Allocation of Scarce Resources, I

1
Allocation of Scarce Resources, I

• Example Production Mix Problem

Television Manufacturing Company
Market 20-inch sets lt 15 sets per
month 27-inch sets lt 40 sets per month
Production Coefficients (hours of labor per TV
set)
Profit per TV set 20-inch --- 80 27-inch ---
120
2
Production Mix Problem

• Linear Programming Formulation

Production Capacity Constraints 6med 15lrg
lt 360 4med 5lrg lt 140 med gt 0 lrg gt
0
(hours/set)sets hours
Market Constraints med lt 15 lrg lt 40
(sets)
Objective Function Maximize Profit 80med
120lrg
(/set)sets
3
Production Mix Problem

• Graphical Solution

lrg
40
27-inch sets
30
20-inch sets
(10,20)
(Optimal Product Mix!)
Profit 8010 12020 3200/mo
20
Electronics
Feasible Region
10
Cabinetry
Profit
0
med
0 10 20 30
40 50 60
4
Linear Programming Formulation (LINGO Model Sets
Data)
MODEL !Product Mix (Television
Manufacturing) SETS PRODUCTS/ MEDIUM, LARGE
/ PROFIT, QUANTITY, SALESMAX RESOURCES
/ELECTRONICS, CABINETRY/ AVAILABLE
PRODUCTION (RESOURCES, PRODUCTS)
CONSUMPTION ENDSETS DATA PROFIT 80 120
SALESMAX 15 40 AVAILABLE 360
140 CONSUMPTION 6 15 4 5 ENDDATA
5
Linear Programming Formulation (LINGO Model
Objective Constraints)
MAX _at_SUM( PRODUCTS PROFIT QUANTITY
) _at_FOR( RESOURCES( I) _at_SUM( PRODUCTS( J)
CONSUMPTION( I, J) QUANTITY( J)) lt
AVAILABLE( I)) _at_FOR( PRODUCTS QUANTITY lt
SALESMAX) END
6
Linear Programming Formulation (LINGO Model
Solution)
Optimal solution found at step 0
Objective value 3200.000
Variable Value QUANTITY(
MEDIUM) 10.0000 QUANTITY(
LARGE) 20.0000
7
Product Mix Problem

• Primal Problem Formulation

Production Capacity Constraints 6med 15lrg
lt 360 4med 5lrg lt 140 med gt 0
lrg gt 0
(hours/set)sets hours
Market Constraints med lt 15 lrg lt 40
(sets)
Objective Function Maximize Profit 80med
120lrg
(/set)sets
8
Product Mix Problem

• Dual Problem Formulation

Production Pricing Constraints 6eprice
4cprice mpricegt 80 15eprice 5cprice
lpricegt 120 eprice gt 0 cprice gt 0
(hours/set)(/hr) /set
Objective Function Minimize Value
360eprice 140cprice 15mprice 40lprice
Hrs(/hr) Sets(/set)
9
Dual Problem

• LINGO Model Sets Data

MODEL !Product Mix (Television
Manufacturing) !Dual Problem SETS
RESOURCES/ ELECTRONICS, CABINETRY,
MEDSALES, LRGSALES / PRICE, AVAILABLE
PRODUCTS /MEDIUM, LARGE/ PROFIT RETURN
(PRODUCTS, RESOURCES) EARNINGS ENDSETS DATA
PROFIT 80 120 AVAILABLE 360
140 15 40 EARNINGS 6 4 1 0 15 5 0
1 ENDDATA
10
Dual Problem

• LINGO Model Objective Constraints

MIN _at_SUM( RESOURCES AVAILABLE PRICE
) _at_FOR( PRODUCTS( I) _at_SUM( RESOURCES( J)
EARNINGS( I, J) PRICE( J)) gt PROFIT( I)) END
11
Dual Problem

• LINGO Model Solution

Optimal solution found at step 3
Objective value 3200.000
Variable Value PRICE(
ELECTRONICS) 2.66667 PRICE(
CABINETRY) 16.00000 PRICE(
MEDSALES) 0.00000 PRICE(
LRGSALES) 0.00000
Row Slack or Surplus Dual Price
1 3200.000
1.000000 2
0.0000000 -10.00000
3 0.0000000 -20.00000
12
Primal Problem (LINGO Model Solution)
Optimal solution found at step 0
Objective value 3200.000
Variable Value QUANTITY(
MEDIUM) 10.0000 QUANTITY(
LARGE) 20.0000
Row Slack or Surplus Dual Price
1 3200.000
1.000000 2
0.0000000 2.666667
3 0.0000000 16.00000
4 0.0000000
0.0000000 5
0.0000000 0.0000000
13
Primal Problem (Interpretation of Dual Prices)

Row Dual Price (Marginal value of an 1
1.000000 additional unit of 2 2.666667
the resource) 3 16.00000 4
0.0000000 5 0.0000000
14
Work-Scheduling Problem
Post-Office Hiring of Full-Time Employees
Number of full-time employees required
Employee Requirements
(a)
Day 1 Monday 17 Day 2 Tuesday 13 Day 3
Wednesday 15 Day 4 Thursday 19 Day 5
Friday 14 Day 6 Saturday 16 Day 7 Sunday
11
(b)
Each employee must work five consecutive days
and then receive two days off.
15
Work-Scheduling Problem

• Formulate as Linear Programming Problem

Define variables
Let xi number of employees beginning work on
day i, i 1,,7
Write objective function
Min z x1 x2 x3 x4 x5 x6 x7
Impose constraints
x1 x4 x5 x6 x7 gt 17
(Monday) x1 x2 x5 x6 x7 gt
13 (Tuesday) x1 x2 x3 x6
x7 gt 15 (Wednesday) x1 x2 x3 x4
x7 gt 19 (Thursday) x1 x2 x3 x4 x5
gt 14 (Friday) x2 x3 x4
x5 x6 x7 gt 16 (Saturday) x3
x4 x5 x6 x7 gt 11 (Sunday) xi gt 0 (i 1,
, 7) (Non-negativity)
16
Work-Scheduling Problem

• (LINGO Model Sets Data)

MODEL !Work-Schedule (Post-Office) SETS
EMPLOYEES/ MONDAY, TUESDAY, WEDNESDAY, THURSDAY,
FRIDAY, SATURDAY, SUNDAY/ NHIRED NEEDS /MON,
TU, WED, THUR, FRI, SAT, SUN/ NREQUIRED
MATRIX (NEEDS, EMPLOYEES) CONSTRAINTS ENDSETS DA
TA NREQUIRED 17 13 15 19 14 16 11
CONSTRAINTS 1 0 0 1 1 1 1 1 1 0 0 1 1
1 1 1 1 0 0 1 1 1 1 1 1 0 0 1 1 1 1 1 1
0 0 0 1 1 1 1 1 0 0 0 1 1 1 1 1 ENDDATA
17
Work-Scheduling Problem

• (LINGO Model Objective Constraints)

MIN _at_SUM( EMPLOYEES NHIRED) _at_FOR( NEEDS(
I) _at_SUM( EMPLOYEES( J) CONSTRAINTS( I, J)
NHIRED( J)) gt NREQUIRED( I)) END
18
Work-Scheduling Problem

• (LINGO Model Solution)

Optimal solution found at step 7
Objective value 22.33333
Variable Value NHIRED( MONDAY)
1.333333 NHIRED( TUESDAY)
3.333333 NHIRED( WEDNESDAY)
2.000000 NHIRED( THURSDAY)
7.333333 NHIRED( FRIDAY)
0.0000000 NHIRED( SATURDAY)
3.333333 NHIRED( SUNDAY)
5.000000
19
Work-Scheduling Problem

• (LINGO Model Dual Prices)

Row Slack or Surplus Dual Price
1 22.33333
1.000000 2
0.0000000 -0.3333333
3 0.0000000
0.0000000 4
0.0000000 -0.3333333
5 0.0000000 -0.3333333
6 0.0000000
-0.5551115E-16
7 0.0000000 -0.3333333
8 6.666667
0.0000000

20
Work-Scheduling Problem

• (LINGO Model with Integer Constraint)

MIN _at_SUM( EMPLOYEES NHIRED) _at_FOR( NEEDS(
I) _at_SUM( EMPLOYEES( J) CONSTRAINTS( I, J)
NHIRED( J)) gt NREQUIRED( I)) ! We want
NHIRED to be integer _at_FOR(EMPLOYEES(I)
_at_GIN(NHIRED(I))) END
21
Work-Scheduling Problem

• (LINGO Solution with Integer Constraint)

Optimal solution found at step 8
Objective value 23.00000 Branch
count 1
Variable Value NHIRED(
MONDAY) 7.000000 1.000000
NHIRED( TUESDAY) 3.000000
1.000000 NHIRED( WEDNESDAY)
2.000000 1.000000
NHIRED( THURSDAY) 7.000000
1.000000 NHIRED( FRIDAY)
1.000000 1.000000
NHIRED( SATURDAY) 3.000000
1.000000 NHIRED( SUNDAY)
0.0000000 1.000000