Lecture
5
Linear Programming (6S)
and
Transportation Problem (8S)
1
Linear Programming
George Dantzig – 1914 -2005
 Concerned with optimal allocation of limited
resources such as


Materials
 Budgets
 Labor
 Machine time

among competitive activities
 under a set of constraints
George Dantzig – 1914 -2005
2
Product Mix Example (from session 1)
Type 1
Type 2
Profit per unit
$60
$50
Assembly time per
unit
4 hrs
10 hrs
Inspection time per
unit
2 hrs
Storage space per
unit
3 cubic ft
Resource
Amount available
Assembly time
100 hours
Inspection time
22 hours
Storage space
39 cubic feet
1 hr
3 cubic ft
3
Linear Programming Example
Variables
Maximize 60X1 + 50X2
Subject to
Objective function
4X1 + 10X2 <= 100
2X1 + 1X2 <= 22
Constraints
3X1 + 3X2 <= 39
Non-negativity Constraints
X1, X2 >= 0
What is a Linear Program?
• A LP is an optimization model that has
• continuous variables
• a single linear objective function, and
• (almost always) several constraints (linear equalities or inequalities)
4
Linear Programming Model

Decision variables



Objective Function



unknowns, which is what model seeks to determine
for example, amounts of either inputs or outputs
goal, determines value of best (optimum) solution among all feasible (satisfy
constraints) values of the variables
either maximization or minimization
Constraints


restrictions, which limit variables of the model
limitations that restrict the available alternatives

Parameters: numerical values (for example, RHS of constraints)
 Feasible solution: is one particular set of values of the decision
variables that satisfies the constraints


Feasible solution space: the set of all feasible solutions
Optimal solution: is a feasible solution that maximizes or minimizes
the objective function

There could be multiple optimal solutions
5
Another Example of LP: Diet Problem

Energy requirement : 2000 kcal
 Protein requirement : 55 g
 Calcium requirement : 800 mg
Food
Energy (kcal)
Protein(g)
Calcium(mg)
Oatmeal
110
4
2
Price per
serving($)
3
Chicken
Eggs
Milk
Pie
205
160
160
420
32
13
8
4
12
54
285
22
24
13
9
24
Pork
260
14
80
13
6
Example of LP : Diet Problem

oatmeal: at most 4 servings/day
 chicken: at most 3 servings/day
 eggs: at most 2 servings/day
 milk: at most 8 servings/day
 pie: at most 2 servings/day
 pork: at most 2 servings/day
Design an optimal diet plan
which minimizes the cost per day
7
Step 1: define decision variables






x1 = # of oatmeal servings
x2 = # of chicken servings
x3 = # of eggs servings
x4 = # of milk servings
x5 = # of pie servings
x6 = # of pork servings
Step 2: formulate objective function
• In this case, minimize total cost
minimize z = 3x1 + 24x2 + 13x3 + 9x4 + 24x5 + 13x6
8
Step 3: Constraints

Meet energy requirement
110x1 + 205x2 + 160x3 + 160x4 + 420x5 + 260x6 2000

Meet protein requirement
4x1 + 32x2 + 13x3 + 8x4 + 4x5 + 14x6  55

Meet calcium requirement
2x1 + 12x2 + 54x3 + 285x4 + 22x5 + 80x6  800

Restriction on number of servings
0x14, 0x23, 0x32, 0x48, 0x52, 0x62
9
So, how does a LP look like?
minimize 3x1 + 24x2 + 13x3 + 9x4 + 24x5 + 13x6
subject to
110x1 + 205x2 + 160x3 + 160x4 + 420x5 + 260x6 2000
4x1 + 32x2 + 13x3 + 8x4 + 4x5 + 14x6  55
2x1 + 12x2 + 54x3 + 285x4 + 22x5 + 80x6  800
0x14
0x23
0x32
0x48
0x52
0x62
10
Optimal Solution – Diet Problem
Using LINDO 6.1

Food
Oatmeal
# of servings
4
Chicken
Eggs
Milk
Pie
0
0
6.5
0
Pork
2
Cost of diet = $96.50 per day
11
Optimal Solution – Diet Problem
Using Management Scientist

Food
Oatmeal
# of servings
4
Chicken
Eggs
Milk
Pie
0
0
6.5
0
Pork
2
Cost of diet = $96.50 per day
12
Guidelines for Model Formulation

Understand the problem thoroughly.
 Describe the objective.
 Describe each constraint.
 Define the decision variables.
 Write the objective in terms of the decision
variables.
 Write the constraints in terms of the decision
variables

Do not forget non-negativity constraints
13
A Transportation Table
1
Factory
Warehouse
3
2
4
4
7
7
1
100
1
3
12
8
8
200
2
10
8
16
Factory 1
can
supply
100
units per
period
5
150
3
450
Demand
80
90
120
160
450
Warehouse B’s demand is 90
units per period
Total
supply
capacity
per
period
Total demand
per period
14
LP Formulation of Transportation Problem

minimize
4x11+7x12+7x13+x14+12x21+3x22+8x23+8x24+
8x31+10x32+16x33+5x34
Minimize total cost of transportation
Subject to
 x11+x12+x13+x14=100
Supply constraint for factories
 x21+x22+x23+x24=200
 x31+x32+x33+x34=150
 x11+x21+x31=80
 x12+x22+x32=90
Demand constraint of warehouses
 x13+x23+x33=120
 x14+x24+x34=160
 xij>=0, i=1,2,3; j=1,2,3,4
15
Solution in Management Scientist
Total transportation cost = 4(80) + 7(0) + 7(10)+ 1(10)
+ 12(0) + 3(90) + 8(110) + 8(0) + 8(0) +10(0) + 16(0)
+5 (150) = $2300
16
Solution using LINDO
Notice multiple
optimal solutions!
17
Product Mix Problem
•
•
•
•
•
•
Floataway Tours has $420,000 that can be used to
purchase new rental boats for hire during the summer.
The boats can be purchased from two different
manufacturers.
Floataway Tours would like to purchase at least 50 boats.
They would also like to purchase the same number from
Sleekboat as from Racer to maintain goodwill.
At the same time, Floataway Tours wishes to have a total
seating capacity of at least 200.
Formulate this problem as a linear program
18
Product Mix Problem
Boat
Maximum
Builder
Cost
Speedhawk Sleekboat
Silverbird Sleekboat
Catman
Racer
Classy
Racer
$6000
$7000
$5000
$9000
Expected
Seating
3
5
2
6
Daily
Profit
$ 70
$ 80
$ 50
$110
19
Product Mix Problem

Define the decision variables
x1 = number of Speedhawks ordered
x2 = number of Silverbirds ordered
x3 = number of Catmans ordered
x4 = number of Classys ordered
 Define the objective function
Maximize total expected daily profit:
Max: (Expected daily profit per unit) x (Number of units)
Max: 70x1 + 80x2 + 50x3 + 110x4
20
Product Mix Problem

Define the constraints
(1) Spend no more than $420,000:
6000x1 + 7000x2 + 5000x3 + 9000x4 < 420,000
(2) Purchase at least 50 boats:
x1 + x2 + x3 + x4 > 50
(3) Number of boats from Sleekboat equals number
of boats from Racer:
x1 + x2 = x3 + x4 or x1 + x2 - x3 - x4 = 0
(4) Capacity at least 200:
3x1 + 5x2 + 2x3 + 6x4 > 200
Nonnegativity of variables:
xj > 0, for j = 1,2,3,4
21
Product Mix Problem - Complete Formulation
Max 70x1 + 80x2 + 50x3 + 110x4
s.t.
6000x1 + 7000x2 + 5000x3 + 9000x4 < 420,000
x1 + x2 + x3 + x4 > 50
Boat
# purchased
x1 + x2 - x3 - x4 = 0
Speedhawk
28
3x1 + 5x2 + 2x3 + 6x4 > 200
Silverbird
0
x1, x2, x3, x4 > 0
Catman
0
Classy

28
Daily profit = $5040
22
Marketing Application: Media Selection
Advertising Media
# of potential
customers reached
Cost ($) per
advertisement
Max times available
per month
Exposure Quality
Units
Day TV
1000
1500
15
65
Evening TV
2000
3000
10
90
Daily newspaper
1500
400
25
40
Sunday newspaper
2500
1000
4
60
Radio
300
100
30
20





Advertising budget for first month = $30000
At least 10 TV commercials must be used
At least 50000 customers must be reached
Spend no more than $18000 on TV adverts
Determine optimal media selection plan
23
Media Selection Formulation

Step 1: Define decision variables






Step 2: Write the objective in terms of the decision variables


DTV = # of day time TV adverts
ETV = # of evening TV adverts
DN = # of daily newspaper adverts
SN = # of Sunday newspaper adverts
R = # of radio adverts
Maximize 65DTV+90ETV+40DN+60SN+20R
Step 3: Write the constraints in terms of the decision variables
DTV
ETV
DN
SN
R
+
25
<=
4
<=
30
<=
30000
0
DN
25
SN
2
R
30
Exposure = 2370 units
Availability of
Media
Budget
+
ETV
>=
10
1500DTV
+
3000ETV
<=
18000
TV Constraints
1000DTV
+
2000ETV
>=
50000
Customers reached
+
100R
<=
ETV
DTV
2500SN
+
10
10
3000ETV
+
1000SN
<=
DTV
+
1500DN
+
15
Value
1500DTV
+
400DN
<=
Variable
300R
DTV, ETV, DN, SN, R >= 0
24
Applications of LP

Product mix planning
 Distribution networks
 Truck routing
 Staff scheduling
 Financial portfolios
 Capacity planning
 Media selection: marketing
25
Possible Outcomes of a LP

A LP is either
Infeasible – there exists no solution which satisfies
all constraints and optimizes the objective function
 or, Unbounded – increase/decrease objective
function as much as you like without violating any
constraint
 or, Has an Optimal Solution



Optimal values of decision variables
Optimal objective function value
26
Infeasible LP – An Example

minimize
4x11+7x12+7x13+x14+12x21+3x22+8x23+8x24+8x31+10x32+16
x33+5x34
Subject to
 x11+x12+x13+x14=100
 x21+x22+x23+x24=200
 x31+x32+x33+x34=150


x11+x21+x31=80
x12+x22+x32=90
x13+x23+x33=120
x14+x24+x34=170

xij>=0, i=1,2,3; j=1,2,3,4


Total demand exceeds total supply
27
Unbounded LP – An Example
maximize 2x1 + x2
subject to
-x1 +
x2  1
x1 - 2x2  2
x1 , x 2  0
x2 can be increased indefinitely without violating any
constraint
=> Objective function value can be increased indefinitely
28
Multiple Optima – An Example
maximize x1 + 0.5 x2
subject to
2x1 + x2  4
x1 + 2x2  3
x1 , x2  0
• x1= 2, x2= 0, objective function = 2
• x1= 5/3, x2= 2/3, objective function = 2
29
Operations Scheduling
Chapter 16
30
Scheduling

Establishing the timing of the use of equipment,
facilities and human activities in an
organization

Effective scheduling can yield

Cost savings

Increases in productivity
31
High-Volume Systems

Flow system: High-volume system with
Standardized equipment and activities

Flow-shop scheduling: Scheduling for highvolume flow system
Work Center #1
Work Center #2
Output
32
High-Volume Success Factors

Process and product design

Preventive maintenance

Rapid repair when breakdown occurs

Optimal product mixes

Minimization of quality problems

Reliability and timing of supplies
33
Scheduling Low-Volume Systems

Loading - assignment of jobs to process
centers
 Sequencing - determining the order in
which jobs will be processed
 Job-shop scheduling

Scheduling for low-volume
systems with many
variations
in requirements
34
Gantt Load Chart
Figure 16.2

Gantt chart - used as a visual aid for loading
and scheduling
Work Mon. Tues. Wed. Thurs. Fri.
Center
1
Job 3
Job 4
2
Job 3 Job 7
3
Job 1
Job 6
Job 7
4
Job 10
35
More Gantt Charts
36
Assignment Problem

Objective: Assign n jobs/workers to n machines
such that the total cost of assignment is minimized
 Special case of transportation problem

When # of rows = # of columns in the
transportation tableau
 All supply and demands =1

Plenty of practical applications

Job shops
 Hospitals
 Airlines, etc.
37
Cost Table for Assignment Problem
Aircraft (j)
1
2
3
4
1
$1
$4
$6
$3
2
$9
$7
$10
$9
3
$4
$5
$11
$7
4
$8
$7
$8
$5
Pilot (i)
All assignment costs in thousands of $
38
Management Scientist Solution
Pilot Assigned to
Cost
aircraft # (`000 $)
1
1
1
2
3
4
3
2
4
10
5
5
39
Formulation of Assignment Problem

minimize x11+4x12+6x13+3x14 + 9x21+7x22+10x23+9x24 +
4x31+5x32+11x33+7x34 + 8x41+7x42+8x43+5x44
subject to
Pilot Assigned to
Cost
 x11+x12+x13+x14=1
aircraft # (`000 $)
 x21+x22+x23+x24=1
1
1
1
 x31+x32+x33+x34=1
 x41+x42+x43+x44=1
2
3
10
3
2
5
 x11+x21+x31+x41=1
4
4
5
 x12+x22+x32+x42=1
Optimal Solution:
 x13+x23+x33+x43=1
x11=1; x23=1; x32=1; x44=1; rest=0
 x14+x24+x34+x44=1
Cost of assignment = 1+10+5+5=$21 (`000)

xij = 1, if pilot i is assigned to aircraft j, i=1,2,3,4; j=1,2,3,4
0 otherwise
40
Sequencing



Sequencing: Determine the order in which jobs at a
work center will be processed.
Workstation: An area where one person works, usually
with special equipment, on a specialized job.
Priority rules: Simple heuristics used to select the
order in which jobs will be processed.

FCFS - first come, first served

SPT - shortest processing time


In-class example
Minimizes mean flow time
EDD - earliest due date
41
Performance Measures


Job flow time

Length of time a job is at a particular workstation

Includes actual processing time, waiting time,
transportation time etc.
Lateness = flow time – due date


Tardiness = max {lateness, 0}
Makespan

Total time needed to complete a group of jobs

Length of time between start of first job and
completion of last job
42
Scheduling Difficulties

Variability in

Setup times
 Processing times
 Interruptions
 Changes in the set of jobs

No method for identifying optimal schedule
 Scheduling is not an exact science
 Ongoing task for a manager
43
Minimizing Scheduling Difficulties

Set realistic due dates

Focus on bottleneck operations

Consider lot splitting of large jobs
44