1. Maximization of Production Output Subject to a Cost Constraint Appendix 7A
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 MPL = CL
LK: Q/K - CK• = 0 MPK = CK
L: C - CL•L - CK•K = 0
Solution MPL/ MPK = CL / CK
Or rearranged: MPK / r = MPL /w }
2005 South-Western Publishing 18

2. 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

3. 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 2

4. 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. 3

5. 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.
* or ForeProfit at : 4

6. 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. 5

7. TWO DIMENSIONAL LINEAR PROGRAMMING
Corner Points
A, B, and C X1
CONSTRAINT # 1 A B
Feasible
Region
Is OABC
CONSTRAINT
# 2 C O X2 6

8. Greatest Output B X1 CONSTRAINT # 1 Optimal Feasible Solution at
Point B A B
CONSTRAINT
# 2 C O X2 7

9. 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 B 8 P3 5 K

10. 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. 10

11. 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.

12. 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) P2 8 B P3 5 K
Figure 7B.3

13. 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. 12