Models - Operations Research Models and Methods

Models

Nonlinear Programming

Manufacturing Example

Linear Objective

Concave Objective

Convex Objective

	Manufacturing Example
	Linear Objective

Problem

We consider a manufacturing problem with three products, P, Q and R, being manufactured on four machines, A through D. The products require four raw materials, M1 through M4. Tables 1, 2 and 3 provide revenue and cost data, as well as information relating production to machine capacity and raw material usage. In this section we show a linear programming model of the problem. After examining its solution, we introduce several nonlinearities and discuss their implications.

Table 1. Product Revenues and Market
Product	P	Q	R
Revenue	$90	100	70
Maximum sales	100	40	60

Table 2. Machine usage and availability
Machine	Processing time (min/unit)			Availability (min)
	P	Q	R
A	20	10	10	2400
B	12	28	16	2400
C	15	6	16	2400
D	10	15	0	2400

Table 3. Raw Material Requirements and Cost
Material (parts)	Requirements (parts/unit)			Cost ($/unit)
	P	Q	R
M1	1	0	0	$20
M2	1	1	0	20
M3	0	1	1	20
M4	1	0	0	5

Linear Programming Model

For the linear programming model, we select the goal of maximizing net operating income, revenue minus raw material cost.

Decision Variables:

: number of units of product P to produce during the week

: number of units of Q to produce during the week

: number of units of R to produce during the week

: number of units of raw material j to purchase, j = 1...4.

Objective: Maximize operating income = Revenue - Raw Material Cost

Subject to:

Machine Time Limit Constraints

Raw Material Constraints

Nonegativity and Upper Bounds

Solution

The figure below shows the linear model created and solved in Excel.

The solution to the LP model is a basic solution. We note that machines A and B are bottlenecks, with their available capacity entirely used. Production of R is limited by the market. In the following pages we modify the objective function to be nonlinear and note the effect on the solution.