Linear Programming Building Good Linear Models And Example 1 Sensitivity Analyses, Unit Conversion, Summation Variables

Building Good Models A Check List Determine in general terms what the objective is (the objective function) and what factors are under the decision maker’s control that can affect this objective (the decision variables). Define decision variables using appropriate units and time frame (cars per month, tons per production run, etc.) List the restrictions (constraints) in short expressions (bulleted list). Do not worry about listing all the variables or all the constraints at the beginning. As the formulation progresses, if you find you need a new variable or another constraint add it at that time.

Building Good Models A Check List First formulate constraints in the form: (Some expression) has (some relation) to (another expression or a constant) Keep units on both sides of the relation the same If the RHS is an expression, do the algebra to rewrite the constraint as: (Some expression involving only linear terms ) has (some relation) to (a constant) Use summation variables and constraints to simplify the input and make it more easily readable. Summation variables are particularly useful when there are many constraints involving percentages. Indicate which variables are: ≥ 0, unrestricted, ≤ 0, integer, binary

Variables and Constraints With Percentages Suppose in the formulation of a particular problem involving the production of four different styles of televisions, the modeler wished to express that no model was to represent more than 30% of the total production. The total production is X1 + X2 + X3 + X4 Valid expressions of the constraints: X1 ≤ .3(X1 + X2 + X3 + X4) X2 ≤ .3(X1 + X2 + X3 + X4) X3 ≤ .3(X1 + X2 + X3 + X4) X4 ≤ .3(X1 + X2 + X3 + X4)

Rewriting the Percentage Constraints These constraints can be rewritten as: .7X1 - .3X2 - .3X3 - .3X4 ≤ 0 -.3X1 + .7X2 - .3X3 - .3X4 ≤ 0 -.3X1 - .3X2 + .7X3 - .3X4 ≤ 0 -.3X1 - .3X2 - .3X3 + .7X4 ≤ 0 Correct but: Input of many coefficients – could make mistakes One of the factors affecting the speed of solving linear programs is the number of non-zero entries in the formulation Looking at these constraints does not instantaneously convey (by inspection) that each TV is to represent no more than 30% of the total production.

Using Summation Variables and Summation Constraints Define the summation variable, X5, to be the total production. Immediately add the following summation constraint that says X5 is the total production X5 = X1 + X2 + X3 + X4 or X1 + X2 + X3 + X4 – X5 = 0 The constraints can now be written as: X1 + X2 + X3 + X4 - X5 = 0 X1 - .3X5 ≤ 0 X2 - .3X5 ≤ 0 X3 - .3X5 ≤ 0 X4 - .3X5 ≤ 0

Summation Variables and Summation Constraints In this form the problem Is easier to input with less chance for input error Involves many 0 coefficients, with many of the remaining coefficients being 1’s – the computer likes this Is easily readable – you can tell the constraints are saying that no model should be more than 30% of the total production But this does add one more variable and one more constraint to the model. This also affects solution speed In the Solver dialogue box, make sure you include: The summation variable as part of the “Changing Cells” The summation constraint as part of the “Add Constraints”

Example 1 Galaxy Industries Expansion Galaxy Industries is planning an expansion and a move to Juarez, Mexico where both material and labor costs are cheaper. It will also produced two additional products – Big Squirts and Soakers Costs/Selling Prices: Plastic – now only $1/lb Other miscellaneous variable costs reduced by 50% Labor Sunk Cost for Regular Time $180 more per hour for each overtime hour (labor, other) Selling Prices for Space Rays/Zappers – reduced by $1/dozen

Example 1 Constraints Constraints: Plastic Availability – 3000 lbs./week Production time (Regular time) – 40 hours/week Overtime Availability – Up to 32 hours/week Must satisfy a Zapper contract – at least 200 dz./week New Product Mix Constraints Space Rays = 50% of total production (Zappers, Big Squirts, Soakers) each ≤ 40% of production Minimum total production – 1000 dz./week

Example 1 Profit/Resource Requirements Selling Price Costs Plastic ($3/lb) Other Variable Costs Total Profit Per Dozen Production Minutes DOZ Space Rays $24 $ 6 (2 lb) $10 ======= $ 8 3 Zappers $26 $ 3 (1 lb) $18 $ 5 4 DOZ Space Rays $23 $ 2 (2 lb) $ 5 ======= $16 3 Zappers $25 $ 1 (1 lb) $ 9 $15 4 DOZ Big Squirts $29 $ 3 (3 lb) $ 6 ======= $20 5 DOZ Soakers $36 $ 4 (4 lb) $10 ======= $22 6 -$1/doz 1 50% Reduction

Decision Variables (Initial) X1 = # dozen Space Rays produced per week X2 = # dozen Zappers produced per week X3 = # dozen Big Squirts produced per week X4 = # dozen Soakers produced per week X5 = # overtime hours scheduled per week

Objective Function Max Total Net Weekly Profit = Max Total Gross Weekly Profit – Weekly Cost of Overtime Gross Weekly Profit Product Profit Per Dozen Doz. Per Week Gross Profit Space Rays $16 X1 16X1 Zappers $15 X2 15X2 Big Squirts $20 X3 20X3 Soakers $22 X4 22X4 Weekly Cost of Overtime Cost Per Overtime Hours Overtime Cost Overtime Hour Scheduled Per Week $180 X5 180X5 OBJECTIVE FUNCTION MAX 16X1 + 15X2 + 20X3 + 22X4 – 180X5

Plastic Constraint Total Amount of Plastic Used Per Week ≤ Plastic Available Per Week 2X1 + 1X2 + 3X3 + 4X4 Total Amount of Plastic Used Per Week ≤ Plastic Available Per Week 3000 2X1 + 1X2 + 3X3 + 4X4 ≤ 3000

Production Time Constraint Total Production Minutes Used Per Week ≤ Total Regular Minutes Available + Total Overtime Minutes Scheduled 3X1 + 4X2 + 5X3 + 6X4 Total Production Minutes Used Per Week ≤ Total Regular Minutes Available + Total Overtime Minutes Scheduled 60(40) = 2400 60X5 3X1 + 4X2 + 5X3 + 6X4 – 60 X5 ≤ 2400

Overtime Availability The Number of Overtime Hours Scheduled/Week ≤ The Number of Overtime Hours Available/Week X5 The Number of Overtime Hours Scheduled/Week ≤ The Number of Overtime Hours Available/Week 32 X5 ≤ 32

Zapper Contract Constraint The number of dozen Zappers produced/wk ≥ The number of dozen required by contract X2 The number of dozen Zappers produced/wk ≥ The number of dozen required by contract 200 X2 ≥ 200

Mix Constraints – Summation Variable/Constraint The next set of constraints involve percentages of the total production. Define X6 = Total Weekly Production Total Weekly Production = X1 + X2 + X3 + X4 Thus the summation constraint is: X1 + X2 + X3 + X4 – X6 = 0

Mix Constraints Space Rays = 50% of total production Zappers ≤ 40% of total production Big Squirts ≤ 40% of total production Soakers ≤ 40% of total production X1 .5X6 X2 .4X6 X3 .4X6 X4 .4X6 X1 - .5X6 = 0 X2 - .4X6 ≤ 0 X3 - .4X6 ≤ 0 X4 - .4X6 ≤ 0

Minimum Total Production The total number of dozen units produced/wk ≥ The minimum production limit The total number of dozen units produced/wk ≥ The minimum production limit X6 1000 X6 ≥ 1000

The Complete Model Including the nonnegativity of the variables the complete linear programming model is: MAX 16X1 + 15X2 + 20X3 + 22X4 - 180X5 s.t. 2X1 + 1X2 + 3X3 + 4X4 ≤ 3000 (Plastic) 3X1 + 4X2 + 5X3 + 6X4 - 60X5 ≤ 2400 (Time) X5 ≤ 32 (Overtime) X2 ≥ 200 (Contract) X1 + X2 + X3 + X4 - X6 = 0 (Sum) X1 - .5X6 = 0 (Sp Ray Mix) X2 - .4X6 ≤ 0 (Zapper Mix) X3 - .4X6 ≤ 0 (Big Sq Mix) X4 - .4X6 ≤ 0 (Soaker Mix) X6 ≥ 1000 (Min Total) All X’s ≥ 0

=SUMPRODUCT($C$3:$H$3,C5:H5) Drag down

Solution/Analysis Produce Weekly All 32 overtime 565 dz. Space Rays 200 dz. Zappers 365 dz. Big Squirts 0 Soakers Total = 1130 dozen All 32 overtime hours scheduled Profit = $13,580 All time and overtime used. Contract met exactly. Exactly 50% Space Rays. 575 lbs. plastic unused.

Sensitivity Anslysis Solution will not change as long profit for doz. Space Rays is between $4 and $20 Profit per dozen Soakers must increase by $2.50 (to $24.50) before it is economically beneficial to produce them. Extra overtime production hours will add $90 each to the profit. This value is valid for total overtime production hours between 23.34 and 47.33.

Review Tips on building mathematical models. Use of summation variables and constraints. Solving a linear program with various constraint types and a summation variable and constraint. Interpreting the output.