Operational ResearchAutonomous work:Exercises aboutlinear programmingPatrick Meyer& Mehrdad MohammadiIMT Atlantique

Part 1 : Linear ProgrammingA few formulation and resolution exercises

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LP-Exercises 3Exercise 1 : Blueberry

Blueberry makes quarterly decisions about their product mix. We consider that theyproduce notebook computers and desktop computers.

There are a number of limits on what Blueberry can produce. The major constraintsare as follows :

- Each computer (either notebook or desktop) requires a Processing Chip. Due totightness in the market, the supplier has allocated 10,000 such chips toBlueberry ;

- Each computer requires memory. Memory comes in 4 GB chip sets. A notebookcomputer has 4 GB memory installed (so needs 1 chip set) while a desktopcomputer has 8 GB (so requires 2 chip sets). Blueberry has a stock of 15,000chip sets to use over the next quarter ;

- Each computer requires assembly time. Due to tight tolerances, a notebookcomputer takes more time to assemble : 4 minutes versus 3 minutes for adesktop. There are 25,000 minutes of assembly time available in the next quarter.

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LP-Exercises 4Exercise 1

Given the current market conditions, material cost, and the productionsystem, each notebook computer produced generates 750 USD profit,and each desktop produces 1000 USD profit.

How many of each type computer should Blueberry produce in thenext quarter in order to maximize its profit?

- Formulate this problem as a linear program;- Solve it graphically ;

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LP-Exercises 5Exercise 1

Solution :- Decision variables :

- x1 : number of 1000 notebooks to produce ;- x2 : number of 1000 desktops to produce.

- Objective :- max750x1 + 1000x2

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LP-Exercises 6Exercise 1

Solution :- Constraints :

- Processing chips : x1 + x2 ≤ 10 ;- Memory chips : x1 + 2x2 ≤ 15 ;- Assembling : 4x1 + 3x2 ≤ 25 ;- Non-negativity : x1, x2 ≥ 0.

- Summary of the LP :

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LP-Exercises 7Exercise 1

Solution :- Graphical resolution :

The constraints and the feasible region

Observation : theprocessing chips constraintis useless !

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LP-Exercises 8Exercise 1

Solution :- Graphical resolution :

Finding the optimal profit

- Optimal solution inx1 = 1 and x2 = 7 ;

- Produce 1000 laptopsand 7000 desktops for aprofit of 7,750,000 USD.

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LP-Exercises 9Exercise 2 (Dantzig 1963)

- Suppose that the two canneries of a distributor are located inSeattle and San Diego. The canneries can fill 350 and 650 casesof tins per day, respectively.

- The distributor operates three warehouses around the country, inNew York, Chicago and Kansas City. Each of the warehouses cansell 300 cases per day.

- The distributor wishes to determine the number of cases to beshipped from the two canneries to the three warehouses so thateach warehouse should obtain as many cases as it can sell dailyat the minimum total transportation cost.

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LP-Exercises 10Exercise 2 (Dantzig 1963)

- The problem is characterized by the following distance matrix (inthousands of miles) :

- The freight per case per thousand miles is 90 USD.

- Formulate this problem as a linear program.

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LP-Exercises 11Exercise 2 : solution

- Decision variables :- x11 ≥ 0 number of cases to be shipped from I to A- x12 ≥ 0 number of cases to be shipped from I to B- x13 ≥ 0 number of cases to be shipped from I to C- x21 ≥ 0 number of cases to be shipped from II to A- x22 ≥ 0 number of cases to be shipped from II to B- x23 ≥ 0 number of cases to be shipped from II to C

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LP-Exercises 12Exercise 2 : solution

- Parameters : transport costs in thousands of USD / case- c11 = 90 ∗ 2.5 = 225 (between I and A)- c12 = 153 (between I and B)- c13 = 162 (between I and C)- c21 = 225 (between II and A)- c22 = 162 (between II and B)- c23 = 126 (between II and C)

- Objective function :

min225 ·x11+153 ·x12+162 ·x13+225 ·x21+162 ·x22+126 ·x23

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LP-Exercises 13Exercise 2 : solution

- Constraints :

- (supply limit at each plant) :

x11 + x12 + x13 ≤ 350

x21 + x22 + x23 ≤ 650

- (demand at each market) :

x11 + x21 ≥ 300

x12 + x22 ≥ 300

x13 + x23 ≥ 300

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LP-Exercises 14Exercise 3 : diet

- An ideal diet would meet or exceed basic nutritional requirements,be inexpensive, have variety and be pleasing to the palate ;

- Suppose the available foods are the following :

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LP-Exercises 15Exercise 3 : diet

- A satisfactory diet has at least 2000kcal of energy, 55g of proteinand 800mg of calcium (vitamins and iron are supplied by pills) ;

- Each food can only be served a certain number of times per day ;

- What is the least cost satisfactory diet ?- Give only the formulation as a LP.

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LP-Exercises 16Exercise 3

Solution :- Decision variables :

- The foods are labelled from 1 to 6 ;- xi , i = 1, . . .6 represents the number of servings of food i in the diet ;

- Objective :- minimize cost ;- min3x1 + 24x2 + 13x3 + 9x4 + 20x5 + 19x6.

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LP-Exercises 17Exercise 3

Solution :- Constraints :

- Energy, protein, calcium, serving / day limit.

- Formulation of the LP :

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LP-Exercises 18Exercise 4 : restaurant

- A restaurant is open seven days a week ;

- Based on past experience, the number of workers needed on aparticular day is given as follows :

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LP-Exercises 19Exercise 4 : restaurant

- Every worker works five consecutive days, and then takes twodays off, repeating this pattern indefinitely ;

- How can the number of workers be minimized?

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LP-Exercises 20Exercise 4

Solution :- Decision variables (bad idea) :

- xi is the number of workers working on day i ;- Typical solution : 15 workers on Monday, 13 on Tuesday, . . . ;- Does not tell us how many workers are needed !

- Decision variables (good idea) :- Let the days be numbered from 1 to 7 ;- Let xi be the number of workers who begin their five day shift on day

i .

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LP-Exercises 21Exercise 4

Solution :- Objective :

- min x1 + x2 + . . .+ x7

- Constraints :- Who works on Monday? Those who start their shift on Monday,

Thursday, Friday, Saturday, Sunday ;- x1 + x4 + x5 + x6 + x7 ≥ 14- . . .

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LP-Exercises 22Exercise 4

Solution :- Formulation of the LP :

- Possibility to add different types of shifts if different costs.

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LP-Exercises 23Exercise 5 : financial portfolio

- A mortgage team has 100,000,000 USD to finance variousinvestments ;

- Five categories of loans with various returns and risks (1-10, 1best) :

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LP-Exercises 24Exercise 5 : financial portfolio

- Uninvested money goes into a savings account with no risk and3% return ;

- Allocate the money to the categories so as to :- Maximize the average return per dollar ;

- Have an average risk of no more than 5 (averages and fractions onthe invested money, not the savings) ;

- Invest at least 20% of the loans in commercial loans ;

- The amount of second mortgages and personal loans combinedshould be no higher than the amount in first mortgages.

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LP-Exercises 25Exercise 5

Solution :- Decision variables :

- Let the investments be numbered 1 to 5 ;

- Let xi be the amount invested in investment i , and xs be the amountin the savings account.

- Objective :- max9x1 + 12x2 + 15x3 + 8x4 + 6x5 + 3xs

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LP-Exercises 26Exercise 5

Solution :- Constraints :

- Average risk :

3x1 + 6x2 + 8x3 + 2x4 + x5

x1 + x2 + x3 + x4 + x5≤ 5

Equivalent linear constraint :

−2x1 + x2 + 3x3 − 3x4 − 4x5 ≤ 0

- Commercial loans :

x4 ≥ 0.2(x1 + x2 + x3 + x4 + x5)

Equivalent linear constraint :

−0.2x1 − 0.2x2 − 0.2x3 + 0.8x4 − 0.2x5 ≥ 0

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LP-Exercises 27Exercise 5

Solution :- Constraints :

- Mortgages :x2 + x3 − x1 ≤ 0

- Total amount :

x1 + x2 + x3 + x4 + x5 + xs ≤ 10.000.000

- Nonnegativity constraints.

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Part 2 : What now?

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What now? 29Autonomous work

X ExercisesX Model decision problems as linear programsX Solve certain of them graphically

- Case studyRead about the two topicsWatch videosDecide which of the two topics you would like to study

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