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

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

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

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

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 4

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

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

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

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

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

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.

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

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