辅导案例-DBA5103-Assignment 2

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DBA5103: Operations Research and Analytics Semester I, 2020/2021, NUS
Assignment 2: Due on Sep.27, 2020
1. (10’) Consider the following optimization problem:
max 10x1 + 12x2 + 12x3
s.t. x1 + 2x2 + 2x3 ≤ 20
2x1 + x2 + 2x3 ≤ 20
2x1 + 2x2 + x3 ≤ 20
x1, x2, x3 ≥ 0.
(a) Write down the dual problem.
(b) Suppose the optimal solution to the primal problem is x1 = x2 = x3 = 4. Solve the optimal
solution to the dual problem using complementary slackness condition.
2. (10’) Portfolio Management Problem: Consider the portfolio management problem discussed in
class. There are 100 stocks. The expected return of stock i is ri per dollar invested (which could be
negative) and the total budget is b dollars, where b > 0. Ignoring the risk of the return, the problem
of maximizing the total expected return is
max
100∑
i=1
rixi
s.t.
100∑
i=1
xi ≤ b
xi ≥ 0, i = 1, ..., 100,
where xi denotes the amount of money invested in stock i.
(a) Write down the dual problem to the above linear programming problem.
(b) What is the optimal solution to the primal and the dual respectively?
(c) Verify the solution you provided in (b) is indeed optimal using the weak duality.
3. (10’) A potter manufacturer can make four different types of dining room service sets: JJP English,
Currier, Primrose, and Bluetail. Furthermore, Primrose can be made by two different methods. Each
set uses clay, enamel, dry room time, and kiln time, and results in a profit shown in Table 2.1.
The manufacturer is currently committed to making the same amount of Primrose using method 1 and
method 2. The formulation of the profit maximization problem is given below. The decision variables
E,C, P1, P2, B are the number of sets of type English, Currier, Primrose Method 1, Primrose Method
2, and Bluetail, respectively. We assume, for the purposes of this problem, that the number of sets of
2-1
Assignment 2: Due on Sep.27, 2020 2-2
Table 2.1: Profit and resources consumed for each product
E C P1 P2 B Capacity
Clay (lbs) 10 15 10 10 20 130
Enamel (lbs) 1 2 2 1 1 13
Dry room (hours) 3 1 6 6 3 45
Klin (hours) 2 4 2 5 3 23
Profit 51 102 66 66 89
each type can be fractional.
max 51E + 102C + 66P1 + 66P2 + 89B
s.t. 10E + 15C + 10P1 + 10P2 + 20B ≤ 130
E + 2C + 2P1 + P2 + B ≤ 13
3E + C + 6P1 + 6P2 + 3B ≤ 45
2E + 4C + 2P1 + 5P2 + 3B ≤ 23
P1 − P2 = 0
E,C, P1, P2, B ≥ 0.
The optimal solution to the primal is given in Table 2.2 and the optimal solution to the dual together
with sensitivity information is given in Table 2.3. Use the information to answer the questions that
follow.
Table 2.2: The optimal solution to the primal
Optimal Value Reduced Cost Objective Coefficient
E 0 -3.571 51
C 2 0 102
P1 0 0 66
P2 0 -37.571 66
B 5 0 89
Table 2.3: The optimal solution to the dual and the sensitivity. The last two columns describe the allowed
changes in the components of b for which the optimal dual solution remains the same
Dual Variable Const. RHS Allowable Increase Allowable Decrease
Clay 1.429 130 23.33 43.75
Enamel 0 13 ∞ 4
Dry room 0 45 ∞ 28
Klin 20.143 23 5.60 3.50
Primrose Methods 11.429 0 3.50 0
(a) What is the optimal quantity of each service set, and what is the total profit?
(b) Give an economic interpretation of the optimal dual variables appearing in the sensitivity report,
for each of the five constraints.
(c) Should the manufacturer buy an additional 20 lbs. of Clay at $1.1 per pound?
Assignment 2: Due on Sep.27, 2020 2-3
(d) In the current model, the number of Primrose produced using method 1 was required to be the
same as the number of Primrose produced by method 2. Consider a revision of the model in which
this constraint is replaced by the constraint P1 − P2 ≥ 0. In the reformulated problem would the
amount of Primrose made by method 1 be positive?
4. (10’) Diet Problem: The “diet problem” concerns with finding a mix of foods that satisfies require-
ments on the amount of nutrition. We consider here the problem of choosing fruits to meet certain
nutritional requirements. The nutrition information of five selected fruits is provided in Table 2.4
along with a minimum required and maximum allowed intake of each type of nutrition in the last two
columns. Your goal is to find the cheapest combination of fruits that will meet nutritional requirements.
Table 2.4: Nutrition and price information per 100 grams
Apple Banana Blueberries Durian Tangerine Min Reqd Max Allowed
Calories 52 89 57 147 53 500 3000
Carbohydrate (g) 14 23 14 27 13 50 400
Fiber (g) 2.4 2.6 2.4 3.8 1.8 20 30
Vitamin A (IU) 54 64 54 44 681 2000 3500
Vitamin C (mg) 4.6 8.7 9.7 19.7 26.7 75 150
Price (S$/100g) 0.5 0.3 2.5 10 0.5
(a) Formulate a mathematical model for this problem. Solve the problem using the software you prefer
and attach your code. Present and interpret the results.
(b) Without solving the problem using the software, can you determine how your optimal solution
and optimal spending in (a) will change if you are also concerned with the total fat intake. The
fat contained in each of the fruit and your nutrition requirement on fat is provided in the table
below. Verify your solution by solving the problem numerically.
Apple Banana Blueberries Durian Tangerine Min Reqd Max Allowed
Fat (g) 0.2 0.3 0.3 5 0.3 0 10
5. (10’) Markdown Management: Consider the benchmark model and solution for the markdown
management problem discussed in class. The initial inventory is 2000 units. The retailer needs to sell
his inventory over a time horizon of 15 weeks by employing in order the four price levels: $60, $54,
$48, $36 over time. The estimated average demands under these price levels are summarized in the
following table. Any left over items at the end of the selling season can be sold to outlet for $25 per
Table 2.5: Benchmark model
Price $60 $54 $48 $36
Demand 125 162.5 217.5 348.8
unit (salvage value). The retailer is required to start with the full price for at least one week and then
he is allowed to decrease his prices over time. We have shown in class the optimal policy is to use for
the first 11.67 weeks of full price $60 and then 3.33 weeks of $54 (ignoring integrality constraints). The
revenue collected under this policy is $116750. Consider the following three scenarios separately and
answer the corresponding questions. Attach the codes you used to derive the solutions.
Assignment 2: Due on Sep.27, 2020 2-4
(a) Suppose in order to maintain the brand reputation, the retailer is only allowed to discount from
week 4. Will the retailer change his markdown strategy in this case?
(b) Due to bad weather, only 25 units were sold in the first week. How should the retailer adjust his
markdown policy? What is the improvement in revenue compared to the case when he sticks with
the original markdown policy (specified above)?
(c) In the traditional brick-and-mortar business, menu cost and consumer price impression are the
two major reasons behind the very few number of price levels. With the rise of e-commerce and
other technologies, it is possible for the retailer to employ more price levels and change the price
more frequently. In the retailer game, what happens if the retailer employs the following price
grids (with a grid of $2) instead of a few discount levels? For simplicity, the demand estimates
at the additional price levels are taken as a linear interpolation of the original demands. What
is the improvement in revenue compared to the benchmark? What is the new markdown policy?
How do you compare with the original policy?
Table 2.6: Price grids model
Price $60 $58 $56 $54 $52 $50 $48 $46 $44 $42 $40 $38 $36
Demand 125 137.5 150 162.5 180.8 199.1 217.5 239.4 261.3 283.2 305.1 327 348.8

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