运筹学练习代写 整数线性规划代写 运筹学作业代写代做

OPERATIONS RESEARCH

Exercises

Integer Linear Programming

FORMULATING PROBLEMS IN INTEGER LINEAR PROGRAMMING

运筹学练习代写 Ex 1 A factory producing radios and recorders carried a market study and found that it could produce at least 1500 radios and 3000 recorders per

Ex 1

A factory producing radios and recorders carried a market study and found that it could produce at least 1500 radios and 3000 recorders per month, respectively. Each radio and recorder requires 5 and 7 hours of work, respectively. There are available 40 000 hours of work per month. The unit profit is 500 u.m. and 1000 u.m., for radios and recorders, respectively. Formulate in integer linear programming, in order to determine the production plan that maximizes profit.

Ex 2

One company sells three products, P1, P2 and P3, in the following weekly quantities: 15 units of P1, 50 units of P2, and 25 units of P3. Any of these products can be supplied by four factories, F1, F2, F3 and F4, whose weekly production capacities are 10, 20, 30 and 30 units, respectively (regardless of the type of products concerned). Unit production costs depend on the factory where the products are produced and are as follows:

	F1	F2	F3	F4
P1	6	4	5	4
P2	7	6	7	3
P3	8	7	6	9

What is the product supply plan that minimizes the total production cost? Formulate in integer linear programming.

Ex 3 运筹学练习代写

A post office needs a different number of employees on each day of the week, as shown in the following table:

Monday	Tuesday	Wednesday	Thursday	Friday	Saturday	Sunday
17	13	15	19	14	16	11

The trade union agreement requires each employee to work five consecutive days and then have two consecutive days off. Formulate the problem in integer linear programming, in order to minimize the number of employees that satisfy the daily needs.

Ex 4

A factory produces glass and has to satisfy an order of, at least:

· 50 panes of glass of 80 cm x 40 cm

· 40 panes of glass of 80 cm x 70 cm

· 60 panes of glass of 80 cm x 50 cm

The factory has panes of glass of 80 cm x 100 cm. How should the panes be cut so as to minimize the waste? Formulate in integer linear programming.

Ex 5 运筹学练习代写

A climber plans to take a trip and would like to take six objects with him. However, the set of objects exceeds the limit of 17 kilograms that he thinks he can carry. To better select the objects to take, he assigned each one a value, which is as greater as its importance for the trip. The weight and value of each object are the following:

	Object 1	Object 2	Object 3	Object 4	Object 5	Object 6
Weight (kg)	10	3	4	3	2	6
Value	20	15	16	9	7	6

Formulate the problem in integer linear programming, in order to maximize the total value of the objects selected for the trip. 

Ex 6

On each of the days of the next week, a company will have to move one of its operators to a specific task. The company employs five operators and intends to assign a move to each of them. The move cost for each operator depends on the day of the week:

	Monday	Tuesday	Wednesday	Thursday	Friday
Operator A	8	5	0	2	6
Operator B	9	6	2	4	5
Operator C	7	3	5	1	4
Operator D	7	2	6	7	2
Operator E	6	5	7	9	1

Formulate the problem in integer linear programming, so that the move plan is as economical as possible.

Ex 7 运筹学练习代写

A company has four investments available: I1, I2, I3 and I4. The following table shows the NPV(*) and the cost of each investment:

	NPV	Cost
Investment I1	16 ´ 103 c.u.	5 ´ 103 c.u.
Investment I2	22 ´ 103 c.u.	7 ´ 103 c.u.
Investment I3	12 ´ 103 c.u.	4 ´ 103 c.u.
Investment I4	8 ´ 103 c.u.	3 ´ 103 c.u.

Knowing that it will not be able to invest more than 14 ´ 10³ c.u., the company intends to select the investments to be made in order to obtain the highest total NPV.

(*)NPV is the Net Present Value and it represents the worth, at present, of the return that will be received

a) Formulate in integer linear programming.

Reformulate the model given in a) in order to include the following conditions:

b) It is only possible to select up to three investments.

c) If investment 3 is selected then investment 1 must also be selected.

d) If investment 2 is selected than investment 3 cannot be selected.

Ex 8 运筹学练习代写

There are three landfills where four population centers have to deposit their waste. Each population center can only use one of the landfills. The transportation costs depend on the location of the population centers and landfills, according to the following table:

	Landfill 1	Landfill 2	Landfill 3
Population Center 1	50	30	80
Population Center 2	50	30	75
Population Center 3	80	50	100
Population Center 4	60	70	90

Each landfill can receive waste from more than one population center, except landfill 1: if landfill 1 receives waste from population center 1 then it cannot receive waste from another population center. Formulate the problem in integer linear programming, in order to minimize the total transportation cost.

Ex 9 运筹学练习代写

An oil exploration company currently has ten possible exploration sites. The estimates for operating costs and profits are as follows (in c.u.):

	Site 1	Site 2	Site 3	Site 4	Site 5	Site 6	Site 7	Site 8	Site 9	Site 10
Cost	9	23	17	15	12	10	11	13	7	19
Profit	51	87	65	81	55	50	47	68	32	93

Management intends to select the five exploration sites with the lowest total cost, guaranteeing a minimum profit of 250 c.u.

a) Formulate the problem in integer linear programming.

For each of the following conditions, indicate the changes to the model presented in a):

b) Among places 5, 6, 7 and 8, only up to two may be selected.

c) The selection of site 1 is incompatible with the selection of site 2.

d) The selection of sites 1 and 7 makes it impossible to select site 8.

e) The selection of sites 3 or 4 makes it impossible to select site 5.

f) Site 9 can only be selected if site 2 is selected.

Ex 10 运筹学练习代写

A company is planning the production of a product for the upcoming 6 months. The company has to fulfill the following orders:

	Month 1	Month 2	Month 3	Month 4	Month 5	Month 6
Quantity (t)	55	40	58	20	20	60

The monthly production requires the installation of an equipment that costs 60 Euros and whose production capacity is bounded to 100 tonnes. As the company has a large warehouse the product can be stored to be sold later. The monthly storage cost is 1 Euro per tonne. When should the equipment be installed? What quantities must be produced in the months in which the equipment is installed? Formulate in linear programming in order to minimize the total costs (production and storage).

Ex 12 运筹学练习代写

A company is planning the monthly production of four products: P1, P2, P3 and P4. The unit profit (c.u.) of each product as well as the units of raw materials required, per unit of product, are listed in the following table:

	Product P1	Product P2	Product P3	Product P4
Profit	10	3	6	7
Raw Material 1	4	1	2	3
Raw Material 2	3	1	3	2

A monthly total of 3000 units of raw material 1 and 2000 units of raw material 2 are available. The company can produce no more than 500 units of P1 or no more than 600 units of P2. Furthermore, it has to produce at least 500 units of P2 or at least 400 units of P3. Formulate in linear programming in order to maximize the total profit.

Ex 13

A company can carry out its production using one from two available processes: process 1 or process 2. Each process requires an equipment. Equipments are process dependent, and so they have different setup costs and production costs as shown in the following table:

	Setup Costs	Production Cost (per unit)
Production process 1	1000	3
Production process 2	1200	2

Moreover, the first process bounds the production to 1500 units, and the second process obliges to produce more than 1500 units. The unit selling price is 10 c.u. Formulate in linear programming in order to maximize the total profit.

Ex 15

A company produces three types of cars, here named A, B and C. The raw materials and labor times required as well as the unit profits are as follows;

	Type A Car	Type B Car	Type C Car
Raw Material (per car)	1.5 tonne	3 tonne	5 tonne
Labor time (per car)	30 h	25 h	40 h
Unit Profit (per car)	200 c.u.	300 c.u.	400 c.u.

There are available 60 000 hours of labor time and 6000 tonnes of raw material. Consider also that each type of car or is not produced or its production is at least 1000 units. Formulate in linear programming in order to maximize the total profit.

Ex 16

A company intends to buy computers and wants to choose at most two among three types of computers. Four departments have graded (from 0 to 10) the three types of computers according to their preferences (the higher is the preference the higher is the grade). The choice of the computer will be based on these grades, which are in the following table:

	Financial Dep.	Sales Dep.	Marketing Dep.	Production Dep.
Computer 1	4	8	9	6
Computer 2	6	5	8	4
Computer 3	7	5	4	8

Due to internal organization, it is intended that the Sales Department and the Marketing Department have the same type of computer. Formulate in integer linear programming in order to maximize the total preference.

Ex 17 运筹学练习代写

A company is planning the production of two products, P1 and P2, for the upcoming month. P1 requires the installation of one equipment, whose monthly cost is 800 c.u. The production of P2 does not require the installation of any equipment. The production costs are 2 c.u./tonne for P1 and 5 c.u./tonne for P2. Due to contracts signed up with clients, the total production must be at least 500 tonnes. The company intends to minimize the total cost.

a) Formulate the problem in linear programming.

Reformulate the model given in a) in order to include the following conditions:

b) The equipment required to produce P1 allows to produce at most 400 tonnes;

c) If the company decides to produce P2 then at least 150 tonnes must be produced;

d) If the company decides to produce P2 then the level of production of P2 must be at least 150 tonnes and at most 350 tonnes;

e) The company must produce at least 200 tonnes of P1 or at least 350 tonnes of P2;

f) The company can produce at most 300 tonnes of P1 or at most 400 tonnes of P2.

Ex 18 运筹学练习代写

The WW factory produces two types of whisky, Glorious Blend and Super Blend, made from three imported malts: Sir Roses, Highland Wind and Old Frenzy. The suppliers can provide at most 2000 liters of Sir Roses, 2500 liters of Highland Wind and 1200 liters of Old Frenzy, at a price of 3.50, 2.50 and 2.00 Euros per liter. A liter of Glorious Blend is sold at 3.40 Euros, whereas a liter of Super Blend is sold for 3.00 Euros. In order to guarantee the taste and the quality it is necessary that Glorious Blend contains at least 60% of Sir Roses and a maximum of 20% of Old Frenzy. Likewise, Super Blend has to contain at least 15% of Sir Roses and at most 60% of Highland Wind. Considering that there are no other costs, the factory intends to determine the production plan of each mixture that maximizes the net profit

a) Formulate the problem in linear programming.

b) Suppose malts and whiskeys are sold in 100-liter barrels. Specifically, Sir Roses, Highland Wind and Old Frenzy malts are purchased at 350, 250 and 200 Euros per barrel, respectively. While selling prices of Glorious Blend and Super Blend whiskeys are 340 and 300 Euros per barrel, respectively. Reformulate the model given in a).

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