamental Duality Theorem • If either the primal Maximize z =cx subject to

Ax b, x 0, or the dual, minimize z* = b/w, subject to A/w c/, w 0, has a finite optimal solution, then the other problem will also have a finite optimal solution.

Furthermore, the the optimal values of the objective functions in both the problems will be the same, i.e., Max z = Min z*.

• If either of the problems has an unbounded solution, then the other problem has no feasible solution.

Linear Programming

• Multiple Optimum Solutions• Degeneracy • Reduced Cost

XYZ is a new firm that is engaged in recycling. Its main facility uses a three-step system to process beverages containers. A consultant has developed the following LP model.

Maximize 14Q + 11R + 15T (revenue)Subject to Sorting : 2.4Q + 3.0R + 4.0T ≤ 960 minutes Crushing:2.5Q +1.8R + 2.4T ≤ 607 minutes Packing : 12Q + 18R + 24T ≤ 3600 minutes Q, R and T ≥ 0.

• Which decision variables are in the final solution? What are the optimal values?

• Find the range of optimality for the variables that are in the final solution.

• Find the range of insignificance of R.• By how much would revenue decrease if sorting time reduced

to 900 minutes? How much would revenue increase if sorting time was increased to 1,269 minutes.

• What effect would an increase of Rs. 2 in the revenue per unit of T have on the optimal solution?

• If you could obtain additional quantities of any one resource at no additional cost, and your goal is to achieve the greatest increase in revenue, what resources would you add and how much you would add.

Knitting Department The manager of a knitting department has developed the following LP model and the following Excel output. x : units of Product 1 y: units of product 2 z: units of product 3 maximize Z1 = 7x + 3y + 9z (profit)subject to Labour 4x + 5y + 6z ≤ 360 hours Machine 2x + 4y + 6z ≤ 300 hours Material 9x + 5y + 6z ≤ 600 pounds x, y, z ≥ 0

Questions

• Why is not product 2 called in for optimal solutions? How much would the pre-unit profit of product 2 have to be in order for it to enter into the optimal solution mix?

• What is the range of optimality for the profit per unit of Product 1?

• What would be the values in the optimal solution be if the objective co-efficient of x were increased by 3?

Questions • What is the range of feasibility of the labour constraint? • What would be the values in the optimal solution be if the

amount of labour available decreased by 10 hours? • If the manager could obtain additional material, how much

more could be used effectively? What would happen if the manager obtains more than this amount?

• If it is possible to obtain an additional amount of one of the resources, which one should be obtained, and how much can be effectively used? Explain.

Questions • If the manager is able to obtain an additional 100

pounds of material at the usual price, what impact would that have on the optimal value in the objective function?

• If the manager is able to obtain an additional 100 pounds of material but has to pay a premium of 5 paisa a pound, what will the net profit be?

• If knitting machine operate 10 hours a day, and one of the machine will be out of service for two and a half days, what impact will this have on the optimal value in the objective function?