The Lego Production Problem
You have a set of legos
8 small bricks
6 large bricks
These are your “raw materials”.
You have to produce tables and chairs out of these legos.
These are your “products”.
The Lego Production Problem
Weekly supply of raw materials:
8 Small Bricks
6 Large Bricks
Products:
Chair
Profit = 15 cents per Chair
Table
Profit = 20 cents per Table
Problem Formulation
X1 is the number of Chairs
X2 is the number of Tables
Large brick constraint
X1+2X2  6
Small brick constraint
2X1+2X2  8
Objective function is to Maximize
15X1+20 X2
X1 ≥ 0
X2 ≥ 0
Linear Programming
We can make Product1 and Product2.
There are 3 resources; Resource1, Resource2, Resource3.
Product1 needs one hour of Resource1, nothing of Resource2, and three
hours of resource3.
Product2 needs nothing from Resource1, two hours of Resource2, and
two hours of resource3.
Available hours of resources 1, 2, 3 are 4, 12, 18, respectively.
Contribution Margin of product 1 and Product2 are $300 and $500,
respectively.
 Formulate the Problem
 Solve the problem using solver in excel
Problem Formulation
Objective Function
Z = 3 x1 +5 x2
Constraints
Resource 1
x1
4
Resource 2
2x2  12
Resource 3
3 x1 + 2 x2  18
Nonnegativity
x1  0, x2  0
Feasible, Infeasible, and Optimal Solution
Given the following problem
Maximize Z = 3x1 + 5x2
Subject to: the following constraints
x1
≤4
2x2 ≤ 12
3x1 + 2x2 ≤ 18
x1, x2 ≥ 0
What combination of x1 and x2 could be the optimal solution?
A)
x1 = 4, x2 = 4 Infeasible; Violates Constraint 3
B)
x1 = -3, x2 = 6 Infeasible; Violates nonnegativity
C)
x1 = 3, x2 = 4 Feasible; z = 3×3+ 5×4 = 29
Infeasible; Violates Constraint 2
D)
x1 = 0, x2 = 7
Feasible; z = 3×2+ 5×6 = 36 and Optimal
E)
x1 = 2, x2 = 6
Optimal Product Mix
The Omega Manufacturing Co. has discontinued the production of a certain
non-profitable product line. This act created considerable excess capacity.
Management is considering devoting this excess capacity to one or more
of three products. The hours required from each resource for each unit of
product, the available capacity (hours per week) of the the three
resources, as well as the profit of each unit of product are given below.
Hours used per unit
Product1 Product2 Product3
9
3
5
5
4
0
3
0
2
$50
$20
$25
Total hours avialable
500
350
150
Profit
Sales department indicates that the sales potentials for products 1 and 2
exceeds maximum production rate, but the sales potential for product 3 is
20 units per week.
Formulate the problem and solve it using excel
Practice (Page 304, Prob. 3)
An appliance manufacturer produces two models of microwave ovens: H and
W. Both models require fabrication and assembly work: each H uses four
hours fabrication and two hours of assembly, and each W uses two hours
fabrication and six hours of assembly. There are 600 fabrication hours
this week and 450 hours of assembly. Each H contributes $40 to profit,
and each W contributes $30 to profit.
a) Formulate the problem as a Linear Programming problem.
b) Solve it using excel.
c) What are the final values?
d) What is the optimal value of the objective function?
Practice (Page 304, Prob. 4)
A small candy shop is preparing for the holyday season. The owner must
decide how many how many bags of deluxe mix how many bags of
standard mix of Peanut/Raisin Delite to put up. The deluxe mix has 2/3
pound raisins and 1/3 pounds peanuts, and the standard mix has 1/2
pound raisins and 1/2 pounds peanuts per bag. The shop has 90 pounds of
raisins and 60 pounds of peanuts to work with. Peanuts cost $0.60 per
pounds and raisins cost $1.50 per pound. The deluxe mix will sell for
2.90 per pound and the standard mix will sell for 2.55 per pound. The
owner estimates that no more than 110 bags of one type can be sold.
a) Formulate the problem as a Linear Programming problem.
b) Solve it using excel.
c) What are the final values?
d) What is the optimal value of the objective function?
Assignment 6a:1 Due at the beginning of next class
The following table summarizes the key facts about two products, A and B,
and the resources, Q, R, and S, required to produce them.
Resource Usage per Unit Produced
Resource
Product A
Product B
Amount of
resource available
Q
2
1
2
R
1
2
2
S
3
3
4
Profit/Unit
$3000
$2000
a) Formulate the problem as a Linear Programming problem.
b) Solve it using excel.
c) What are the final values?
d) What is the optimal value of the objective function?
Assignment 6a:2 Due at the beginning of next class
The Apex Television Company has to decide on the number of 27” and 20”
sets to be produced at one of its factories. Market research indicates that
at most 40 of the 27” sets and 10 of the 20” sets can be sold per month.
The maximum number of work-hours available is 500 per month. A 27”
set requires 20 work-hours and a 20” set requires 10 work-hours. Each
27” set sold produces a profit of $120 and each 20” set produces a profit
of $80. A wholesaler has agreed to purchase all the television sets
produced if the numbers do not exceed the maximum indicated by the
market research.
a) Formulate the problem as a Linear Programming problem.
b) Solve it using excel.
c) What are the final values?
d) What is the optimal value of the objective function?
Assignment 6a:3 Due at the beginning of next class
Ralph Edmund loves steaks and potatoes. Therefore, he has decided to go on a steady
diet of only these two foods (plus some liquids and vitamins supplements) for all his
meals. Ralph realizes that this isn’t the healthiest diet, so he wants to make sure that
he eats the right quantities of the two foods to satisfy some key nutritional
requirements. He has obtained the following nutritional and cost information:
Grams of Ingredient per Serving
Ingredient
Steak
Potatoes
Daily Requirements
(grams)
Carbohydrates
5
15
≥ 50
Protein
20
5
≥ 40
Fat
15
2
≤ 60
Cost per serving
$4
$2
Ralph wishes to determine the number of daily servings (may be fractional of steak
and potatoes that will meet these requirements at a minimum cost.
Formulate the problem as a Linear Programming problem. Solve it using
excel. What are the final values? What is the optimal value of the
objective function?
Assignment 6a:4 Due at the beginning of next class
You are given the following linear programming model in algebraic form,
where, X1 and X2 are the decision variables and Z is the value of the
overall measure of performance.
Maximize Z = X1 +2 X2
Subject to
Constraints on resource 1: X1 + X2 ≤ 5 (amount available)
Constraints on resource 2: X1 + 3X2 ≤ 9 (amount available)
And
X1 , X2 ≥ 0
Assignment 6a:4 Due at the beginning of next class
a) Identify the objective function, the functional constraints, and the nonnegativity constraints in this model.
b) Incorporate this model into a spreadsheet.
c) Is (X1 ,X2) = (3,1) a feasible solution?
d) Is (X1 ,X2) = (1,3) a feasible solution?
e) Use the Excel Solver to solve this model.
Assignment 6a:5 Due at the beginning of next class
You are given the following linear programming model in algebraic form,
where, X1 and X2 are the decision variables and Z is the value of the
overall measure of performance.
Maximize Z = 3X1 +2 X2
Subject to
Constraints on resource 1: 3X1 + X2 ≤ 9 (amount available)
Constraints on resource 2: X1 + 2X2 ≤ 8 (amount available)
And
X1 , X2 ≥ 0
Assignment 6a:5 Due at the beginning of next class
a) Identify the objective function, the functional constraints, and the nonnegativity constraints in this model.
b) Incorporate this model into a spreadsheet.
c) Is (X1 ,X2) = (2,1) a feasible solution?
d) Is (X1 ,X2) = (2,3) a feasible solution?
e) Is (X1 ,X2) = (0,5) a feasible solution?
f) Use the Excel Solver to solve this model.
Example
A Production System Manufacturing Two Products, P and Q
$90 / unit
P: 110 units / week
Q:
$100 / unit
60 units / week
D
5 min.
D
10 min.
Purchased Part
$5 / unit
C
10 min.
C
5 min.
B
25 min.
A
15 min.
B
10 min.
A
10 min.
RM1
$20 per
unit
RM2
$20 per
unit
Time available at each work center: 2,400 minutes per week
Operating expenses per week: $6,000
RM3
$25 per
unit