limited finance test 2.5 hour

3QA3/2DA3

Management Science for

Lecture 03

C01: June 28, 2021

C02: June 29, 2021

Instructors:

Seyyed Hossein Alavi Zdravko Dimitrov

Practice Question P2-13?

2

 Let us look at Handout 1.1

LP Models in the World

Develop a written LP model on paper before

attempting to implement it on Excel!

 We deal with problems:

 with more than 2 variables

 require more complicated modelling techniques

✓ Understand the logic behind a model before implementing it on the

computer!

✓ If the model is incorrect or incomplete from a logical perspective

(even if it is correct from a mathematical perspective), Excel cannot

recognize the logical error!

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1- Blending Problem

Determining the optimal (least costly) mix of ingredients

 Oil Companies
 Mix of different crude oils and other chemicals for gasoline

 n Care Companies
 Mix of chemicals and other products to make fertilizers

 Agriculture Industry (Feed mix problem)
 Mix of grain, corn, and minerals to make different feeds for various farm

animals

 Diet Problem
 Select a set of foods that will satisfy a set of daily nutritional requirement

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1- Blending Problem

 Handout 1.2.

 An agriculture company received an order for 8,000 pounds of
feed

 At least 20% corn, 15% grain, 15% minerals

 What should the company do to minimize the cost?

6

Nutrient
Percent of Nutrition in

Feed 1 Feed 2 Feed 3 Feed 4

Corn 30% 5% 20% 10%

Grain 10% 30% 15% 10%

Minerals 20% 20% 20% 30%

Cost/pound $0.25 $0.30 $0.32 $0.15

1- Blending Problem

 Decision Variables

 ?1 = amount of feed 1 in pounds to use in the blend

 ?2 = amount of feed 2 in pounds to use in the blend

 ?3 = amount of feed 3 in pounds to use in the blend

 ?4 = amount of feed 4 in pounds to use in the blend

 Objective Function

Minimize 0.25?1 + 0.30?2 + 0.32 ?3 + 0.15?4

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1- Blending Problem

 Constraints

 Pounds of feed required

?1 + ?2 + ?3 + ?4 = 8,000

 Min % of corn required

0.30?1 + 0.05?2 + 0.20?3 + 0.10?4 ≥ 0.20(8,000)

 Min % of grain required

0.10?1 + 0.30?2 + 0.15?3 + 0.10?4 ≥ 0.15(8,000)

 Min % of minerals required

0.20?1 + 0.20?2 + 0.20?3 + 0.30?4 ≥ 0.15(8,000)

 Non-negativity constraints

?1, ?2, ?3, ?4 ≥ 0
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2- Investment/Portfolio Problem

 Problems

 Maximizing the expected return of an investment

 Selecting from a variety of alternatives (companies)

 Satisfying certain cash flow requirements

 Satisfying certain risk constraints

 Satisfying certain policy requirements

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2- Investment/Portfolio Problem

 Investing $750,000 in below companies

 No more than 25% of money in any company

 At least half of the money be invested in long-term bonds that mature in
10 or more years

 No more than 35% in low rating companies (3 and 4)

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Company Rate of return Years to Maturity Rating

1 8.65% 11 1-Excellent

2 9.50% 10 3-Good

3 10.00% 6 4-Fair

4 8.75% 10 1-Excellent

5 9.25% 7 3-Good

6 9.00% 13 2-Very Good

2- Investment/Portfolio Problem

 Decision Variables

 ?1 = amount of money to invest in Company 1

 ?2 = amount of money to invest in Company 2

 ?3 = amount of money to invest in Company 3

 ?4 = amount of money to invest in Company 4

 ?5 = amount of money to invest in Company 5

 ?6 = amount of money to invest in Company 6

 Objective Function

Maximize .0865?1 + .095?2 + .10?3 + .0875?4 + .0925?5 + .09?6

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2- Investment/Portfolio Problem

 Constraints

 Total investment

?1 + ?2 + ?3 + ?4 + ?5 + ?6 = 750,000

 Max amount in one bond

?1, ?2, ?3, ?4 , ?5, ?6 ≤ 0.25 750,000

 Min amount invested in long-term bond

?1 + ?2 + ?4 + ?6 ≥ 0.5 750,000

 Max amount invested in low rating bond

?2 + ?3 + ?5 ≤ 0.35 750,000

 Non-negativity constraints

?1,?2, ?3,?4 ,?5,?6 ≥ 0
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3- Transportation Problem

Distribution of goods from several supply points to a number of

demand points

 Each supply point has a capacity for production.

 Each demand point has a certain demand.

 Minimizing the distance/cost

 Can also be defined as maximization of profit

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3- Transportation Problem

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Sources
Destinations

Supply
A B C

D $5 $4 $3 200

E $6 $7 $5 300

F $3 $8 $5 400

Demand 350 150 400 900=900

3- Transportation Problem

Sources Destinations

15

D

E

F

A

C

B

200 350

150

400

300

400

3- Transportation Problem

 Decision Variables

??? = number of goods shipped from source ? to destination ?

where ? = D, E or F, and ? = A, B or C

 Objective Function

Minimize 5?DA + 4?DB + 3?DC

+ 6?EA + 7?EB + 5?EC

+ 3??? + 8?FB + 5?FC
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3- Transportation Problem

 Constraints (Flow in = Flow out)

 Net flow at supply node D

200 = ?DA + ?DB +?DC

 Net flow at supply node E

300 = ?EA + ?EB +?EC

 Net flow at supply node F

400 = ?FA + ?FB +?FC

 Net flow at demand node A

?DA + ?EA +?FA = 350

 Net flow at demand node B

?DB + ?EB +?FB = 150

 Net flow at demand node C
?DC + ?EC +?FC = 400

 Non-negativity constraints

??? ≥ 017

3- Transportation Problem

 Supply > Demand

 Supply equality must change to inequality

 ?????? ?????????(???? ??) ≥ ???? ???

e.g. 300 ≥ ?DA + ?DB +?DC

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Sources
Destinations

Supply
A B C

D $5 $4 $3 300

E $6 $7 $5 300

F $3 $8 $5 400

Demand 350 150 400 900/1000

3- Transportation Problem

 Supply < Demand  Demand equality must change to inequality  ???? ?? ≤ ?????? ????????(???? ???) e.g. ?DA + ?EA +?FA ≤ 350 19 Sources Destinations Supply A B C D $5 $4 $3 200 E $6 $7 $5 300 F $3 $8 $5 400 Demand 350 150 500 1000/900 4- Crew/Labor Scheduling  Determining the optimal number of staff for a specific planning horizon (month, week, shift, flight, …)  Staffing needs could be different during different time periods of the planning horizon 20 4- Crew/Labor Scheduling  A bank requires between 10 and 18 tellers  Can be full-time (at most 12) and part-time  Part-time salary: $28/day full-time: $90/day  Part-time work is 4 hours, can start from 9 am to1:00 pm  Full-time tellers work from 9 am to 5 pm with 1 hour for lunch  50% have lunch at 11:00 am, others at noon  Part-time hours cannot exceed 50% of the total required 21 Time Period # Required 9 a.m.–10 a.m. 10 10 a.m.–11 a.m. 12 11 a.m.–Noon 14 Noon–1 p.m. 16 1 p.m.–2 p.m. 18 2 p.m.–3 p.m. 17 3 p.m.–4 p.m. 15 4 p.m.–5 p.m. 10 4- Crew/Labor Scheduling  Decision Variables  ? = number of full−time tellers to use (starting from 9 am to 5 pm)  ?1 = number of part-time tellers starting from 9 am to 1 pm  ?2 = number of part-time tellers starting from 10 am to 2 pm  ?3 = number of part-time tellers starting from 11 am to 3 pm  ?4 = number of part-time tellers starting from 12 pm to 4 pm  ?5 = number of part-time tellers starting from 1 pm to 5 pm  Objective Function Minimize 90? + 28(?1 + ?2 + ?3 + ?4 + ?5) 22 4- Crew/Labor Scheduling  Constraints  ?+?1 ≥ 10 (9 am-10 am requirement)  ?+?1+?2 ≥ 12 (10 am-11am requirement)  0.5?+?1+?2+?3 ≥ 14 (11 am-12 pm requirement)  0.5?+?1+?2+?3+?4 ≥ 16 (12 pm-1 pm requirement)  ?+?2+?3+?4+?5 ≥ 18 (1 pm-2 pm requirement)  ?+?3+?4+?5 ≥ 17 (2 pm-3 pm requirement)  ?+?4+?5 ≥ 15 (3 pm-4 pm requirement)  ?+?5 ≥ 10 (4 pm-5 pm requirement)  ? ≤ 12 (Max # of full-timers)  4(?1+?2+?3+?4+?5) ≤ 0.5(10+12+14+16+18+17+15+10) = 56 (Max hours of part-timers)  ?,?1,?2,?3,?4,?5 ≥ 0 (Non-negativity constraints) 23 5- Multi-period Applications  Determine the optimal decisions for several periods  It is challenging because the decisions are not independent!  The decision of period ? depends on the decision of periods ? − 1, ? − 2, …  Applications in:  Production   etc. 24 5- Multi-period Production Planning  A company is trying to plan its production, with the objective of minimizing the total production and total inventory carrying cost.  Inventory Capacity: 6,000 (Current inventory: 2,000)  Safety stock: 1,500  Produce no less than half of max capacity  Cost of carrying inventory: 1.5% of unit cost  Inventory carrying cost is accounted based on the ending inventory each month (after demand is met)  What is the optimal quantity to produce for each month? 25 Month 1 2 3 Unit cost $240 $250 $265 Demand 1,000 4,500 4,000 Max Prod. Capacity 4,000 3,500 4,000 5- Multi-period Production Planning  Decision Variables • ?1 = number of units to produce in month 1 • ?2 = number of untis to produce in month 2 • ?3 = number of units to produce in month 3 • ?1 = ending inventory in month 1 • ?2 = ending inventory in month 2 • ?3 = ending inventory in month 3  Objective Function Minimize 240?1 + 250?2 + 265?3 +3.6?1 + 3.75?2 + 3.98?3 26 5- Multi-period Production Planning  Constraints  Production constraints:  2,000 ≤ ?1≤ 4,000 (production level for month 1)  1,750 ≤ ?2≤ 3,500 (production level for month 2)  2,000 ≤ ?3≤ 4,000 (production level for month 3) 27 5- Multi-period Production Planning  Constraints  Balance Constraints: Ending Inventory = Beginning Inventory + Units Produced − Demand  ?1 = ?0 + ?1 − 1,000 (balance constraint for month 1)  ?2 = ?1 + ?2 − 4,500 (balance constraint for month 2)  ?3 = ?2 + ?3 − 4,000 (balance constraint for month 3)  Warehouse Capacity & Safety Stock  1,500 ≤ ?1≤ 6,000 (ending inventory for month 1)  1,500 ≤ ?2≤ 6,000 (ending inventory for month 2)  1,500 ≤ ?3 ≤ 6,000 (ending inventory for month 3)28 6- Vehicle Loading Problem  Deciding on items to load on a vehicle  Maximizing the value of the load shipped  Constraints  Weight and volume limit of vehicle  Minimum level of certain items 29 6- Vehicle Loading problem  The company has 2 vehicles with  Weight capacity 10,000 pounds  Volume capacity 900 cubic feet  The weight of both vehicles must be equal.  The objective is to maximize the value of loaded items. 30 ITEM VALUE WEIGHT (pounds) Value per pound VOLUME (cu. ft. per pound) 1 $15,500 5,000 3.10 0.125 2 $14,400 4,500 3.20 0.064 3 $10,350 3,000 3.45 0.144 4 $14,525 3,500 4.15 0.448 5 $13,000 4,000 3.25 0.048 6 $9,625 3,500 2.75 0.018 Practice Problems  CHAPTER 3  Problems 3, 7, 9, 12, 13, 17, 21  CHAPTER 5  Problem 17 31