Advanced Modeling LP Applications

1
Advanced Modeling LP Applications
2
LEO Dairy. Inc. produces three products liquid
milk, yogurt and milk powder. In February, LEO
Dairy has 100 tons of raw milk available. It
takes one unit of raw milk to produce one unit of
liquid milk, two units of raw milk to produce one
unit of yogurt, and eight units of raw milk to
produce one unit of milk powder. The maximum
demand for yogurt is 20 tons per month. The
maximum demand for milk powder is 10 tons per
month. The demand of liquid milk is unlimited. In
order to keep the market share, LEO Dairy at
least produces 5 tons of liquid milk each month.
The profit of the one ton liquid milk is 1,000,
one ton yogurt is 2,200 and one ton milk powder
is 10,000 LEO Dairy wants to maximize the total
profit in February. Solve in Excel. Decision
Variables Let x1 denote the number of tons of
liquid milk produced Let x2 denote the number of
tons of yogurt produced Let x3 denote the number
of tons of milk powder produced
3
Model Maximize Total Profit Max
1000x12200x210000x3 s.t. x12x28x3 lt 100
(Availability of Raw Milk) x2
lt 20 (Max demand of yogurt)
x3 lt 10 (Max demand of milk powder) x1 gt 5
(Min req. on liquid milk) x1, x2, x3, x4 gt 0
4
(No Transcript)
5
(No Transcript)
6
Q1 If the liquid milk profit now is 1080 per
ton, should we produce more liquid milk? Q2 If
the profit of milk powder now is 8000 per ton,
should we change our production plan? Q3 (I am
tricky) If we have 1500 Investment which can be
used in one of the following options 1. Invest
in TV commercial and increase the demand of
milk powder by 1.5 tons. 2. Purchase 2 tons of
raw milk. 3. Increase the demand of yogurt by 4
units Q4. Would it be beneficial if we decrease
the level of liquid milk market share?
7
Work Schedule Multi-period Investment Multi-pe
riod Inventory
8
The work schedule problem occurs in many
management contexts. In general, it is the
problem of determining the optimal assignment of
n agents or objects to n tasks. The agents or
objects to be assigned are indivisible in the
sense that no agent can be divided among several
tasks. Basic Characteristics Objective
Function - Minimize cost Constraints Meet
Shift Requirements
9
Security Force Scheduling The personnel manager
at IBMs Fishkill plant must schedule
the security force in such a way as to satisfy
the following staffing requirements
10
Make sure that a particular set of values for x1
,, x6 satisfies the staffing requirements.
11
Model Decision Variables Let xi, i1,2,3,4,5,6
denote the of officers working shift
i Minimize cost Z x1 x2 x3 x4 x5 x6
s.t. x1 x6 gt 5 (Shift1) x1 x2 gt 7
(Shift2) x2 x3 gt 15 (Shift3) x3 x4 gt 7
(Shift4) x4 x5 gt 12 (Shift5) x5 x6 gt 9
(Shift6) x1,x2,x3,x4,x5,x6 gt 0
(Non-negativity)
12
Work Schedule Finance Application Multi-period
Inventory
13

• Welte Mutual Funds, Inc. is located in NYC. Welte
  just obtained 100,000 by converting industrial
  bounds to cash and is now looking for other
  investment opportunities for these funds. Based
  on Weltes current investments, the firms top
  financial analyst recommends that all new
  investments be made in the oil industry, steel
  industry, or in government bounds. Specifically,
  the analyst identified five investment
  opportunities and projected their annual rates of
  return. The investment and rates of return are
  shown below

14

• Management of Welte imposed the following
  investment guidelines
• Neither industry (oil or steel) should receive
  more than 50,000.
• Government bounds should be at least 25 percent
  of the steel industry in investments.
• The investment in Pacific Oil, the high-return
  but high-risk investment, cannot be more than 60
  of the total oil industry investment.

15
Decision Variable A dollars invested in
Atlantic Oil P dollars invested in Pacific
Oil M dollars invested in Midwest Steel H
dollars invested in Huber Steel G dollars
invested in government bonds Objective Max
0.073A 0.103P 0.064M 0.075H
0.045G Subject to APMHG lt 100,000 (Available
Funds) A P lt 50,000 (Oil industry
maximum) MH lt 50,000 (Steel industry
maximum) G gt 0.25 (MH) ? -.25M-0.25HG gt 0
(Gov. bonds req.) P lt 0.60 (AP) ? -0.6A0.4P lt
0 (Pacific Oil req.) A,P,M,H,G gt 0
16
(No Transcript)
17
(No Transcript)
18

• What is the marginal rate of return on the
  portfolio?
• Obviously the firm didnt choose to invest in
  Midwest Steel. How much improvement the
  management would like to see in order to invest
  in Midwest Steel.
• As an investment consultant, what changes would
  you suggest to make the portfolio more
  profitable?
• The government bond constraint has a shadow price
  of -0.024. Is there any way to justify it?

19

• Lets modify the constraints limiting investments
  in the oil and steel industries as follows
• No more than 50 of the total funds invested in
  stock (oil and steel) may be invested in the oil
  industry
• No more than 50 of funds invested in stock (oil
  and steel) may be invested in the steel industry.
• How would you modify the model?

20

• A wealthy investor has 3 investment opportunities
  available at the beginning of each of the next 4
  years and also has a total of 500 available for
  investment at the beginning of year 1 (all is
  invested). Immediate reinvestment is possible
  for each alternative. The investor wants to
  determine a plan that will maximize the amount of
  money that he will have at the beginning of year
  5. Formulate the model and determine what are
  the decision variables, objective function,
  constraints, and parameters.

21
Beg of Yr 1
Beg of Yr 2
Beg of Yr 3
Beg of Yr 4
Beg of Yr 5

• x1
• x2
• x3
• x4
• x5
• x6
• x7
• x8
• x9

22

• The model is
• Let x1 the amount of invested in beginning of
  Yr 1 in Alt 1
• x2 the amount of invested in beginning of
  Yr 1 in Alt 2
• x3 the amount of invested in beginning
  of Yr 1 in Alt 3
• x4 the amount of invested in beginning of
  Yr 2 in Alt 1
• x5 the amount of invested in beginning of
  Yr 2 in Alt 2
• x6 the amount of invested in beginning
  of Yr 2 in Alt 3
• x7 the amount of invested in beginning of
  Yr 3 in Alt 1
• x8 the amount of invested in beginning of
  Yr 3 in Alt 2
• x9 the amount of invested in beginning of
  Yr 4 in Alt 1
• Max Return 1.1x6 1.06x8 1.05x9
• s.t. x1x2x3 lt 500 (Available Allocation at
  beginning of year 1)
• x4x5x6 lt 1.05x1 (Available Allocation at
  beginning of year 2)
• x7x8 lt 1.05x4 1.06x2 (Available Allocation
  at beginning of year 3)
• x9 lt 1.05x7 1.06x5 1.1x3 (Available
  Allocation at beginning of year 4)
• x1,x4,x7,x9 lt 100 (Max Investment in
  Alternative 1)

Beg of Yr 1
Beg of Yr 2
Beg of Yr 3
Beg of Yr 4
23
Work Schedule Finance Application Multi-period
Inventory
24
The multi-period inventory problem is concerned
with producing or buying products over a time
horizon to meet demand and deal with limited
capacity, etc. Basic Characteristics Objective
Function - Minimize cost Constraints Meet
Demand 1st period current production
current inv Demand Middle Periods previous
inv current prod current inv
Demand Last Period previous inv current prod
Demand Capacity
25
The multi-period inventory problem is concerned
with producing products over a time horizon to
meet demand and deal with limited capacity,
etc. Basic Characteristics Objective Function
- Minimize cost Constraints Meet Demand
1st period current production current inv
Demand Middle Periods previous inv current
prod current inv Demand Last
Period previous inv current prod
Demand Capacity
If you had a two-period model, which Demand
constraints would you use?
26
The multi-period inventory problem is concerned
with producing products over a time horizon to
meet demand and deal with limited capacity,
etc. Basic Characteristics Objective Function
- Minimize cost Constraints Meet Demand
1st period current production current inv
Demand Middle Periods previous inv current
prod current inv Demand Last
Period previous inv current prod
Demand Capacity
If you had a two-period model, which Demand
constraints would you use? The 1st and Last
27
The multi-period inventory problem is concerned
with producing products over a time horizon to
meet demand and deal with limited capacity,
etc. Basic Characteristics Objective Function
- Minimize cost Constraints Meet Demand
1st period current production current inv
Demand Middle Periods previous inv current
prod current inv Demand Last
Period previous inv current prod
Demand Capacity
If you had a one-period model, which Demand
constraints would you use?
28
The multi-period inventory problem is concerned
with producing products over a time horizon to
meet demand and deal with limited capacity,
etc. Basic Characteristics Objective Function
- Minimize cost Constraints Meet Demand
1st period current production current inv
Demand Middle Periods previous per inv
current prod current inv
Demand Last Period previous per inv current
prod Demand Capacity
If you had a one-period model, which Demand
constraints would you use? None no inventory
model!
29
LEO. com needs to meet the demands of product 1
and product 2 in January, February, and March.
The demands are shown as follows LEO.com
can purchase the products from manufacturers, and
prices in the two months are shown above. LEO.com
can store the products in its warehouse the
capacity of which is 10000 ft2. Storing one unit
of product 1 needs 50 ft2 space in the warehouse.
Storing one unit of product 2 needs 80 ft2 space
in the warehouse. The inventory cost of product 1
is 2 per unit per month and product 2 is 3 per
month LEO.com wants to minimize the total cost of
satisfying the demands in Jan, Feb, and March.
30
Model Decision Variables Let xij amt of Product
i purchased at the BEG in Month j i1,2
j1,2,3. Let iij amount of Product i held in
inventory at the END of Month j.
31
Model Decision Variables Let xij amt of Product
i purchased at the BEG of Month j i1,2
j1,2,3. Let iij amount of Product i held in
inventory at the END of Month j. Minimize Z
32
Model Decision Variables Let xij amt of Product
i purchased at the BEG of Month j i1,2
j1,2,3. Let iij amount of Product i held in
inventory at the END of Month j. Minimize Z
Total purchasing cost Total inventory cost
33
Model Decision Variables Let xij amt of Product
i purchased at the BEG of Month j i1,2
j1,2,3. Let iij amount of Product i held in
inventory at the END of Month j. Minimize Z
Total purchasing cost Total inventory cost
10x11 13x12 30x13 14x21
21x22 12x23 2i113i212i12 3i22
34
Model Decision Variables Let xij amt of Product
i purchased at the BEG of Month j i1,2
j1,2,3. Let iij amount of Product i held in
inventory at the END of Month j. Minimize Z
Total purchasing cost Total inventory cost
10x11 13x12 30x13 14x21
21x22 12x23 2i113i212i12 3i22 Subject
to x11 - i11 200 (January Demand, Product
1)
Meet Demand 1st period current production
current inv Demand
35
Model Decision Variables Let xij amt of Product
i purchased at the BEG of Month j i1,2
j1,2,3. Let iij amount of Product i held in
inventory at the END of Month j. Minimize Z
Total purchasing cost Total inventory cost
10x11 13x12 30x13 14x21
21x22 12x23 2i113i212i12 3i22 Subject
to x11 - i11 200 (January Demand, Product
1) x21 - i21 300 (January Demand, Product
2)
Meet Demand 1st period current production
current inv Demand
36
Model Decision Variables Let xij amt of Product
i purchased at the BEG of Month j i1,2
j1,2,3. Let iij amount of Product i held in
inventory at the END of Month j. Minimize Z
Total purchasing cost Total inventory cost
10x11 13x12 30x13 14x21
21x22 12x23 2i113i212i12 3i22 Subject
to x11 - i11 200 (January Demand, Product
1) x21 - i21 300 (January Demand, Product
2) x12 i11 i12 400 (February Demand,
Product 1)
Meet Demand Middle period current production
last period inv current inv Demand
37
Model Decision Variables Let xij amt of Product
i purchased at the BEG of Month j i1,2
j1,2,3. Let iij amount of Product i held in
inventory at the END of Month j. Minimize Z
Total purchasing cost Total inventory cost
10x11 13x12 12x13 14x21
21x22 12x23 2i113i212i12 3i22 Subject
to x11 - i11 200 (January Demand, Product
1) x21 - i21 300 (January Demand, Product
2) x12 i11 i12 400 (February Demand,
Product 1) x22 i21 i22 500 (February
Demand, Product 2)
Meet Demand Middle period current production
last period inv current inv Demand
38
Model Decision Variables Let xij amt of Product
i purchased at the BEG of Month j i1,2
j1,2,3. Let iij amount of Product i held in
inventory at the END of Month j. Minimize Z
Total purchasing cost Total inventory cost
10x11 13x12 12x13 14x21
21x22 12x23 2i113i212i12 3i22 Subject
to x11 - i11 200 (January Demand, Product
1) x21 - i21 300 (January Demand, Product
2) x12 i11 i12 400 (February Demand,
Product 1) x22 i21 i22 500 (February
Demand, Product 2) x13 i12 300 (March
Demand, Product 1) x23 i22 300 (March
Demand, Product 2)
Meet Demand Last period current production
last period inv Demand
39
Model Decision Variables Let xij amt of Product
i purchased at the BEG of Month j i1,2
j1,2,3. Let iij amount of Product i held in
inventory at the END of Month j. Minimize Z
Total purchasing cost Total inventory cost
10x11 13x12 30x13 14x21
21x22 12x23 2i113i212i12 3i22 Subject
to x11 - i11 200 (January Demand, Product
1) x21 - i21 300 (January Demand, Product
2) x12 i11 i12 400 (February Demand,
Product 1) x22 i21 i22 500 (February
Demand, Product 2) x13 i12 300 (March
Demand, Product 1) x23 i22 300 (March
Demand, Product 2) 50i11 80i21 lt 10,000
(Warehouse Capacity-January)
40
Model Decision Variables Let xij amt of Product
i purchased at the BEG of Month j i1,2
j1,2,3. Let iij amount of Product i held in
inventory at the END of Month j. Minimize Z
Total purchasing cost Total inventory cost
10x11 13x12 30x13 14x21
21x22 12x23 2i113i212i12 3i22 Subject
to x11 - i11 200 (January Demand, Product
1) x21 - i21 300 (January Demand, Product
2) x12 i11 i12 400 (February Demand,
Product 1) x22 i21 i22 500 (February
Demand, Product 2) x13 i12 300 (March
Demand, Product 1) x23 i22 300 (March
Demand, Product 2) 50i11 80i21 lt 10,000
(Warehouse Capacity-January) 50i12 80i22 lt
10,000 (Warehouse Capacity-February)
41
Model Decision Variables Let xij amt of Product
i purchased at the BEG of Month j i1,2
j1,2,3. Let iij amount of Product i held in
inventory at the END of Month j. Minimize Z
Total purchasing cost Total inventory cost
10x11 13x12 30x13 14x21
21x22 12x23 2i113i212i12 3i22 Subject
to x11 - i11 200 (January Demand, Product
1) x21 - i21 300 (January Demand, Product
2) x12 i11 i12 400 (February Demand,
Product 1) x22 i21 i22 500 (February
Demand, Product 2) x13 i12 300 (March
Demand, Product 1) x23 i22 300 (March
Demand, Product 2) 50i11 80i21 lt 10,000
(Warehouse Capacity-January) 50i12 80i22 lt
10,000 (Warehouse Capacity-February)
x11,x12,x13,x21,x22,x23,i11,i21,i12,i22 gt 0
(Non-negativity)
42
Model Decision Variables Let xij amt of Product
i purchased at the BEG of Month j i1,2
j1,2,3. Let iij amount of Product i held in
inventory at the END of Month j. Minimize Z
Total purchasing cost Total inventory cost
10x11 13x12 12x13 14x21
21x22 12x23 2i113i212i12 3i22 Subject
to x11 - i11 gt 200 (January Demand, Product
1) x21 - i21 gt 300 (January Demand, Product
2) x12 i11 i12 gt 400 (February Demand,
Product 1) x22 i21 i22 gt 500 (February
Demand, Product 2) x13 i12 gt 300 (March
Demand, Product 1) x23 i22 gt 300 (March
Demand, Product 2) 50i11 80i21 lt 10,000
(Warehouse Capacity-January) 50i12 80i22 lt
10,000 (Warehouse Capacity-February)
x11,x12,x13,x21,x22,x23,i11,i21,i12,i22 gt 0
(Non-negativity)
What if the demand given by the market is the
minimum amount that has to be satisfied?
43
(No Transcript)
44
(No Transcript)
45

• Questions
• If the firm has an opportunity to rent a
  warehouse at a cost of .25 per sq. ft. at the
  end of Jan., would you recommend the firm to do
  so? Why? What would you recommend if the firm has
  the same option at the end of Feb.?
• If in Jan., the firm has an option to stock out
  for a small amount for one product. However, in
  return, the firm has to increase the same amount
  of sales for the other product. Which one would
  you recommend to be stock out and which one to
  be increased?
• The shadow price for the warehouse capacity in
• Feb. is 0.3. Can you justify it?

46
From the textbook 3.13 4.8 and 4.10 (For the
last two questions, you need to solve the
problem by using Excel Solver).
47
If you need helpemail me, see me, or call me.
Have a great day!