Maximization of Production Output
Subject to a Cost Constraint
Appendix 6A
• Max output (Q) subject to a cost constraint. Let
CK be the cost of capital and CL the cost of labor
• Max L = Q -{CL•L + CK•K - C}
LL: Q/L - CL• = 0
LK: Q/K - CK• = 0
L: C - CL•L - CK•K = 0
}
MPL = CL
MPK = CK
• Solution MPL/ MPK = CL / CK
• Or rearranged:
2005 South-Western Publishing
MPK / r = MPL /w
Slide 1
Production Decisions and Linear Programming
Appendix 7B
• Manufacturers have alternative production processes,
some involving mostly labor, others using machinery
more intensively.
• The objective is to maximize output from these
production processes, given constraints on the inputs
available, such as plant capacity or union labor
contract constraints.
• A number of business problems have inequality
constraints, as in a machine cannot work more than 24
hours in a day. Linear programming works for these
types of constraints
Slide 2
Using Linear Programming
• Constraints of production capacity, time,
money, raw materials, budget, space, and
other restrictions on choices.
• These constraints can be viewed as
inequality constraints,  or .
• A "linear" programming problem assumes a
linear objective function, and a series of
linear inequality constraints
Slide 3
Linearity implies:
1.
constant prices for outputs (as in a
perfectly competitive market).
2.
constant returns to scale for
production processes.
3.
Typically, each decision variable
also has a non-negativity constraint. For
example, the time spent using a machine
cannot be negative.
Slide 4
Solution Methods
• Linear programming problems can be solved
using graphical techniques, SIMPLEX
algorithms using matrices, or using software,
such as Lindo or ForeProfit software*.
• In the graphical technique, each inequality constraint
is graphed as an equality constraint. The Feasible
Solution Space is the area which satisfies all of the
inequality constraints.
• The Optimal Feasible Solution occurs along the
boundary of the Feasible Solution Space, at the extreme
points or corner points.
*www.lindo.com or ForeProfit at :
www.swlearning.com/economics/mcguigan/learning_resources.html
Slide 5
• The corner point that maximize the objective
function is the Optimal Feasible Solution.
• There may be several optimal solutions. Examination of
the slope of the objective function and the slopes of the
constraints is useful in determining which is the optimal
corner point.
• One or more of the constraints may be slack, which
means it is not binding.
• Each constraint has an implicit price, the shadow price of
the constraint. If a constraint is slack, its shadow price is
zero.
• Each shadow price has much the same meaning as a
Lagrangian multiplier.
Slide 6
TWO DIMENSIONAL LINEAR PROGRAMMING
Corner Points
A, B, and C
X1
CONSTRAINT # 1
A
Feasible
Region
Is OABC
O
B
CONSTRAINT
#2
C
X2
Slide 7
X1
CONSTRAINT # 1
Greatest
Output
Optimal Feasible
Solution at
Point B
A
B
CONSTRAINT
#2
O
C
X2
Slide 8
Process Rays: Figure 7B.3
• Each lamp is a different
production process (a
combination of labor & capital)
• P1 requires 1 hour of capital
and 4 hours of labor
• P2 requires 2 hours of capital
and 2 hours of work
• P3 requires 5 hours of capital
and 1 hour of work
• Combinations of labor and
capital produce lamps: Q1 +
Q2 + Q3
• The shaded box is the constraint
on time for L and K
P1
L
P2
8
B
P3
5
K
Slide 9
Maximization Problem
There are three types of lamps produced each day.
There are but 8 hours of labor available a day and
There are only 5 hours of capital machine hours.
Maximize Q1 + Q2 + Q3 subject to:
Q1 + 2·Q2 + 5Q3 < 5 The capital
constraint of 5 hours per day.
4Q1 + 2·Q2 + Q3 < 8 The labor
constraint of 8 hours per day.
where Q1, Q2 and Q3 > 0
Nonnegativity
constraint.
Slide 10
Feasible Region
• If all inputs were used in making process 1,
which takes 4 hours of labor and 1 hour of
machine time, we’d make 2 lamps, but have
slack machine time. This is feasible, but not
optimal.
• At point B on Figure 7B.3, all inputs are
used. It involves some of Process 1 and
some of Process 2.
• Using the two rays, Point B can be reached
by creating a parallelogram of the two rays.
Slide 11
A Parallelogram of Process Rays
• Using 4 hours of labor
and 1 hour capital
makes 1 using process
P1
• Using the remaining 4
hours of labor and 4
hours of machine time
makes 2 lamps using
process P2.
• Solution: 3 lamps (one
of type 1 and 2 of type
2)
P1
L
P2
8
B
P3
K
5
Figure 7B.3
Slide 12
Complexity
and the
Method of Solution
• The solutions to linear programming
problems may be solved graphically, so
long as this involves two dimensions.
• With many products, the solution involves
the SIMPLEX algorithm, or software
available in FOREPROFIT or LINDO.
Slide 13