The company produces high-tech products, the production of which. The mechanism of the enterprise's functioning. Optimal production problem

Analyze the situation and choose the most beneficial solution for the enterprise:

A. Make parts yourself.

B. Purchase parts externally and use your own released equipment to produce other products that can bring a profit in the amount of 18,000 rubles.

Problem 24

The company has 1,000 obsolete computer parts that were previously purchased for 200,000 rubles. Which is more profitable: processing parts at a cost of 40,000 rubles. and sell them for 64,000 rubles. or sell them for 17,000 rubles. without any processing?

Problem 25

The company produces technically complex products, the production of which requires a significant number of components. An enterprise can buy these parts at a price of 100 rubles. / PC. or produce them yourself. Determine what is more profitable for the enterprise (the profitability threshold has been crossed).

Cost of production of parts at the enterprise itself:

direct variable costs - 85 rub. / PC.;

Initial data for calculations:

fixed costs - 60,000 rubles; selling price per unit of production - 20 rubles; variable costs per unit of production - 12 rubles; current sales volume - 8200 units; acceptable range of production volumes is 4500–12,500 units.

Problem 46

Based on those given in table. 1 and 2 of the initial data, calculate the retail price of the old and new goods.

Table 1 Expert assessments of quality parameters of old and new products

Table 2 Conditions for calculating the retail price

The factory produces two types of products: P 1 and P 2. Both types of products are sold wholesale. For the production of these products, three initial products are used - A, B, C. The maximum possible daily reserves of these products are 6, 8 and 5 tons, respectively. The costs of raw materials A, B, C per 1 thousand products P 1 and P 2 are given in table. A study of the sales market showed that the daily demand for products P 2 never exceeds the demand for products P 1 by more than 1 thousand units. In addition, it has been established that the demand for P 2 products never exceeds 2 thousand units. per day.
Wholesale prices 1 thousand pcs. products P 1 are equal to 3 thousand rubles, 1 thousand pcs. P 2 - 2 thousand pcs.
How many products (in thousands) of each type should the factory produce in order for the income from product sales to be maximized?

The construction of a mathematical model should begin with the identification of variables (the desired quantities). After this, the objective function and constraints are determined through the corresponding variables.

In the example under consideration we have the following:
Variables.
Since it is necessary to determine the production volumes of each type of product, the variables are:
X 1 - daily production volume of product P 1 in thousand units;
X 2 - daily production volume of product P 2 in thousand units.

Objective function. Since the cost of 1 thousand products P 1 is equal to 3 thousand rubles, the daily income from its sale will be 3X 1 thousand roubles. Similarly, income from sales of X 2 thousand units. P 2 will be 2X 2 thousand rubles. per day. Assuming independence of sales volumes of each product, total income is equal to the sum of two terms - income from the sale of products P 1 and income from the sale of products P 2.
Denoting income (in thousand rubles) through f(X), we can give the following mathematical formulation of the objective function: determine the (admissible) values of X 1 and X 2 that maximize the amount of total income:
f(X) = 3X 1 + 2X 2, X = (X 1, X 2)
Restrictions. When solving the problem under consideration, restrictions on the consumption of initial products A, B and C and the demand for manufactured products must be taken into account, which can be written as follows: Consumption of the initial product for the production of both types of products
≤
The maximum possible supply of a given input product

This leads to three limitations:
X 1+ 2Х 2 ≤ 6 (for A),
2X 1+ X 2 ≤ 8 (for B),
X 1+ 0.8Х 2 ≤ 5 (for C).
Restrictions on the amount of demand for products have the form:
X 2 – X 1 ≤ 1 (ratio of demand for products P 1 And P 2),
X 2 ≤ 2 (maximum demand for products P 2).
Conditions for the non-negativity of variables are also introduced, i.e. restrictions on their sign:
X 1 ≥ 0 (production volume P 1)
X 2 ≥ 0 (production volume P 2).
These restrictions are that production volumes cannot take negative values.
Therefore, the mathematical model is written as follows.
Determine the daily production volumes (X 1 and X 2) of products P 1 and P 2 in thousand units, at which
max f(X) = 3 X 1 + 2 X 2 (objective function)
with restrictions:
X 1 + 2X 2 ≤ 6
2Х 1 + Х 2 ≤ 8
X 1 + 0.8Х 2 ≤
5 - X 1 + X 2 ≤ 1
X 2 ≤ 2
X 1 ≥0, X 2 ≥ 0

Resource task

For the manufacture of products of type A and B, three types of raw materials are used, the reserves of each of which are P1, P2, P3. To produce one product of type A, it is required to spend a1 kg of raw materials of the first type, a2 kg of raw materials of the second type, and a3 kg of raw materials of the third type. For one product of type B, b1, b2, b3 kg of raw materials of each type are consumed, respectively. The profit from the sale of a unit of product A is α monetary units, and of product B is β monetary units. Draw up a plan for the production of products A and B, ensuring maximum profit from their sale. Solve the problem using the simplex method. Give a geometric interpretation of the problem.

Mathematical model of the problem
2x 1 + x 2 ≤438
3x 1 + 6x 2 ≤747
4x 1 + 7x 2 ≤812
F(X) = 7x1 + 5x 2 => max

Resource utilization problem (production planning problem)

To manufacture two types of products P 1 and P 2, three types of resources S 1, S 2, S 3 are used. Resource reserves, the number of resource units spent on producing a unit of product, are given in Table. 1.

Table 1

Resource type	Number of units of production required to produce a unit of product		Resource reserves
Resource type	P 1	R 2	Resource reserves
S 1	2	3	20
S 2	3	–	18
S 3	1	4	10

Profit received from a unit of production P 1 and P 2 – respectively 2 and 3 units. It is necessary to draw up a production plan in which the profit from its sale will be maximum.
Solution. Let's create an economic and mathematical model of the problem. Let us denote by x 1, x 2 the number of units of production P 1 and P 2, respectively. Then the total profit F will be 2x 1 unit. from sales of products P 1 and 3x 2 d.e. from sales of products P 2, that is

F = 2x 1 + 3x 2. (1)
Since the amount of resources required for production is limited, we will draw up a system of resource restrictions. To manufacture products you will need (2x 1 + 3x 2) units of resource S 1 , 3x 1 units of resource S 2 and (x 1 + 4x 2) units of resource S 3 . Since the consumption of resources S 1, S 2, S 3 should not exceed their reserves, 20, 18, 10 units, respectively, the connection between the consumption of resources and their reserves will be expressed by a system of inequality restrictions:

(2)
So, the economic-mathematical model of the problem: find a production plan that satisfies the system of constraints (2), under which the objective function (1) takes on the maximum value.
The problem of using resources can be generalized to the case of producing n types of products using m types of resources.
Let us denote by x (j = 1, 2,…,n) the number of units of product P j planned for production; b 1 (i = 1, 2,…,m) – resource reserves Si, a ij – number of units of resource Si spent on producing a unit of product P j; c j – profit from the sale of a unit of production P j . Then the economic and mathematical model of the problem in the general formulation will take the form:

F=c 1 x 1 +c 2 x 2 +…+c n x n →(max) (3)

(4)
Find such a plan output that satisfies system (4), at which function (3) takes its maximum value.
Comment. This problem is also called the problem of determining the optimal product range.

Minimum production target

The team accepted an order for the production of 57 pieces. products P1, 68 pcs. products P2 and 80 pcs. P3 products. Products are produced on machines A and B. To produce a unit of product P1 on machine A, it takes 15 minutes, a unit of product P2 - 50 minutes, a unit of product P3 - 27 minutes, on machine B - 11, 15 and 13 minutes, respectively.

Build a mathematical model of the problem, on the basis of which you can find how many products and what type should be produced on machines A and B so that the order is completed in the minimum time.

Solution. Mathematical model of the problem.
x 1 - products P1 were manufactured on machine A, pcs.
x 2 - products P1 were manufactured on machine B, pcs.
x 3 - products P2 were manufactured on machine A, pcs.
x 4 - P2 products manufactured on machine B, pcs.
x 5 - P3 products manufactured on machine A, pcs.
x 6 - P3 products manufactured on machine B, pcs.

Quantity restrictions:
x 1 + x 2 ≥ 57
x 3 + x 4 ≥ 68
x 5 + x 6 ≥ 80

Objective function:
15x 1 + 11x 2 + 50x 3 + 15x 4 + 27x 5 + 13x 6 = min

Optimal production problem

Example No. 1. The company plans to produce two types of products I and II, the production of which requires three types of raw materials A, B, And WITH. The need a ij for each unit of the j-th type of product of the i-th type of raw material, the stock b i of the corresponding type of raw material and the profit c j from the sale of a unit of the j-th type of product are given by the table:

To produce two types of products I and II with a plan of x 1 and x 2 units, create an objective profit function Z and a corresponding system of restrictions on raw material reserves, assuming that it is required to produce a total of at least n units of both types of products.
In the conditions of task 1, draw up an optimal plan (x 1, x 2) for production, ensuring maximum profit Z max. Determine the remains of each type of raw material. (Solve the problem using the simplex method)
Using the resulting system of constraints, construct a polygon of feasible solutions and find the optimal production plan geometrically. Determine the corresponding profit Z max.

Example No. 2. A farmer can grow 4 crops on an area of 80 hectares. He has already entered into agreements for the sale of certain products (sales volume) and can purchase 250 quintals of mineral fertilizers.
The area of row crops (sunflower, sugar beets, potatoes, corn) should be 20 hectares.
Labor and fertilizer costs, profit per 1 hectare are shown in Table 2.
Determine how much space should be allocated to each crop to maximize profits.
Develop an economic and mathematical model and solve the problem.
Take the output data according to the option (Table 2)

Solution.
x 1 – area under buckwheat, ha; x 2 – area under barley, ha; x 3 – area under millet, ha; x 4 - area for potatoes, ha
Objective function: 140x 1 + 110x 2 + 120x 3 + 380x 4 → max
Limitations on fertilizer costs

Sales volume restrictions

Area restrictions
x 1 + x 2 + x 3 + x 4 ≤ 80
Restrictions on the area of row crops
x 4 ≤ 20
In total we have the following ZLP
3x 1 + 3x 2 + 2x 3 + 5x 4 ≤ 250
10x 1 + 30x 2 + 25x 3 + 180x 4 ≤ 200
x 1 + x 2 + x 3 + x 4 ≤ 80
x 4 ≤ 20
x 1 , x 2 , x 3 , x 4 ≤ 0
Objective function: 140x 1 + 110x 2 + 120x 3 + 380x 4 → max

Coursework - improvement of enterprise activities (Coursework)

Questions and answers on organizing commercial activities of an enterprise (Question)

Coursework - Organization of commercial activities and ways to improve it (Coursework)

Pambukhchiyants O.V. Organization and technology of commercial activities (Document)

Spurs for commercial activities (Crib sheet)

Diploma - Planning of commercial activities at the enterprise Spetstrans Chulman LLC (Diploma work)

Coursework - Trade secrets of the organization (Coursework)

Kiseleva E.N., Budanova O.G. Organization of commercial activities by industry and application (Document)

Tyutyushkina G.S. Organization of commercial activities of an enterprise (Document)

n1.docx

Problem 1

1. In the planned year, an industrial enterprise must produce machine tools with numerical control. The following parts are cast from carbon steel shaped castings for machine tools:

Table 4

2. In accordance with the supply plan, the enterprise must ship 12,500 brackets numbers K-26 - K-35. There is no detailed program for the production of brackets at the time of drawing up the plan. Data on consumption rates and production of individual types of brackets in the reporting period are given in Table 5.

Table 5.

Indicators	Bracket numbers
Indicators	K-26	K-27	K-28	K-29	K-30	K-31	K-32	K-33	K-34	K-35
Rough casting mass, kg	12.4	13,4	14,8	16,5	17,3	19,9	22,4	26,7	38,2	40,1
Production volume, pcs	260	270	290	1140	680	1330	6020	170	180	2960

Batch recipe:

Pig iron - 35%

Cast iron scrap - 62.4%

45% ferrosilicon –0.9%

76% ferromanganese -1.7%

Take the yield rate of suitable casting equal to 85%.

5. The consumption rate of molding materials per 1 ton of suitable casting is, in kg:

Quartz sand - 870.0
Raw brick -150.0
Molding clay - 50.0

4) Fireproof clay - 100.0

5) Fireclay brick. - 50.0

6) Silver graphite - 0.5

7) Graphite black - 10.0

8) Talc -1.0

9) Bentonite -5.0

Solution

Product type	Production plan	Rough casting mass, kg	Casting requirement, t	Coef. Exit	Batch requirement
					Total	Including
					Total	cast iron	scrap	Ferros	Ferrom-ts
1.machines				85%
A. body	1200	1634	1960800		1666680	583338	1040008,3	15000,1	28333,6
B. plate	1200	873	1047600		890460	311661	555647	8014,1	15137,8
B.holder	2400	78	187200		159120	55692	99290,9	1432	2705
G.bracket	15000	26,7	400500		340425	119148,8	212425,2	3063,8	5787,2
Total	19800	2611,7	3596100		3056685

Production plan for A, B, C = to plan. Year they produce (1200 pieces) * number of parts, not machine (tab4)

Casting requirement = production plan* rough casting mass (Table 4)

A=1200*1634=1960800

B=1200*873=1047600

B=2400*78=187200

Batch requirement (total) = casting requirement * 85%

A=1960800* 85%= 1666680

B=1047600* 85%=890460

B=187200* 85%=159120

Batch requirement (last 4th column) = total% (point 3)

Cast iron=1666680*35%=583338

Scrap=1666680*62.4%=1040008.3

Ferros th =1666680*0.9%=15000.1

Ferrom-ts = 1666680*1.7%=28333.6

Cast iron=890460*35%=311661

Scrap=890460*62.4%=555647

Ferros th =890460*0.9%=8014.1

Ferrom-ts = 890460*1.7%=15137.8

Cast iron=159120*35%=55692

Scrap=159120*62.4%=99290.9

Ferros th =159120*0.9%=1432

Ferrom-ts = 159120*1.7%=2705

For Brackets:

Production plan= 12500+12500=15000

Requirement for casting (brackets) = rough mass * production plan = 26.71 * 15000 = 400500

Requirement for charge (bracket) = total*% in the 3rd point

Batch requirement (total) = sweat casting * 85% = 400500 * 85% = 340425

Blend recipe for bracket

Cast iron=340425*35%=119148.8

Scrap=340425*62.4%=212425.2

Ferros th =340425*0.9%=3063.8

Ferrom-ts = 340425*1.7%=5787.2

Material requirement

Quartz sand =3596100*870=3128607000
Raw brick =3596100*150=539415000
Molding clay =3596100*50=179805000

4) Fireproof clay =3596100*100=359610000

5) Fireclay brick. =3596100*50=179805000

6) Silver graphite =3596100*0.5=178050

7) Graphite black =3596100*10=35961000

8) Talc =3596100*1=3596100

9) Bentonite =3596100*5= 17980500

Problem 2

Determine the enterprise’s need for medium-grade rolled steel and the order quantity based on the following data:

Annual production volume - 13 thousand units.

The consumption rate of rolled steel per product is 1150 kg.

The standard backlog of work in progress at the end of the planning period is 920 units.

The expected balance of work in progress at the beginning of the planning period is 750 products.

The need for repair and maintenance needs is 2650 tons.

The standard value of carryover stock is 10 days.

The actual stock of rolled stock as of the first day of the month in which the supply plan was developed was 1,450 tons.

The enterprise's need for rolled products for the period remaining until the planned year is 1050 tons .

Rolled products will be imported according to plan for the period remaining before the start of the planning year - 1200 tons.

Materials on the way - 150 tons.

Mobilization of internal resources - 3% of the total rental demand.

Solution.

Rp+Rnt+Rren+ Rnp+Rz = Oozh + Onp + Mwe + Zs

Rnt - the need for materials for the implementation of new technology

Rren - the need for materials for repair and maintenance needs

Oozh - expected balance at the beginning of the planning period

SP materials in work in progress at the beginning of the planning period.

MVE - mobilization of internal resources

3s - procurement from outside.

Let’s determine the enterprise’s need for average daily rentals:

Rp+Rnt+Rren+ Rnp+Rz

Let's convert 1150kg=1.15t

Рп = annual production volume* consumption rate

Рп = 13000 * 1.15 = 14950t.

Rren=2650t.

Rnp= 920*1.15=1058t.

Рз = (14950+2650+1058-862.5)/360 = 49.4 for 1 day

For 10 days = 494

Enterprise need = 14950+2650+1058+494=19152

Determining the order quantity

Mwe = 3%*19152 = 574.5 t

Zs = 14950 –(1450-1050+150+1200+574.5) = 2324.5

Problem 3

Determine the optimal size of the purchased batch, the number of batches and the time for renewing orders at the lowest costs for warehousing and servicing purchases based on the data below. Using the data obtained, construct a graph of the optimal size of the purchased lot. Initial data: 1. Annual consumption of product “B” - 30 thousand units.

2. The price ex-warehouse of the buyer per unit of product “B” is 15 dollars.

3. Discounts for order sizes starting from 10 thousand pieces.

4. Costs for servicing procurement - 1875 dollars. for each batch.

5. Warehousing costs - 20% of the cost of the average annual production inventory 6. Number of working days per year - 260.

Solution

Determine the optimal size of the purchased batch:

Сг - total annual costs

S- annual demand

i is the annual cost of maintaining a unit of production in a warehouse.

C=15dol-(5% discount)=14.25

Better value with a discount.

Number of batches= 30000/10000= 3 batches

Number of batches = 30000/6123 = 5 batches

Order renewal time = 260/3=86

Order renewal time = 260/5=52

Size

number of batches

Batch size – 10000; number of batches 3
Batch size – 6123; number of batches 5

Problem 4

Develop a quarterly supply plan, determine the degree of its provision with profile rolled products (in percentage, tons, days of work) and the amount of the order, using the data below.

Table 23

Products	Plan Production On the block thousand pieces	Norm Expenses for products, kg	Expected balance at the beginning of the quarter, t		Standards
			Work in progress	Production stock	Work in progress	Production stock, days
B	110.0	2.5	108.0	30.8	75.0	10
C	250.0	1.8

In the planned quarter, according to a previously placed order, rolled products are expected to arrive in
quantity 150 tons. It is also planned to use production waste in the amount of 48 tons.

Solution.

Determine the need for profile products:

Rp+ Rnp+Rz

Рп-requirement for materials production

Rnp - the need for materials to form a backlog of work in progress

Рз - the need for materials for the formation of carry-over stocks

Рп = 110*2.5+250*1.8=275+450=725t.

Рз=(725+75-108)/90=27t. – in 1 day

10 days – 270

Demand for profile rolled products = 725+75+270=1070t.

Determine the balance at the enterprise = 108 + 30.8 + 150 + 48 = 336.8 tons.

Security level:

X=(336.8*100)/1070= 31.5%

In days: 336.8/27=12.5

Determining the order quantity: 1070-336.8=733.2t.

Problem 5

Determine the form of commodity distribution (warehouse or transit) for long structural steel and the maximum annual volume of consumption at which the warehouse form of supply is appropriate.

Initial data:

1.Annual steel consumption - 396 tons

2.Wholesale price (ex-supplier warehouse) - 1430 rubles

3. Costs for importing 1 ton of steel:

for transit delivery - 35 rubles.
with warehouse delivery form -95 rub.

4. The annual cost of storing one ton of steel in the enterprise’s warehouse is 72 rubles.
5.Inventory standard:

- for transit delivery - 25 days

For warehouse delivery - 5 days

6. Capital costs for creating a storage capacity for storing 1 ton of steel - 1300 rubles.

7. Standard coefficient of efficiency of capital investments (Yong)- 0,15

Solution

Annual steel consumption is 396t.

with the transit form 396/360*25=27.5
with warehouse form 396/360*5=5.5

Determination of costs:

Ztr=P*Rtr+Wtr*C*K+Wtr*S+Wtr*Q*K

Zsklad = P*Rsklad+Vsklad*Ts*K+ Vsklad*S+ Vsklad*Q*K

P-annual demand for this type of material

Rtr, Rwarehouse - costs for the delivery of units of products to the warehouse of the consumer enterprise.

Tue, In warehouse - the amount of production stock

C - wholesale price of this type of material

K is the coefficient of effective use of capital investments.

S - annual costs for storing units. stock in the warehouse of the consumer enterprise

Q - capital investment for the creation of a storage capacity for storing units. stock

in transit form:

Ztr=396*35+27.5*1430*0.15+27.5*72+27.5*1300*0.15=13860+5898.75+1980+5362.5=27101.25

with warehouse form:

Zwarehouse=396*95+5.5*1430*0.15+5.5*72+5.5*1300*0.15=37620+1179.75+396+1072.5=40268.25

Maximum annual consumption

35x+5898.75+1980+5362.5=95x+1179.75+396+1072.5

60x=1179.75+396+1072.5-5898.75-1980-5362.5

Transit will be more profitable if the number is greater than 10593, and if less, then warehouse.

Problem 6

The supply service of an industrial enterprise needs to develop a supply strategy regarding component product “A”. It can be purchased from a supplier at a price (including delivery costs) of 710 rubles, or you can make it yourself. Variable costs for enterprises independently manufacturing a unit of component product “A” will amount to 605 rubles. Fixed expenses of the enterprise are equal to 6900 thousand rubles.

Using the given data, it is necessary to determine what is the feasibility of producing a component product at home and at what volume of output. Determine how a 5% reduction in customer costs will affect the possible production volume.

Solution

710x=605x+6900000

710x-605x=6900000

X=65714 (quantity) output volume

With decreasing variable costs

605 – 30 = 575

710x=575x+690000

710x-575x=690000

x=690000/135=5111

Reduced variable costs

575-29=546 (more profitable)

Problem 7

The company produces high-tech products, the production of which requires components. You can buy these components from a supplier at a price of 50 thousand rubles. per unit or make them yourself. Variable expenses of the enterprise amount to 43 thousand rubles. per unit of components. Fixed expenses amount to 64,500 thousand rubles.

Determine the “supply strategy” of the enterprise with components: purchase components from a supplier specializing in their production or manufacture them at the enterprise?

Solution

(solved in a similar way)

50000x=43000x+64500

Problem 8

The company purchases raw materials from the supplier. The enterprise's annual demand for raw materials is 6,400 tons. When the order batch size increases, the supplier provides buyers with price discounts in order to encourage them to purchase in larger quantities.

Order batch Price 1 ton of raw materials, thousand rubles.

Q t 1 T up to 499 t 40.0

From 500 t to 999 t 2% discount

From 1000 tons and above 3% discount

The cost of placing and fulfilling an order is 100 thousand. rub., and the cost of storing 1 ton of raw materials in a warehouse per year is 8,000 rubles.

Calculate:

an economical batch of raw materials without taking into account price discounts, as well as full costs, including purchasing costs, placing an order, and storing raw materials;
the total costs of the enterprise in accordance with those volumes of orders of raw materials that exceed the economic order size and the selling price of which includes discounts;
select the value of the order batch that provides the minimum amount of total costs, including the costs of purchasing raw materials, placing and fulfilling the order, as well as storing raw materials in the enterprise warehouse.

Solution

Сг - total annual costs

A - costs for the supply of the purchased batch

S- annual demand

C - unit price of purchased products

qo is the number of units of product contained in the purchased batch.

annual costs of maintaining a unit of product in a warehouse.

From 1t to 499t (without discount)

From 500t to 999t 2% discount

Discount=40000-2%=39200

From 1000t and above 3% discount

Discount=40000-3%=38800

Problem 9

The plant's machine shop is scheduled to produce hot-rolled
hexagonal steel 24120 parts. The steel consumption rate for 1 part is 0.8 kg. Standard
work in progress at the end of November - 9% of the monthly parts production program, the expected availability at the beginning of November is 1,700 parts. The norm for workshop stocks is 3 days. The expected balances of steel at the beginning of November are assumed to be equal to 70% of the standard workshop reserves. Set a limit for the supply of materials to the workshop for the month of November.

Solution

L=Zk+PpPnp-Zn

Product name	Production plan	Consumption rate			Material requirement			Average daily requirement	Zk
					pp	Pnp	generally		days	kg
details	24120	0,8	1700	2170,8	19296	376,64	19672,64	655,7	3	1967,2

Limit	Zn (kg)
20262,84	1377

Pp= 24120*0.8=19296

24120*9%= 2170,8

Pnp = (2170.8-1700)*0.8=376.64

Average daily requirement = 19672.64/30 = 655.7

Total consumption=19296+376.64=19672.64

Inventory at the beginning: Zn=70%*19672.64=1377

Limit =1967.2+19296+376.64-1377=20262

Problem 10

Determine the workshop limit for the supply of sheet metal for the month of April based on the data below.

Table 24

	Planned volume	Consumption rate	Actual balance as of 01.04		Unfinished standard	Shop standard
	Planned volume	Consumption rate	unfinished	workshop	Unfinished standard	Shop standard
Products	production,	rental at	production,	stock	production,	stock
	PC.	one	PC.	rental,	V % from plan	leafy
		product, kg		kg	production	rental, days
A	1060	15	85	517,0	6,5	2,0
B	2200	7	160		7,0

Solution

Product name		Production plan	Consumption rate			Material requirement			Average daily requirement		Zk
						pp	Pnp	generally		days		kg
A		1060	15	86	68,9	15900	-241,5	15658,5	521,9	2		2067,6
B		2200	7	160	154	15400	-42	15358	511,9
Total				245	222,9	31300	-283,5	31016,5	1033,8
Limit	Zn (kg)
32567,1	517

L=Zk+PpPnp-Zn

Zk - the amount of stock for a specific planning period

Pp - the workshop's need for material resources

Pnp - the workshop's need for material resources to change the backlog of work in progress

Zn - the amount of stock of material resources at the beginning of the planning period

Pp (A)= 1060*15=15900

Pp (B)=2200*7=15400

Pp (total)=15900+15400=31300

(A)=6.5%*1060=68.9

(B)=7%*2200=154

Pnp (A)=(68.9-85)*15=-241.5

Pnp (B)=(154-160)*7=-42

Average daily consumption:

15658,5/30=521,9

Ed(B)=15358/30=511.9

L=-517+31300-283.5+2067.6=32567.1

The maximum permissible operating time for devices I, II, III, IV is 84, 42, 21 and 42 hours, respectively.

Solution

Let's place the table with the initial data in cells A1:G9 of the Excel worksheet and perform the necessary preliminary calculations (see Fig. 4.3).

Rice. 4.3 – Initial data of the optimization problem

Find a solution to the problem, accepting the following conditions:

The final form of the problem formulation is presented in Fig. 4.4:

Fig. 4.4 – Formulation of the problem in terms of an Excel worksheet

The final result is presented in Fig. 4.5:

Fig.4.5 – Optimization result

Analysis of the solution shows that all the requirements of the optimization problem are met without exception. It is clear that in order to obtain maximum profit it is not advisable to produce product C.

The calculation results are presented in the results report (Fig. 4.6):

Fig.4.6 – Results report

The Solution Search utility can also be used to solve more complex optimization problems.

Task 2. Optimization of transportation plan (transport problem).

The company has 4 factories and 5 distribution centers for its goods. Factories are located in the city. Slutsk, Borisov, Molodechno and Bobruisk with production capabilities of 200, 150, 225 and 175 units of production daily, respectively.

Distribution centers are located in Vitebsk, Minsk, Orsha, Mogilev and Gomel with requirements of 100, 200, 50, 250 and 150 units of products daily, respectively.

Storing at the factory a unit of product that is not delivered to the distribution center costs 0.75 USD. per day, and the penalty for late delivery of a unit of product ordered by the consumer at the distribution center, but not located there, is equal to 2.5 cu. in a day.

The cost of transporting a unit of product from factories to distribution points is shown in table 4.1

Table 4.1. Transportation plan

It is necessary to plan transportation in such a way as to minimize total transportation costs.

It is important to note that since this model is balanced, i.e. Since the total volume of products produced is equal to the total volume of needs for them, then in this model there is no need to take into account the costs associated with both warehousing and short deliveries of products. Otherwise, you need to enter into the model:

In case of overproduction - a fictitious distribution point; the cost of transporting a unit of product to this fictitious point is assumed to be equal to the cost of storage, and the volume of transport to this point is equal to the volume of storage of surplus products in factories;

In case of shortage - a fictitious factory; the cost of transporting a unit of product from a fictitious factory is assumed to be equal to the cost of fines for short-delivery of products, and the volume of transportation from this factory is equal to the volume of short-delivery of products to distribution points.

To solve this problem, we will build a mathematical model. The unknown here is the volume of traffic. Let xij be the volume of transportation from the i-th factory to the j-th distribution center. The goal function is the total transport costs, i.e.

where cij is the cost of transporting a unit of product from the i-th factory to the j-th distribution center. In addition, the unknowns must satisfy the following restrictions:

Non-negativity of traffic volume;

Because If the model is balanced, then all products must be removed from factories and the needs of all distribution centers must be fully satisfied.

Thus we have the following model:

Minimize:

With restrictions:

IО, jО,

where a i is the production volume at the i-th factory, b j is the demand at the j-th distribution center.

Statement of the problem in terms of an Excel worksheet for using the “Solution Search” utility.

1. Place the initial data as shown in Fig. 4.7, 4.8.

2. Assign cells B8:F11 to the values of the unknowns (transport volumes).

3. Enter production volumes at factories in cells H8:H11.

4. Enter in cells B13:F13 the demand for products at distribution points.

5. In cell B16, enter the goal function = SUMPRODUCT(B3:F6;B8:F11).

6. In cells G8:G11, enter formulas that calculate production volumes at factories, in cells B12:F12 - volumes of products delivered to distribution points.

Fig.4.7 – Initial data

Rice. 4.8– Source data in formula mode

In the “Search for Solution” utility window, set the target cell, changeable cells and restrictions (see Fig. 4.9).

Rice. 4.9– Parameters of the “Search for a solution” window

The optimal plan that ensures minimal costs for transporting products from producers to consumers, found using the “Solution Search” utility, is presented in Fig. 4.10.

Fig. 4.10 – Results of the Search for a solution

As described above, the “Search for Solution” utility can generate a report on the results.

Tasks for independent work

Exercise 1. Solve a linear programming problem using the “Search for Solution” add-in of TP MS Excel.

To produce two types of products A and B, three types of technological equipment are used. To produce a unit of product A, equipment of the first type uses 1 hour, equipment of the second type - 2 hours, equipment of the third type - 3 hours. To produce a unit of product B, equipment of the first type is used in 1 hour, equipment of the second type - in 2 hours, equipment of the third type - in 3 hours.

For the manufacture of all products, the enterprise administration can provide equipment of the first type for no more than t 1 hours, equipment of the second type for no more than t 2 hours, equipment of the third type for no more than t 3 hours.

The profit from the sale of a unit of finished product A is α rub., and product B is β rub.

Draw up a plan for the production of products A and B, ensuring maximum profit from their sale.

Options for tasks are given in Table 4.2.

Table 4.2. Task options

Option	a 1	a 2	a 3	in 1	at 2	at 3	t 1	t 2	t 3	α	β

Task 2. Solve the problem of optimizing the transportation plan (transport problem).

There are n production points and m distribution points. The cost of transporting a unit of product from the i-th production point to the j-th distribution center c ij is given in the table, where the row is the production point, and the column is the distribution point. In addition, in this table, the i-th row indicates the production volume at the i-th production point, and the j-th column indicates the demand at the j-th distribution center. It is necessary to draw up a transportation plan for the delivery of the required products to distribution points, minimizing the total transportation costs.

Option 1.

Option 2.

Option 3.

Option 4.

Option 5.

Option 6.

Option 7.

Option 8.

Option 9.

5. What is the ABC analysis method based on?

6. What are the transport terms of delivery?

7. What solutions are used to optimize the purchase batch?

8. What conditions does the supply agreement include?

Questions and tasks for independent work:

1. Expand methods for studying the market for raw materials and supplies.

2. Analyze domestic and foreign experience in rationing enterprise inventories.

3. Expand the content of the concept “strategy for supplying an enterprise with material resources.”

4. Inventory management in the enterprise.

5. Give a scheme for calculating an economically feasible batch of purchases of material resources.

6. Expand the content of the ABC analysis method.

7. Task: The company produces products, the production of which requires components. An enterprise can buy these components at a price of 50 rubles. per unit, or make it yourself. Variable expenses are 43 rubles. per unit of components. Fixed expenses of the enterprise are equal to 490 thousand rubles. Based on the given initial data, it is necessary to determine at what volume of components it is more profitable to manufacture them at the enterprise?

8. Task: Enterprise A produces products X in the amount of 550 thousand. things. In 2007, 500 thousand were sold. pieces of products. In 2007, two more competing enterprises began producing similar products. In 2007, enterprise A was unable to sell 80 thousand units of product X.

As a result of marketing products, enterprise A determined that customer demand for product X in 2008 will remain at the same level.

Determine: the most effective production program for the production of products X in 2008, taking into account the projected customer demand for these products.

9. Task: The enterprise's need for components is 1,500 pieces per year. The cost of placing and fulfilling one order is 5,500 rubles. Warehouse storage costs 1 pc. products per year - 300 rubles. Calculate the economical batch of components purchased and the total costs, including the costs of purchasing, placing and fulfilling an order and storing inventories of components. (Make calculations and provide a graphical solution).

Reports on the topic:

2. Basic provisions of the contract for the supply of goods.

3. Development of a plan for the purchase of material resources at the enterprise.

4. Planning of production needs at the enterprise.

5. Construction of an information system for the procurement of raw materials and supplies at the enterprise.

Topic 3. Organization of sales activities at the enterprise.

(lectures 4 hours, seminars 4 hours, independent work 15 hours)

Target– study the sales activities of the enterprise, master methods of forecasting and planning the product range at the enterprise.

Determine the role of product sales in the enterprise;

Study methods for forecasting the market capacity of manufactured products;

Master the methods of planning the range of products in accordance with market requirements;

Study methods for forecasting the sales volume of products released to the market;

Study methods for drawing up a sales plan for an enterprise’s products;

Learn to analyze the choice of sales channels for manufactured products.

Topic content. The concept and tasks of determining the role of product sales in an enterprise. Formation of a portfolio of orders at the enterprise. Planning the range of products in accordance with market requirements. Methods for researching the product market, determining the capacity of the product market. Methods for forecasting the sales volume of products released to the market. Product positioning. Selecting a product distribution channel.

Methods:

Study of the essence of the role of sales of manufactured products at the enterprise;

Study and analysis of methods for planning the range of products, taking into account market requirements, analysis of methods for forecasting sales volume.

Literature:

Basic textbook: No. 1 chapter 2 p. 39 – 41, chapter 3 p. 44 – 62; No. 2 chapter 7-11 p. 187 – 360.

Basic literature: No. 5 chapter 5 p. 193 – 266; No. 4 Ch. 7-9 pp. 145 – 179, ch. 11-14 p.183 – 335.