1 © 2006 by Nelson, a division of Thomson Canada Limited Slides developed by: William Rentz & Al Kahl University of Ottawa Web Appendix 4A Production Decisions and Linear Programming

2 © 2006 by Nelson, a division of Thomson Canada Limited Manufacturers have alternative production processes, some using mostly labour, 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 labour 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 Production Decisions and Linear Programming

3 © 2006 by Nelson, a division of Thomson Canada Limited 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

4 © 2006 by Nelson, a division of Thomson Canada Limited 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. Linearity Implies:

5 © 2006 by Nelson, a division of Thomson Canada Limited Linear programming problems can be solved using graphical techniques, SIMPLEX algorithms using matrices, or using software, such as Lindo*. 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. * Solution Methods

6 © 2006 by Nelson, a division of Thomson Canada Limited The corner point that maximizes 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.

7 © 2006 by Nelson, a division of Thomson Canada Limited TWO DIMENSIONAL LINEAR PROGRAMMING X1X1 X2X2 A B C CONSTRAINT # 1 CONSTRAINT # 2 Corner Points A, B, and C Feasible Region Is OABC O

8 © 2006 by Nelson, a division of Thomson Canada Limited X1X1 X2X2 A B C CONSTRAINT # 1 CONSTRAINT # 2 Optimal Feasible Solution at Point B Greatest Output O

9 © 2006 by Nelson, a division of Thomson Canada Limited Each ray is a different production process (a combination of labour & capital) P1 requires 1 hour of capital and 4 hours of labour P2 requires 2 hours of capital and 2 hours of work P3 requires 5 hours of capital and 1 hour of work Combinations of labour and capital produce lamps: Q 1 + Q 2 + Q 3 The shaded box is the constraint on time for L and K K L P1 P2 P3 8 5 B labour Process Rays

10 © 2006 by Nelson, a division of Thomson Canada Limited There are 3 types of lamps produced each day. There are but 8 hours of labour available a day and There are only 5 hours of capital machine hours. Maximize Q 1 + Q 2 + Q 3 subject to: Q 1 + 2·Q 2 + 5Q 3 < The capital constraint of 5 hours per day. 4Q 1 + 2·Q 2 + Q 3 < The labour constraint of 8 hours per day. where Q 1, Q 2 and Q 3 > 0Nonnegativity constraint. Maximization Problem

11 © 2006 by Nelson, a division of Thomson Canada Limited If all inputs were used in making process 1, which takes 4 hours of labour and 1 hour of machine time, wed make 2 lamps, but have slack machine time. This is feasible, but NOT optimal. At point B on Slide 9, 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. Feasible Region

12 © 2006 by Nelson, a division of Thomson Canada Limited A Parallelogram of Process Rays Using 4 hours of labour and 1 hour of capital makes 1 lamp using process P1 Using the remaining 4 hours of labour and 4 hours of machine time makes 2 lamps using process P2. Solution: 3 lamps (one of type 1 and 2 of type 2) K L P1 P2 P3 8 5 B

13 © 2006 by Nelson, a division of Thomson Canada Limited 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. Complexity and the Method of Solution