Computational Methods for Management and Economics Carla Gomes

1
Computational Methods forManagement and
EconomicsCarla Gomes

• Module 7b
• Duality and Sensitivity Analysis
• Economic Interpretation of Duality

(slides adapted from M. Hilliers, J. Orlins,
and H. Sarpers)
2
Post-optimality Analysis

• Post-optimality very important phase of
  modeling.
• Duality plays and important role in
  post-optimality analysis
• Simplex provides several tools to perform
  post-optimality analysis

3
Post-optimality analysis for LP
4
Economic Interpretation of Duality

• LP problems quite often can be interpreted as
  allocating resources to activities.
• Lets consider the standard form

xi 0 , (i 1,2,,n)
5
What if we change our resources can we improve
our optimal solution?

• Resources m (plants)
• Activities n (2 products)
• Wyndor Glass problem? optimal product mix ---
  allocation of resources to activities i.e.,
  choose the levels of the activities that achieve
  best overall measure of performance
•  

6
Sensitivity Analysis

• How would changes in the problems objective
  function coefficients or right-hand side values
  change the optimal solution?

7
Dual Variables (Shadow Prices)

• y1 0 ? dual variable (shadow price) for
  resource 1
• y2 1.5 ? dual variable (shadow price) for
  resource 2
• y3 1 ? dual variable (shadow price) for
  resource 3

How much does Z increase if we increase resource
2 by 1 unit (i.e., b2 12 ? b213)?
8
Graphical Analysis of Dual variables Variation
in RHS Increasing level of resource 2 (b2)
(5/3,13/2)
2w13 ? Z3(5/3)5(13/2)37.5
? Z1.5 y2
Z3(2)5(6)36
(2,6)
Why is y10?
9
Economic Interpretation of Dual Variables

• The dual variable associated with resource i
• (also called shadow price), denoted by yi,
  measures
• the marginal value of this resource, i.e., the
  rate at
• which Z could be increased by (slightly)
  increasing
• the amount of this resource (bi), assuming
  everything
• else stays the same. The dual variable yi is
  identified
• by the simplex method as the coefficient of the
  ith slack
• variable in row 0 of the final simplex tableau.

10
Dual Variables binding and non-binding
constraints

• The shadow prices (dual variables) associated
  with non-binding constraints are necessarily 0
  (complementary optimal slackness) ? there is a
  surplus of non-binding resource and therefore
  increasing it will not increase the optimal
  solution. Economist refer to such resources as
  free resources (shadow price 0)
• Binding constraints on the other hand correspond
  to scarce resources there is no surplus. In
  general they have a positive shadow price.

11
Does Z always increase at the same rate if we
keep increasing the amount of resource 2?
(0,9)
b218
(5/3,13/2)
2w13 ? Z3(5/3)5(13/2)37.5
? Z1.5 y2
Z3(2)5(6)36
(2,6)
What if b2 18 (i.e., 2W18)?
? the optimal solution will stay at (0,9) for
b218
12
Does Z always decrease at the same rate if we
decrease resource 2?
(5/3,13/2)
(2,6)
If b2 proportionally. The optimal solution varies
proportionally to the variation in b2 only if 6
remains optimal for 6 b2 18, but the
decision variable values and z-value will
change.
13

• A dual variable yi gives us the rate at which Z
  could be increased by increasing the amount of
  resource i slightly.
• However this is only true for a small increase in
  the amount of the resource. I.e., this definition
  applies only if the change in the RHS of
  constraint i leaves the current basis optimal. It
  also assumes everything else stays the same.
• Another interpretation of yi is if a premium
  price must be paid for the resource i in the
  market place, yi is the maximum premium (excess
  over the regular price) that would be worth
  paying.

14
Optimal Basis in the Wyndor Glass Problem

• How can we characterize (verbally) the optimal
  basis of the Wyndor Glass problem?
• Plant 1 unutilized capacity (non-binding
  constraint)
• Plant 2 fully utilized capacity (binding
  constraint)
• Plant 3 - fully utilized capacity (binding
  constraint)

15
How do we interpret the intervals?

• If we change one coefficient in the RHS, say
  capacity of plant 2, by D the basis remains
  optimal, that is, the same equations remain
  binding.
• So long as the basis remains optimal, the shadow
  prices are unchanged.
• The basic feasible solution varies linearly with
  D. If D is big enough or small enough the basis
  will change.

16

• The dual price or shadow price for the i th
  constraint
• of an LP is the amount by which the optimal
  z-value
• is improved (increased in a max problem or
• decreased in a min problem) if the rhs of the i
  th
• constraint is increased by one. This definition
• applies only if the change in the rhs of
  constraint i
• leaves the current basis optimal.
• The dual variables or shadow prices are valid in
  a
• given interval.

17
Sensitivity analysis for c1

• How much can we vary c1 without changing
• the current basic optimal solution?

18
Sensitivity analysis for c1
Our objective function is Z c1
D5Wk ?slope of iso-profit line is
isoprofit line
How much can c1 vary until the slope of the
iso-profit line equals the slope of constraint 2
and constraint 3?
19

• How much can c1 vary until the slope of the
  iso-profit line equals the slope of constraint 2
  and constraint 3?
• Slope of constraint 2? 0
• Slope of constraint 3 ? -3/2

20
Importance of Sensitivity Analysis

• Sensitivity analysis is important for several
  reasons
• Values of LP parameters might change. If a
  parameter changes, sensitivity analysis shows it
  is unnecessary to solve the problem again. For
  example in the Wyndor problem, if the profit
  contribution of product 1 changes to 5,
  sensitivity analysis shows the current solution
  remains optimal.
• Uncertainty about LP parameters. In the Wyndor
  problem for example, if the capacity of plant 1
  decreases to 2, the optimal solution remains a
  weekly rate of 2 doors and 6 windows. Thus, even
  if availability of capacity of plant 1 uncertain,
  the company can be fairly confident that it is
  still optimal to produce a weekly rate of 2 doors
  and 6 windows.

21
Does the shadow price always have an economic
interpretation?

• Not necessarily
• For example,there is no economic interpretation
  for dual variables associated with ratio
  constraints

22
Glass Example

• x1 of cases of 6-oz juice glasses (in 100s)
• x2 of cases of 10-oz cocktail glasses (in
  100s)
• x3 of cases of champagne glasses (in 100s)
• max 5 x1 4.5 x2 6 x3 (100s)
• s.t 6 x1 5 x2 8 x3 ? 60
  (prod. cap. in hrs)
• 10 x1 20 x2 10 x3 ? 150
  (wareh. cap. in ft2)
• x1 ?
  8 (6-0z. glass dem.)
• x1 ? 0, x2 ? 0, x3 ? 0

(from AMP and slides from James Orlin)
23

• Z 51.4286
• Decision Variables
• x1 6.4286 ( of cases of 6-oz juice glasses (in
  100s))
• x2 4.2857 ( of cases of 10-oz cocktail glasses
  (in 100s))
• x3 0 ( of cases of champagne glasses (in
  100s))
• Slack Variables
• s1 0
• s2 0
• s3 1.5714
• Dual Variables
• y1 0.7857
• y2 0.0286
• y3 0

Complementary optimal slackness conditions
24

• Consider constraint 1. 6 x1 5 x2 8 x3
  ? 60 (prod. cap. in hrs)
• Lets look at the objective function if we
  change the production time from 60 and keep all
  other values the same.

The dual /shadow Price is 11/14.
25
More changes in the RHS
The shadow Price is 11/14 until production 65.5
26
What is the intuition for the shadow price
staying constant, and then changing?

• Recall from the simplex method that the simplex
  method produces a basic feasible solution. The
  basis can often be described easily in terms of a
  brief verbal description.

27
The verbal description for the optimum basis for
the glass problem

• Produce Juice Glasses and cocktail glasses only
• Fully utilize production and warehouse capacity
• z 5 x1 4.5 x2
• 6 x1 5 x2 60
• 10 x1 20 x2 150

x1 6 3/7 (6.4286) x2 4 2/7 (4.2857) z 51
3/7 (51.4286)
28
The verbal description for the optimum basis for
the glass problem

• Produce Juice Glasses and cocktail glasses only
• Fully utilize production and warehouse capacity
• z 5 x1 4.5x2
• 6 x1 5 x2 60 D
• 10 x1 20 x2 150

For D 5.5, x1 8, and the constraint x1 ? 8
becomes binding.
x1 6 3/7 2D/7 x2 4 2/7 D/7 z 51 3/7
11/14 D
29
How do we interpret the intervals?

• If we change one coefficient in the RHS, say
  production capacity, by D the basis remains
  optimal, that is, the same equations remain
  binding.
• So long as the basis remains optimal, the shadow
  prices are unchanged.
• The basic feasible solution varies linearly with
  D. If D is big enough or small enough the basis
  will change.

30
Illustration with the glass example

• max 5 x1 4.5 x2 6 x3 (100s)
• s.t 6 x1 5 x2 8 x3 ? 60
  (prod. cap. in hrs)
• 10 x1 20 x2 10 x3 ? 150
  (wareh. cap. in ft2)
• x1 ?
  8 (6-0z. glass dem.)
• x1 ? 0, x2 ? 0, x3 ? 0

The shadow price is the increase in the optimal
value per unit increase in the RHS.
If an increase in RHS coefficient leads to an
increase in optimal objective value, then the
shadow price is positive.
If an increase in RHS coefficient leads to a
decrease in optimal objective value, then the
shadow price is negative.
31
Illustration with the glass example

• max 5 x1 4.5 x2 6 x3 (100s)
• s.t 6 x1 5 x2 8 x3 ? 60
  (prod. cap. in hrs)
• 10 x1 20 x2 10 x3 ? 150
  (wareh. cap. in ft2)
• x1 ?
  8 (6-0z. glass dem.)
• x1 ? 0, x2 ? 0, x3 ? 0

Claim the shadow price of the production
capacity constraint cannot be negative.
Reason any feasible solution for this problem
remains feasible after the production capacity
increases. So, the increase in production
capacity cannot cause the optimum objective value
to go down.
32
Illustration with the glass example

• max 5 x1 4.5 x2 6 x3 (100s)
• s.t 6 x1 5 x2 8 x3 ? 60
  (prod. cap. in hrs)
• 10 x1 20 x2 10 x3 ? 150
  (wareh. cap. in ft2)
• x1 ?
  8 (6-0z. glass dem.)
• x1 ? 0, x2 ? 0, x3 ? 0

Claim the shadow price of the x1 ? 0
constraint cannot be positive.
Reason Let x be the solution if we replace the
constraint x1 ? 0 with the constraint x1 ?
1. Then x is feasible for the original
problem, and thus the original problem has at
least as high an objective value.
33
Signs of Shadow Prices for maximization problems

• ? constraint . The shadow price is
  non-negative.
• ? constraint . The shadow price is
  non-positive.
• constraint. The shadow price could be
  zero or positive or negative.

34
Signs of Shadow Prices for minimization problems

• The shadow price for a minimization problem is
  the increase in the objective function per unit
  increase in the RHS.
• ? constraint . The shadow price is
  non-positive.
• ? constraint . The shadow price is
  non-negative
• constraint. The shadow price could be
  zero or positive or negative.
• Please answer with your partner.

35
The shadow price of a non-binding constraint is
0. Complementary Slackness.

• max 5 x1 4.5 x2 6 x3 (100s)
• s.t 6 x1 5 x2 8 x3 ? 60
  (prod. cap. in hrs)
• 10 x1 20 x2 10 x3 ? 150
  (wareh. cap. in ft2)
• x1 ?
  8 (6-0z. glass dem.)
• x1 ? 0, x2 ? 0, x3 ? 0

In the optimal solution, x1 6 3/7.
Claim The shadow price for the constraint x1 ?
8 is zero.
Intuitive Reason If your optimum solution has
x1 permitting x1 8.
36
Is the shadow price the change in the optimal
objective value if the RHS increases by 1 unit.

• That is an excellent rule of thumb! It is true
  so long as the shadow price is valid in an
  interval that includes an increase of 1 unit.

37
The shadow price is valid if only one right hand
side changes. What if multiple right hand side
coefficients change?

• The shadow prices are valid if multiple RHS
  coefficients change, but the ranges are no longer
  valid.

38
Reduced Costs
39
Do the non-negativity constraints also have
shadow prices?

• Yes. They are very special and are called
  reduced costs?
• Look at the reduced costs for
• Juice glasses reduced cost 0
• Cocktail glasses reduced cost 0
• Champagne glasses red. cost -4/7

40
What is the managerial interpretation of a
reduced cost?

• There are two interpretations. Here is one of
  them.
• We are currently not producing champagne glasses.
  How much would the profit of champagne glasses
  need to go up for us to produce champagne glasses
  in an optimal solution?
• The reduced cost for champagne classes is 4/7.
  If we increase the revenue for these glasses by
  4/7 (from 6 to 6 4/7), then there will be an
  alternative optimum in which champagne glasses
  are produced.

41
Why are they called the reduced costs? Nothing
appears to be reduced

• The reduced costs can be obtained by treating the
  shadow prices are real costs. This operation is
  called pricing out.

42
Pricing Out
shadow price 11/14 1/35 .0
max 5 x1 4.5 x2 6 x3 (100s)
s.t 6 x1 5 x2 8 x3 ? 60
10 x1 20 x2 10 x3 ? 150 1
x1 ? 8
x1 ? 0, x2 ? 0, x3 ? 0
Pricing out treats shadow prices as though they
are real prices. The result is the reduced
costs.
43
Pricing Out of x1
shadow price 11/14 1/35 .0
max 5 x1 4.5 x2 6 x3 (100s)
s.t 6 x1 5 x2 8 x3 ? 60
10 x1 20 x2 10 x3 ? 150 1
x1 ? 8
x1 ? 0, x2 ? 0, x3 ? 0

• 5
• - 6 x 11/14
• - 10 x 1/35
• - 1 x 0
• 5 33/7 2/7 0

Reduced cost of x1
44
Pricing Out of x2
shadow price 11/14 1/35 .0
max 5 x1 4.5 x2 6 x3 (100s)
s.t 6 x1 5 x2 8 x3 ? 60
10 x1 20 x2 10 x3 ? 150 1
x1 ? 8
x1 ? 0, x2 ? 0, x3 ? 0

• 4.5
• - 5 x 11/14
• - 20 x 1/35
• - 0 x 0
• 4.5 55/14 4/7 0

Reduced cost of x2
45
Pricing Out of x3
shadow price 11/14 1/35 .0
max 5 x1 4.5 x2 6 x3 (100s)
s.t 6 x1 5 x2 8 x3 ? 60
10 x1 20 x2 10 x3 ? 150 1
x1 ? 8
x1 ? 0, x2 ? 0, x3 ? 0

• 6
• - 8 x 11/14
• - 10 x 1/35
• - 0 x 0
• 6 44/7 2/7 -4/7

Reduced cost of x3
46
Can we use pricing out to figure out whether a
new type of glass should be produced?
shadow price 11/14 1/35 .0
max 5 x1 4.5 x2 7 x4 (100s)
s.t 6 x1 5 x2 8 x4 ? 60
10 x1 20 x2 20 x4 ? 150 1
x1 ? 8
x1 ? 0, x2 ? 0, x4 ? 0

• 7
• - 8 x 11/14
• - 20 x 1/35
• - 0 x 0
• 7 44/7 4/7 1/7

Reduced cost of x4
47
Pricing Out of xj
shadow price y1 y2 ym
max 5 x1 4.5 x2 cj xj (100s)
s.t 6 x1 5 x2 a1j xj ? 60
10 x1 20 x2 a2j xj ? 150
.. . amjxj bm
x1 ? 0, x2 ? 0, x3 ? 0
Reduced cost of xj ?
48
Brief summary on reduced costs

• The reduced cost of a non-basic variable xj is
  the increase in the objective value of
  requiring that xj 1.
• The reduced cost of a basic variable is 0.
• The reduced cost can be computed by treating
  shadow prices as real prices. This operation is
  known as pricing out.
• Pricing out can determine if a new variable would
  be of value (and would enter the basis).

49
Summary

• The shadow price is the unit change in the
  optimal objective value per unit change in the
  RHS.
• The shadow price for a ? 0 constraint is called
  the reduced cost.
• Shadow prices usually but not always have
  economic interpretations that are managerially
  useful.
• Non-binding constraints have a shadow price of
  0.
• The sign of a shadow price can often be
  determined by using the economic interpretation
• Shadow prices are valid in an interval.
• Reduced costs can be determined by pricing out

50
Reduced Costs

• The reduced cost of a variable x is the shadow
  price of the x ? 0 constraint. It is also the
  negative of cost coefficient for x in the final
  tableau.
• Suppose in the previous example that we required
  that x3 ? 1? What is the impact on the optimal
  objective value? What is the resulting solution?
• By the previous slide, the impact is -4/7.

51
More on reduced costs

• In a pivot, multiples of constraints are added to
  the cost row.
• We will use this fact to determine explicitly how
  the cost row in the final tableau is obtained.

52
Implications of Reduced Costs

• Implication 1 increasing the cost coefficient
  of a non-basic variable by D leads to an increase
  of its reduced cost by D.

53
Implications of Reduced Costs

• Implication 2 We can compute the reduced cost
  of any variable if we know the original column
  and if we know the prices for each constraint.

FACT We can compute the reduced cost of a new
variable. If the reduced cost is positive, it
should be entered into the basis.
54

• Every tableau has prices. These are usually
  called simplex multipliers.
• The prices for the optimal tableau are the shadow
  prices.

55
Quick Summary

• Connection between shadow prices and reduced
  cost. If xj is the slack variable for a
  constraint, then its reduced cost is the negative
  of the shadow price for the constraint.
• The reduced cost for a variable is the negative
  of its cost coefficient in the final tableau

56

• Sensitivity Analysis
• Computer Analysis

57
The Computer and Sensitivity Analysis

• If an LP has more than two decision variables,
  the range of values for a rhs (or objective
  function coefficient) for which the basis remains
  optimal cannot be determined graphically.
• These ranges can be computed by hand but this is
  often tedious, so they are usually determined by
  a packaged computer program. MPL and LINDO will
  be used and the interpretation of its sensitivity
  analysis discussed.
• Note sometimes Excel provides erroneous results

58
MPL Sensitivity analysis info
c1
Reduced cost is the amount the objective
function coefficient for variable i would have to
be increased for there to be an alternative
optimal solution. More later
Dual or Shadow prices are the amount the optimal
z-value improves if the rhs of a constraint is
increased by one unit (assuming no change in
basis).
Dual variables
b2
59
MPL Sensitivity analysis info
Allowable ranges (w/o changing basis) for the
x1 coefficient (c1) is 0 c1 7.5
c1
Allowable range (w/o changing basis) for the rhs
(b2) of the second constraint is 6 b2 18
b2
What about c2? And b1 and b3?
60
Lindo Sensitivity Analysis
Allowable ranges in terms of increase and
decrease (w/o changing basis) for the x1
coefficient (c1) is 0 c1 7.5
61
The Computer and Sensitivity Analysis

• Consider the following maximization problem.
  Winco sells four types of products. The
  resources needed to produce one unit of each are

To meet customer demand, exactly 950 total units
must be produced. Customers demand that at least
400 units of product 4 be produced. Formulate an
LP to maximize profit. Let xi number of units
of product i produced by Winco.
62

• The Winco LP formulation

max z 4x1 6x2 7x3 8x4 s.t. x1 x2
x3 x4 950 x4 400 2x1
3x2 4x3 7x4 4600 3x1 4x2 5x3
6x4 5000 x1,x2,x3,x4 0
63
MAX 4 X1 6 X2 7 X3 8 X4 SUBJECT TO
2) X1 X2 X3 X4 950 3)
X4 400 4) 2 X1 3 X2 4 X3
7 X4 6 X4 4 OBJECTIVE FUNCTION VALUE 1)
6650.000 VARIABLE VALUE
REDUCED COST X1
0.000000 1.000000 X2
400.000000 0.000000 X3
150.000000 0.000000 X4
400.000000 0.000000 ROW
SLACK OR SURPLUS DUAL PRICES
2) 0.000000 3.000000
3) 0.000000 -2.000000
4) 0.000000 1.000000
5) 250.000000 0.000000 NO.
ITERATIONS 4
LINDO output and sensitivity analysis example(s).
Reduced cost is the amount the objective
function coefficient for variable i would have to
be increased for there to be an alternative
optimal solution.
64

RANGES IN WHICH THE BASIS IS UNCHANGED
OBJ COEFFICIENT RANGES
VARIABLE CURRENT ALLOWABLE
ALLOWABLE COEF
INCREASE DECREASE
X1 4.000000 1.000000
INFINITY X2 6.000000
0.666667 0.500000
X3 7.000000 1.000000
0.500000 X4 8.000000
2.000000 INFINITY
RIGHTHAND SIDE RANGES ROW
CURRENT ALLOWABLE
ALLOWABLE RHS
INCREASE DECREASE
2 950.000000 50.000000
100.000000 3 400.000000
37.500000 125.000000
4 4600.000000 250.000000
150.000000 5
5000.000000 INFINITY
250.000000
LINDO sensitivity analysis example(s).
Allowable range (w/o changing basis) for the x2
coefficient (c2) is 5.50 c2 6.667
Allowable range (w/o changing basis) for the rhs
(b1) of the first constraint is 850 b1 1000
65

MAX 4 X1 6 X2 7 X3 8 X4 SUBJECT TO
2) X1 X2 X3 X4 950 3)
X4 400 4) 2 X1 3 X2 4 X3
7 X4 6 X4 4 OBJECTIVE FUNCTION VALUE 1)
6650.000 VARIABLE VALUE
REDUCED COST X1
0.000000 1.000000 X2
400.000000 0.000000 X3
150.000000 0.000000 X4
400.000000 0.000000 ROW
SLACK OR SURPLUS DUAL PRICES
2) 0.000000 3.000000
3) 0.000000 -2.000000
4) 0.000000 1.000000
5) 250.000000 0.000000 NO.
ITERATIONS 4
Shadow prices are shown in the Dual Prices
section of LINDO output.
Shadow prices are the amount the optimal z-value
improves if the rhs of a constraint is increased
by one unit (assuming no change in basis).
66

• Interpretation of shadow prices for the Winco LP

ROW SLACK OR SURPLUS DUAL PRICES
2) 0.000000
3.000000 (overall demand) 3)
0.000000 -2.000000
(product 4 demand) 4) 0.000000
1.000000 (raw material
availability) 5) 250.000000
0.000000 (labor availability)
Assuming the allowable range of the rhs is not
violated, shadow (Dual) prices show 3 for
constraint 1 implies that each one-unit increase
in total demand will increase net sales by 3.
The -2 for constraint 2 implies that each unit
increase in the requirement for product 4 will
decrease revenue by 2. The 1 shadow price for
constraint 3 implies an additional unit of raw
material (at no cost) increases total revenue by
1. Finally, constraint 4 implies any additional
labor (at no cost) will not improve total
revenue.
67

• Shadow price signs

• Constraints with ³ symbols will always have
  nonpositive shadow prices.
• Constraints with will always have nonnegative
  shadow prices.
• Equality constraints may have a positive, a
  negative, or a zero shadow price.

68
Managerial Use of Shadow Prices
The managerial significance of shadow prices is
that they can often be used to determine the
maximum amount a manager should be willing to pay
for an additional unit of a resource. Reconsider
the Winco to the right. What is the most Winco
should be willing to pay for additional units of
raw material or labor?
MAX 4 X1 6 X2 7 X3 8 X4 SUBJECT TO
2) X1 X2 X3 X4 950 3)
X4 400 4) 2 X1 3 X2 4 X3
7 X4 6 X4 4 OBJECTIVE FUNCTION VALUE 1)
6650.000 VARIABLE VALUE
REDUCED COST X1
0.000000 1.000000 X2
400.000000 0.000000 X3
150.000000 0.000000 X4
400.000000 0.000000 ROW
SLACK OR SURPLUS DUAL PRICES
2) 0.000000 3.000000
3) 0.000000 -2.000000
4) 0.000000 1.000000
5) 250.000000 0.000000 NO.
ITERATIONS 4
raw material
labor
69
Managerial Use of Shadow Prices
MAX 4 X1 6 X2 7 X3 8 X4 SUBJECT TO
2) X1 X2 X3 X4 950 3)
X4 400 4) 2 X1 3 X2 4 X3
7 X4 6 X4 4 OBJECTIVE FUNCTION VALUE 1)
6650.000 VARIABLE VALUE
REDUCED COST X1
0.000000 1.000000 X2
400.000000 0.000000 X3
150.000000 0.000000 X4
400.000000 0.000000 ROW
SLACK OR SURPLUS DUAL PRICES
2) 0.000000 3.000000
3) 0.000000 -2.000000
4) 0.000000 1.000000
5) 250.000000 0.000000 NO.
ITERATIONS 4
The shadow price for raw material constraint (row
4) shows an extra unit of raw material would
increase revenue 1. Winco could pay up to 1 for
an extra unit of raw material and be as well off
as it is now. Labor constraints (row 5) shadow
price is 0 meaning that an extra hour of labor
will not increase revenue. So, Winco should not
be willing to pay anything for an extra hour of
labor.