`OPIM101 Decision Analysis
Semester 2, 2021/2022
Assignment 1: Due 6 Feb 2022
Solution Key
Topics Covered: Linear Programming and Sensitivity Analysis, Graphical and Excel Solutions,
Question Max Marks
1
8
2
8
3
8
4
12
5
6
6
16
7(a)
4
8
10
9
4
10
6
11
8
12
10
TOTAL
100
Awarded
Question 1* (8 marks)
Blacktop Refining extracts minerals from ore mined at two different sites in Montana. Each ton of ore
type 1 contains 20% copper, 20% zinc, and 15% magnesium. Each ton of ore type 2 contains 30% copper,
25% zinc, and 10% magnesium. Ore type 1 costs $90 per ton while ore type 2 costs $120 per ton.
Blacktop would like to buy enough ore to extract at least 8 tons of copper, 6 tons of zinc, and 5 tons of
magnesium in the least costly manner.
a. Formulate an LP model for this problem. [4 marks]
b. Sketch the feasible region for this problem and find the optimal solution. [4 marks]
Answer:
X1 = tons of ore purchased from mine 1, X2 = tons of ore purchased from mine 2
MIN
90 X1 + 120 X2
(cost)
ST
0.2 X1 + 0.3 X2 > 8
(copper)
0.2 X1 + 0.25 X2 > 6
(zinc)
0.15 X1 + 0.1 X2 > 5
(magnesium)
X1, X2  0
(4 marks)
(4 marks) – graphical analysis – 2 marks; Feasible region - 1 marks; optimal solution – 1 marks
Question 2 * (8 marks)
GSK Pharma produces a drug from two ingredients (ingredients 1 and 2). Each ingredient contains the same
3 antibiotics in different proportions. One gram of ingredient 1 contributes 3 units of antibiotic 1, and one
gram of ingredient 2 contributes 1 unit of the same antibiotic. The drug requires at least 6 units of antibiotic
1. At least 4 units of antibiotic 2 are required in the drug. Both ingredients 1 and 2 contribute 1 unit of
antibiotic 2 each per gram. At least 12 units of antibiotic 3 are required in the drug. One gram of ingredient
1 contributes 2 units of antibiotic 3, whereas one gram of ingredient 2 contributes 6 units of antibiotic 3.
The cost for one gram of ingredient 1 is $80, whereas the cost for one gram of ingredient 2 is $50. GSK
wishes to formulate a linear programming model to determine the number of grams of each ingredient to
procure in order to meet the antibiotic requirements of the drug at the minimum cost. Assume mass
conservation, answer the following:
(a) Formulate the linear programming model for this problem.
(b) Solve this model by using graphical analysis. List down all the corner points.
Answer:
(a)
Let:
X1 : Amount of ingredient 1 to procure (in grams)
X2 : Amount of ingredient 2 to procure (in grams)
[1 mark]
Minimize Z = 80x1 + 50x2
(1 mark)
Subject to
3x1 + x2 ≥ 6 (antibiotic 1, units)
x1 + x2 ≥ 4 (antibiotic 2, units)
2x1 + 6x2 ≥ 12 (antibiotic 3, units)
x1, x2 ≥ 0
[2 marks]
(b)
[4 marks - 1 mark for feasible region; 2 marks for graphical analysis and 1 mark for optimal solution]
Question 3 * (8 marks)
AIC processes insurance applications for large insurance companies. Insurance application processing is
done by a large number of workers. There are permanent staff as well as contractual staff for the
processing of these applications. A permanent staff can process 16 applications per day, whereas a
temporary staff can process 12 applications per day. On the average, AIC processes at least 450
applications each day.
AIC has 40 desktop workstations. The permanent staff generates 0.5 applications with errors each day,
whereas a temporary staff averages about 1.4 defective applications per day. The company wants to limit
the application processing with errors to 25 per day. The permanent staff is paid $64 per day and the
temporary staff is paid $42 per day. Each staff uses 1 desktop.
The company wants to determine the number of permanent and temporary staff to deploy in order to
minimize costs.
(a) Formulate a linear programming model for this problem [4 marks]
(b) Solve this model by using graphical analysis [4 marks]
Answer:
(a)
x1 = Number of permanent staff to employ
x2 = Number of temporary staff to employ
[1 mark]
Min z = 64 x1 + 42 x2
[1 mark]
Subject to:
16 x1 + 12 x2 ≥ 450 (claims per day)
x1 + x2 ≤ 40 (desktops)
0.5 x1 + 1.4 x2 ≤ 25 (defective applications)
x1 , x2 ≥ 0
[2 mark]
(b)
[4 marks - 1 mark for feasible region; 2 marks for graphical analysis and 1 mark for optimal solution]
Question 4 [Product Mix] * (12 marks)
F&N beverages produces 2 mixtures of fruit drinks: Berry Health (BM) and Berry Delight (BD). The profit
contributions are $0.60 per gallon for BH and $1 per gallon for BD. Each gallon of BH contains 0.5 gallons
of strawberry mix and each gallon of BD contains 1 gallon of strawberry mix. For the next production
period, F&N has 30,000 gallons of strawberry mix available. The production line used to produce the BM
and BD has a production capacity of 40,000 gallons for the next production period. F&N distributors have
indicated that demand for the BD for the next production period will be at most 25,000 gallons.
a) Formulate a linear programming model that can be used to determine the number of gallons of BH
and the number of gallons of BD that should be produced in order to maximize total profit
contribution using graphical analysis. What is the optimal solution? [8 marks]
b) What are the values and interpretations of the slack variables? [3 marks]
c) What are the binding constraints? [1 mark]
(a) Let
H = number of gallons of BH produced
D = number of gallons of BD produced
[1 mark]
Max z = 0.60H + 1D
[1 mark]
s.t.
0.50H + 1D ≤ 30,000
1H + 1D ≤ 40,000
1D ≤ 25,000
H,D ≥ 0
[2 marks]
(F&N Strawberry mix available)
(Production capacity)
(Demand for BD)
Optimal solution:
20,000 gallons of BH
20,000 gallons of BD
[4 marks] [feasible region -1; graphical methods – 2; optimal solution – 1]
b)
Constraint
1
2
3
Value of slack
variable
0
0
5,000
Interpretation
All available strawberry mix is used
Total production capacity is used
BD is 5,000 less than the maximum demand
[3 marks]
c) Strawberry mix and the production capacity are the binding constraints.
[1 mark]
* Question 5 [6 marks]
Given the following linear program:
Max
z = x1 – 2x2
Subject to:
-4x1 + 3x2 ≤ 3
x1 – x2 ≤ 3
x 1, x 2 ≥ 0
(a) Solve the model using graphical analysis for this problem. [3 marks]
(b) Is the feasible region unbounded? Explain [1 mark]
(c) Does an unbounded region imply that there is no optimal solution? Explain your answer. [2
marks]
Solution:
(a)
The optimal solution is x1 = 3, x2 = 0, z = 3
[3 marks] [methods – 1 mark; feasible region – 1 mark; optimal solution – 1 mark]
(b) The feasible region is unbounded
[1 mark]
(c)
An unbounded region does not imply the problem is unbounded. This will only be the case when it
is unbounded in the direction of improvement of the objective function.
[2 marks]
* Question 6 [16 marks]
The manager of a Cookies and Crumble (C&C) wants to determine how many Chocolate and Fruit Tarts to
prepare each morning for breakfast customers. The two types of tarts require the following:
Tarts
Labour (hr.) Chocolate Paste (lb.) Fruit Paste (lb.) Flour (lb.)
Chocolate 0.010
0.10
—
0.04
Fruits
0.024
—
0.15
0.04
C&C has 6 hours of labor available each morning. The manager has a contract with a local grocer for 30
pounds of fruit paste and 30 pounds of chocolate paste each morning. The manager also purchases 16
pounds of flour. The profit for a chocolate tart is $0.60 per pc; the profit for a fruit tart is $0.50 per pc. The
manager wants to know the number of each type of tart to prepare each morning in order to maximize profit.
a. Formulate a linear programming model for this problem and solve it graphically. [6 marks]
b. How much extra chocolate and fruits is left over with the optimal production plan? Is there any idle
labor time? [2 marks]
c. What would the solution be if the profit for a fruit tart was increased from $0.50 to $0.60? Explain
your reasoning and any special observation from the graphical analysis. [2 marks]
d. Formulate and solve the LP model in Excel. Show the relevant tables. Identify and explain the
shadow prices for each of the resource constraints. [4 marks]
e. Based on the Excel solution, identify the sensitivity ranges for the profit of a chocolate tart and the
amount of chocolate paste available. Explain these sensitivity ranges. [2 marks]
Answer:
(a)
X1: Number of chocolate tarts to produce
X2: Number of fruit tarts to produce
Minimize Z = 0.6X1 + 0.5X2
subject to:
0.01X1 + 0.024X2 ≤ 6 hours [Labour]
0.1X1 ≤ 30 lbs [Chocolate paste]
0.15X2 ≤ 30 lbs [Fruit paste]
0.04X1 + 0.04X2 ≤ 16 lbs [Flour]
X1 , X2 ≥ 0
[3 marks]
[3 marks] [ methods: 1; optimal: 1; feasible: 1]
(b)
x1 = 300, x2 = 100, Z = $230
.10(300) + s1 = 30
s1 = 0 leftover Chocolate Paste
.15(100) + s2 = 30
s2 = 15 lbs. leftover Fruit Paste
.01(300) + .024(100) + s4 = 6
s4 = 0.6 hr labour time
0 lb of chocolate paste and 15lb of fruit paste are left over. There is 0.6 hours of idle labor hours.
[2 marks]
(c) Increase of $0.10 is within sensitivity range.
The slope of the objective function, −6/5, must become flatter (i.e., less) than the slope of the
constraint line, .04x1 + .04x2 = 16, for the solution to change. The profit for fruit, c2, which would
change the solution point is,
−0.6/c2 = −1
c2 > .60
Thus, an increase in profit for fruit tarts to 0.60 will create multiple alternative optimal along the line
segment CD with Z = $240.
[2 marks]
(d)
[2 marks]
[1 mark]
The shadow price for Chocolate Paste is $1. For every additional pound of chocolate paste that can be
obtained, profit will increase by $1.
The shadow price for flour is $12.50. For each additional pound of flour that can be obtained, profit
will increase by this amount.
There are extra fruit paste and labor hours available, so their shadow prices are zero, indicating
additional amounts of those resources would add nothing to profit.
[1 marks]
(e)
.50 ≤ c1 ≤ ∞
25.7 ≤ b2 ≤ 40
The sensitivity range for profit indicates that the optimal mix of chocolate and fruit tarts will
remain optimal as long as profit for chocolate tart does not fall below $0.50.
The sensitivity range for chocolate paste indicates the shadow price of $1 will be maintained as
long as the available chocolate paste is between 25.7 and 40 lbs (solution mix remains the same).
[2 marks]
QUESTION 7 * [4 marks for Part (a); other parts are optional for practice]
Xara Stores in the United States stocks a particular type of designer denim jeans that is manufactured in
China and Brazil and imported to the Xara distribution center in the United States. It orders 500 pairs of
jeans each month from its two suppliers. The Chinese supplier charges Xara $11 per pair of jeans, and the
Brazilian supplier charges $16 per pair (and then Xara marks them up almost 1,000%). Although the jeans
from China are less expensive, they also have more defects than those from Brazil. Based on past data,
Xara estimates that 7% of the Chinese jeans will be defective compared to only 2% from Brazil, and Xara
does not want to import any more than 5% defective items. However, Xara does not want to rely only on
a single supplier, so it wants to order at least 20% from each supplier every month.
(a) Formulate a linear programming model for this problem. [4 marks]
(b) Solve the linear programming model for Xara Stores graphically.
(c) If the Chinese supplier were able to reduce its percentage of defective pairs of jeans from 7% to
5%, what would be the effect on the solution?
(d) If Xara Stores decided to minimize its defective items while budgeting $7,000 for purchasing the
jeans, how would you deal with this situation? What is the new decision problem?
Solution:
(a)
X1: Number to order from China
X2: Number to order from Brazil
Minimize Z = 11x1 + 16x2
subject to
x1 + x2 = 500
.07x1 +.02x2 ≤ 25
x1
 .20
x1 + x2
x2
 .20
x1 + x2
x1, x2 ≥ 0
[4 marks]
(b)
Graphically:
(c)
Change in optimal solution to the following:
x1 = 400,
x2 = 100,
z = $6,000
(d)
Reformulate and solve again the following model:
Minimize Z = .07x1 + .02x2
subject to
11x1 + 16x2 ≤ 7,000
x1 + x2 = 500
x1
 .20
x1 + x2
x2
 .20
x1 + x2
x 1, x 2 ≥ 0
x1 = 200
x2 = 300
Z = 20
Note: Those with * are the graded questions.
* QUESTION 8 [10 marks]
The Mercedes Club of Singapore sponsors driver education events that provide high performance driving
instruction on actual race tracks. Because safety is a primary consideration at such events, many owners
elect to install roll bars in their cars for the multilink suspension system.
F1 Auto manufactures two types of roll bars for Mercedes. Model CLA is bolted to the car using existing
holes in the car’s frame. Model CLB is a heavier roll bar that must be welded to the car’s frame. Model
CLA requires 10 pounds of a special high alloy steel, 30 minutes of manufacturing time, and 80 minutes
of assembly time. Model CLB requires 30 pounds of the special high alloy steel, 90 minutes of
manufacturing time, and 40 minutes of assembly time.
F1 Auto steel supplier indicated that at most 50,000 pounds of high-alloy steel will be available next
quarter. In addition, F1 estimates that 1,500 hours of manufacturing time and 2,000 hours of assembly
time will be available next quarter. The profit contributions are $100 per unit for model CLA and $150
per unit for model CLB.
a. Formulate the Linear Programming model for the production problem. [3 marks]
b. Solve the LP model using Excel and show the relevant reports. What are the optimal solution and
total profit contribution? [3 marks]
c. F1 is considering using overtime for manufacturing time to meet a new order. The Union proposed
a rate of $1 over the normal rate per hour for additional 700 hours of manufacturing time. What
would you advise F1 Auto Management to do regarding this option? Explain. [2 marks]
d. Because of increased competition, F1 is considering reducing the price of model CLA such that the
new contribution to profit is $75 per unit. How would this change in price affect the optimal
solution? Explain. [2 marks]
Solution
(a) The linear programming model of this problem is as follows:
𝑀𝑎𝑥 100𝐶𝐿𝐴 + 150𝐶𝐿𝐵
𝑠. 𝑡.
10𝐶𝐿𝐴 + 30𝐶𝐿𝐵 ≤ 50,000 𝑆𝑡𝑒𝑒𝑙 𝑎𝑣𝑎𝑖𝑙𝑎𝑏𝑙𝑒 𝑖𝑛 𝑙𝑏𝑠
30𝐶𝐿𝐴 + 90𝐶𝐿𝐵 ≤ 90,000 𝑀𝑎𝑛𝑢𝑓𝑎𝑐𝑡𝑢𝑟𝑖𝑛𝑔 𝑚𝑖𝑛𝑢𝑡𝑒𝑠
80𝐶𝐿𝐴 + 40𝐶𝐿𝐵 ≤ 120,000 𝐴𝑠𝑠𝑒𝑚𝑏𝑙𝑦 𝑚𝑖𝑛𝑢𝑡𝑒𝑠
𝐶𝐿𝐴, 𝐶𝐿𝐵 ≥ 0
[3 marks]
(b)
[1 mark]
Produce 1200 units of CLA and 600 units of CLB
Total profit contribution = $210,000
[1 mark]
[1 mark]
(c) The shadow price for constraint 2 (Manufacturing hours) is $1.33. Thus, each additional
manufacturing hour will increase by the profit by $1.33.
At $1 overtime premium per hour, F1 should still increase the overtime hours as he will still make
a profit.
The upper limit on the right hand side for constraint 2 is 150,000 minutes. Thus, the shadow price
of $1.33 is applicable for an increase of as much as 60,000 minutes (1000 hours), so the shadow
price is applicable for the additional 700 hours.
[2 marks]
(d) The objective coefficient range for model CLA shows a lower limit of $50. Thus, the optimal
solution will not change; the value of the optimal solution will be $75(1,200) + $150(600) =
$180,000.
[2 marks]
* QUESTION 9 [4 marks]
SR Company manufactures two types of electric cars: model 1 and model 2. However, SR’s
production capability is limited in three departments: production, assembly, and packaging. The
following table summarizes the hours of processing time available and the processing time
required by each department, for both models:
Hours Required per Car model
model 1 model 2 Hours Available
Production
2
4
10,000
Assembly
3
5
15,000
Packaging
10
14
5,000
The company sells each model 1 for $15,000 and each model 2 for $25,000. The sale of model 1
and model 2 are predicted to be not more than 10,000 and 20,000 respectively.
How many cars should SR make? Formulate a model to help SR make the decision.
Solution
[4 marks]
* QUESTION 10 [6 marks]
The following model is a resultant model of a production problem for 2 products (represented by the
decision variables) with 2 constraints relating to the resources used. The decision maker seeks to
increase the profit of the products.
𝑚𝑎𝑥 𝑍 = 𝑥1 + 2𝑥2
𝑠. 𝑡.
𝑥1 + 𝑥2 ≤ 10
2𝑥1 + 𝑥2 ≤ 16
𝑥1 , 𝑥2 ≥ 0
(Resource 1)
(Resource 2)
a) Draw the feasible region of the following linear programming model, and solve it graphically. [4
marks]
b) What is the sensitivity range for 𝑐2 (the coefficient of 𝑥2 in the objective function)?
Solution:
(a)
[Feasible region: 1m; Method/solution: 2m; optimal: 1m]
(b)
2
𝑍
2
2
The objective function is 𝑍 = 2𝑥1 + 𝑐2 𝑥2 , and we rewrite as 𝑥2 = − 𝑐 𝑥1 + 𝑐
As 𝑐2 increases, the gradient gets less negative. Optimal solution remains the same.
As 𝑐2 decreases, the gradient gets more negative. Optimal solution remains the same until
2
2
2
2
− 𝑐 = gradient of 𝑥1 + 𝑥2 ≤ 10 ⇒ − 𝑐 = -1 ⇒ 𝑐2 = 2.
Hence, sensitivity range is 𝑐2 ≥ 2.
[2 marks]
QUESTION 11 * (8 marks)
Consider the following linear programming model:
Minimize 𝑍 = 8𝑥1 + 2 𝑥2
s.t. 2𝑥1 − 6𝑥2 ≤ 12
5𝑥1 + 4𝑥2 ≥ 40
𝑥1 + 2𝑥2 ≥ 12
𝑥2 ≤ 6
𝑥1 , 𝑥2 ≥ 0
a) Solve the linear programming model graphically. (6 marks)
b) What happens if we remove the third constraint? (2 marks)
Solution:
(a)
Feasible region: 2m; Corner points: 2m; Solution: 2m
(b)
The feasible region changes but the optimal solution will remain the same.
* Question 12 [10 marks]:
You have just joined Plato Ventures (PV), a technology venture capitalist (VC) specializing in high tech
platform companies. The CEO of PV has directed you to evaluate 3 potential start-up platform companies
for an investment program. The amount that PV has for this investment is $10 million ($10M). In addition,
PV can only assign at most 600HRS of consulting manpower for these start-ups.
Given the maturity of each start-up, the amount of capital and manpower commitments to invest by PV to
maximize their success of these startups varies. All the start-ups are flexible and would agree to any fraction
of the partnership commitments that were proposed by PV. However, the profits will be multiplied
according to same fraction of capital and manpower invested. The proposed commitments and estimated
profits for the 3 startups are as follows:
Start-up 1 (SU1)
Capital Investment $10M
by PV
Manpower
400HRS
Investment by PV
Estimated Profit
$9M
Start-up 2 (SU2)
$8M
Start-up 3 (SU3)
$5M
500HRS
300HRS
$9M
$7M
With the growing success of platform companies in the industry, PV CEO’s directive is to invest the entire
$10M of VC funds in these start-ups to maximize the profits.
a) Formulate a linear programming model to determine the optimal fraction of capital and manpower
to be allocated to each of the startup.
[4 marks]
b) Using graphical analysis, solve this linear programming model considering the following:
• Reformulate the LP if needed,
• Draw and label all the constraints.
• Shade the feasible region in the graphical analysis and indicate redundant constraints.
[6 marks]
Answer:
(1)
x1 = Fraction of capital and manpower allocated to SU1
x2 = Fraction of capital and manpower allocated to SU2
x3 = Fraction of capital and manpower allocated to SU3
[1 mark]
maximize Z = 9x1 + 9x2 + 7x3 [in $M]
[1 mark]
subject to
CAPITAL:
10x1 + 8x2 + 5x3 = 10
MANPOWER: 400x1 + 500x2 + 300x3 ≤ 600
Fractions:
x1 ≤ 1
x2 ≤ 1
x3 ≤ 1
x1,x2,x3 ≥ 0
[2 marks]
[TOTAL: 4 marks]
(2)
x3 = 2 - 2x1 – 8/5x2
Objective Function Reformulation:
Z = 9x1 + 9x2 + 7(2 - 2x1 – 8/5x2) = -x1 + x2 + 10
Manpower Reformulation:
400x1 + 500x2 + 300(2 - 2x1 – 8/5x2) ≤ 600
 -200x1 + 20x2 ≤ 0
 -10x1 + x2 ≤ 0
Fractions Reformulation:
x1 ≤ 1
x2 ≤ 1
2 - 2x1 – 8/5x2 ≤ 1
 2x1 + 8/5x2 ≥ 1
Non-negative Reformulation:
2 - 2x1 – 8/5x2 ≥ 0
 2x1 + 8/5x2 ≤ 2
[Present Re-formulated LP: 2 marks]
Note that there are other possible reformulation in terms of X1, X3, and X2, X3. All can be given
full marks
-10x1+x2=0
x2 = 1
x1 = 1
[method: 2 marks; Shade/Indicate Feasible Region: 1 marks; optimal solution: 1 mark]
[Total Graphical Analysis: 4 marks]
[TOTAL: 10 marks]
EXTRA PRACTICE QUESTION (OPTIONAL)
[Ref: Textbook 13th Edition; Chapter 3, Question 52-53]
Note: Use Excel Solver for Sensitivity Analysis. Extract the relevant Excel reports for the answer script.
Exeter Mines produces iron ore at four different mines; however, the ores extracted at each mine are
different in their iron content. Mine 1 produces magnetite ore, which has a 70% iron content; mine 2
produces limonite ore, which has a 60% iron content; mine 3 produces pyrite ore, which has a 50% iron
content; and mine 4 produces taconite ore, which has only a 30% iron content. Exeter has three customers
that produce steel—Armco, Best, and Corcom. Armco needs exactly 400 tons of pure (100%) iron, Best
requires 250 tons of pure iron, and Corcom requires 290 tons.
It costs $37 to extract and process 1 ton of magnetite ore at mine 1, $46 to produce 1 ton of limonite ore at
mine 2, $50 per ton of pyrite ore at mine 3, and $42 per ton of taconite ore at mine 4. Exeter can extract
350 tons of ore at mine 1; 530 tons at mine 2; 610 tons at mine 3; and 490 tons at mine 4. The company
wants to know how much ore to produce at each mine in order to minimize cost and meet its customers’
demand for pure (100%) iron.
a) Formulate a linear programming model for this problem.
b) Solve the linear programming model for Exeter Mines by using Excel. Extract the Answer and
Sensitivity Excel reports for the answer.
c) Do any of the mines have slack capacity? If yes, which one(s)?
d) If Exeter Mines could increase production capacity at any one of its mines, which should it be?
Why?
e) If Exeter decided to increase capacity at the mine identified in (d), how much could it increase
capacity before the optimal solution point (i.e., the optimal set of variables) would change?
f) If Exeter determined that it could increase production capacity at mine 1 from 350 tons to 500
tons, at an increase in production costs to $43 per ton, should it do so?
Solution
(a)
Xij: Amount of pure iron ore to produce from Mine i for Customer j
i ={1,2,3} where 1: Mine 1; 2: Mine 2; 3: Mine 3
and
j = {1, 2, 3} where 1: Armco; 2: Best; 3: Corcom
Minimize Z = $37x11 + 37x12 + 37x13 + 46x21 + 46x22 + 46x23 + 50x31 + 50x32 + 50x33 + 42x41 + 42x42 +
42x43
subject to:
.7x11 + .6x21 + .5x31 + .3x41 = 400 tons
.7x12 + .6x22 + .5x32 + .3x42 = 250 tons
.7x13 + .6x23 + .5x33 + .3x43 = 290 tons
x11 + x12 + x13 ≤ 350 tons
x21 + x22 + x23 ≤ 530 tons
x31 + x32 + x33 ≤ 610 tons
x41 + x42 + x43 ≤ 490 tons
xij ≥ 0
(b)
x13 = 350 tons
x21 = 158.333 tons
x22 = 296.667 tons
x23 = 75 tons
x31 = 610 tons
x42 = 240 tons
Z = $77,910
Mine 1 = 350 tons
Mine 2 = 530 tons
Mine 3 = 610 tons
Mine 4 = 240 tons
Multiple optimal solutions exist – Check solutions
Alternative Solution [Solution 2] :
Decisions j
Xij
i
1
1
2
3
0 38.57143 311.4286
2 158.3333 371.6667
0
3
610
0
0
4
0
0
240
Excel Formulation :
Parameters
Cost
Constraints
1
2
3
1
2
3
4
37
46
50
42
37
46
50
42
37
46
50
42
1
2
3
4
1
0.7
0.6
0.5
0.3
2
0.7
0.6
0.5
0.3
3
0.7
0.6
0.5
0.3
Quality
Decisions j
Xij
i
1
2
3
1
0 38.57143 311.4286
2 158.3333 371.6667
0
3
610
0
0
4
0
0
240
1
2
3
Demand
400
=
400
250
=
250
290
=
290
Supply
350
530
610
240
<=
<=
<=
<=
350
530
610
490
Demand
LHS
Objective
77910
(c) Mine 4 has 250 tons of “slack” capacity.
(d) The shadow price for the 4 constraints representing the capacity at the 4 mines show that Mine 1
has the highest shadow price of $61, so its capacity is the best one to increase.
(e) The sensitivity range for Mine 1 is 242.86 ≤ c1 ≤ 452.86, thus capacity could be increased by
107.142857 tons before the optimal solution point would change.
(f) The effect of simultaneous changes in objective function coefficients and constraint quality values
cannot be analyzed using the sensitivity ranges provided by the computer output. It is necessary to
make both changes in the model and solve it again. Doing so results in a new solution with Z =
$73,080, which is $4,830 less than the original solution, so Exeter should make these changes.
Constraints
Parameters
Cost
1
2
3
4
1
43
46
50
42
2
43
46
50
42
3
43
46
50
42
1
2
3
4
1
0.7
0.6
0.5
0.3
2
0.7
0.6
0.5
0.3
3
0.7
0.6
0.5
0.3
Quality
Decisions
i
j
1
2
3
4
Objective
Xij
1
2
3
0 85.71429 414.2857
213.3333 316.6667
0
544
0
0
0
0
0
73080
----- End of Assignment -----
Demand LHS
Demand
1
400
=
400
2
250
=
250
<=
<=
<=
<=
500
530
610
490
Supply
500
530
544
0
3
290
=
290