Linear Programming Problems
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Introduction
Today many of the resources needed as inputs to operations are in limited supply. Operations managers must understand the impact of this situation on meeting their objectives. Linear programming (LP) is one way that operations managers can determine how best to allocate their scarce resources, under constraints. The constraints can be time, labor, energy, material, money, engineering specifications etc. Most frequently the objective of business firm is to maximize profit whereas the objective of individual operational units within firm is often minimize cost.
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Introduction
Today many of the resources needed as inputs to operations are in limited supply. Operations managers must understand the impact of this situation on meeting their objectives. Linear programming (LP) is one way that operations managers can determine how best to allocate their scarce resources, under constraints. The constraints can be time, labor, energy, material, money, engineering specifications etc. Most frequently the objective of business firm is to maximize profit whereas the objective of individual operational units within firm is often minimize cost.
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Linear Programming (LP) in OM
There are five common types of decisions in which LP may play a role
Product mix Ingredient mix Transportation Production plan Assignment
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LP Problems in OM: Product Mix
Objective To select the mix of products or services that results in maximum profits for the planning period Decision Variables How much to produce and market of each product or service for the planning period Constraints Maximum amount of each product or service demanded; Minimum amount of product or service policy will allow; Maximum amount of resources available 4
LP Problems in OM: Ingredient Mix
Objective To select the mix of ingredients going into products that results in minimum operating costs for the planning period Decision Variables How much of each ingredient to use in the planning period Constraints Amount of products demanded; Relationship between ingredients and products; Maximum amount of ingredients and production capacity available 5
LP Problems in OM: Transportation
Objective To select the distribution plan from sources to destinations that results in minimum shipping costs for the planning period Decision Variables How much product to ship from each source to each destination for the planning period Constraints Minimum or exact amount of products needed at each destination; Maximum or exact amount of products available at each source 6
LP Problems in OM: Production Plan
Objective To select the mix of products or services that results in maximum profits for the planning period Decision Variables How much to produce on straight-time labor and overtime labor during each month of the year Constraints Amount of products demanded in each month; Maximum labor and machine capacity available in each month; Maximum inventory space available in each month 7
LP Problems in OM: Assignment
Objective To assign projects to teams so that the total cost for all projects is minimized during the planning period Decision Variables To which team is each project assigned Constraints Each project must be assigned to a team; Each team must be assigned a project
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Recognizing LP Problems Characteristics of LP Problems in OM
A well-defined single objective must be stated.
There must be alternative courses of action.
The total achievement of the objective must be constrained by scarce resources or other restraints. The objective and each of the constraints must be expressed as linear mathematical functions.
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Steps in Formulating LP Problems 1. Define the objective. 2. Define the decision variables. 3. Write the mathematical function for the objective. 4. Write a 1- or 2-word description of each constraint. 5. Write the right-hand side (RHS) of each constraint. 6. Write <, =, or > for each constraint. 7. Write the decision variables on LHS of each constraint. 8. Write the coefficient for each decision variable in each constraint. 9. Write non-negativity restriction 10
Example:1. LP Formulation Cycle Trends is introducing two new lightweight bicycle frames, the Deluxe and the Professional, to be made from aluminum and steel alloys. The anticipated unit profits are 10 units for the Deluxe and 15 units for the Professional. The number of pounds of each alloy needed per frame is summarized on the next slide. A supplier delivers 100 pounds of the aluminum alloy and 80 pounds of the steel alloy weekly. How many Deluxe and Professional frames should Cycle Trends produce each week? 11
Example: LP Formulation
Aluminum Alloy
Steel Alloy
Deluxe
2
3
Professional
4
2
12
Example: LP Formulation Define the objective
Maximize total weekly profit
Define the decision variables
x1 = number of Deluxe frames produced weekly x2 = number of Professional frames produced weekly
Write the mathematical objective function
Max Z = 10x1 + 15x2
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Example:1 LP Formulation Write a one- or two-word description of each constraint
Aluminum available Steel available
Write the right-hand side of each constraint
100 80
Write <, =, > for each constraint
< 100 < 80 14
Example: LP Formulation
Write all the decision variables on the left-hand side of each constraint
x1 x2 < 100
x1 x2 < 80
Write the coefficient for each decision in each constraint
+ 2x1 + 4x2 < 100
+ 3x1 + 2x2 < 80
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Example 2: Product-Mix Problem
Consider a small manufacturer making two products A and B. Two resources R1 and R2 are required to make these products. Each unit of product A requires 1 unit of R1 and 3 units of R2. Each unit of product B requires 1 unit of R1 and 2 unit of R2. the manufacturer has 5 units of R1 and 12 units of R2 available. He makes profit of Rs 6 per unit of A and Rs 5 per unit of B sold. Formulate this problem as linear programming problem. 16
Example 3: Production Planning Problem
A textile company can use any or all of three different processes for weaving its standard white polyester fabric. The different costs and capacities of three production process are shown in the table. The daily demand forecast for its white polyester fabric is 4000 sq. metre. Develop a linear programming model to determine combination of production process to be utilized to meet daily demand forecast at minimum total production cost.
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Example 3:Production Planning Problem cont…
Cost of Production Process
Process Weaving M/c Processing Maximum Daily No. Set up cost Cost/sq. capacity (Sq. m.) (Rs) m. (Rs)
1
150
15
2000
2
240
10
3000
3
300
8
3500
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Example 4: Product Mix Problem
A company is planning to determine its product mix out of three different products:P1, P2 and P3. The monthly sales of the product P1 is limited to maximum of 500 units. For every two units of P2 produced, there will be one unit of by-product which can be sold at the rate of Rs.20 per unit. The highest monthly demand for this by-product is 200 units. The contributions per unit of the products P1, P2 and P3 are Rs 50, Rs 70,and Rs.60 respectively. The processing requirements of these products are shown in table. Formulate a linear programming model of this problem to find the optimum product mix such 19 that the total contribution is minimized.
Example 4: Product Mix Problem Cont..
Process
Hours per units
P1
P2
P3
Available Hours
I
3
5
2
1000
II
4
-
3
700
III
4
3
2
1300
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Example 5:
A company manufactures three products, P1, P2 and P3. The machine hours requirements and labor hour requirements to process the three products and the maximum machine hours per week for each machine type and maximum available labour hours per week are summarized in table along with selling and cost price. The company wants to limit the production volume per week of the product P3 to utmost 35 units. Formulate a linear programming model to find the production volume per week of each product such that the total profit is maximum. 21
Example 5 : cont. Machine hours required
P1
P2
P3
Maximum available hours per week
Machine 1
4
6
3
500
Machine 2
3
-
2
300
Machine 3
5
7
8
600
Labour
3
2
4
200
Selling Price/unit (Rs)
500
400
550
Product Cost Price/unit (Rs)
350
280
390 22
Example 6:
A computer company procures cabinet from three different suppliers (A, B and C) located at three different cities. The company has production plants (P,Q and Q) in three different cities. The cost of transportation per cabinet for different combination of supplier and production plant are given in the table. Formulate a linear programming model to find optimal procurement plan for the cabinets such that the total cost is minimized.
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Example 6: cont..
Supplier
Production Plant
P
Q
R
Supply
Price per cabinet
A
10
15
8
300
85
B
20
21
15
500
90
C
12
16
13
100
75
200
300
Demand 100
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Example 7: Manpower Scheduling Problem
In a multi-speciality hospital, nurses report to duty at the end of every 4 hour as shown in the table. Each nurse, after reporting, will work for 8 hours continuously. The minimum number of nurses required during various periods are also given in the table. Develop a mathematical model to determine the number of nurses to report at the begining of each period such that the total number of nurses who have to report to duty in a day is minimized.
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Example 7: Manpower Scheduling Problem cont.. Interval Number
Time Period
From
to
Minimum number of nurses required
1
12 midnight
4 a.m.
20
2
4 a.m.
8 a.m.
25
3
8 a.m.
12 noon
35
4
12 noon
4 p.m.
32
5
4 p.m.
8 p.m.
22
6
8 p.m.
12 midnight
15
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LP Problems in General
Units of each term in a constraint must be the same as the RHS Units of each term in the objective function must be the same as Z
Units between constraints do not have to be the same
LP problem can have a mixture of constraint types
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Graphical Solution Approach 1. Formulate the objective and constraint functions. 2. Draw the graph with one variable on the horizontal axis and one on the vertical axis. 3. Plot the no-negativity restriction ( restrict yourself the quadrant where both X1 and X2 are ≥ 0. 4. Plot each of the constraints as if they were lines or equalities. 5. Outline the feasible solution space that contain the set of points satisfying all the constraints. 28
Graphical Solution Approach cont.. 6. Identify corner points. 7. Evaluate the objective function at all the corner points 8. Substitute each of the potential solution point values of the two decision variables into the objective function and solve for Z. 9. Select the solution point that optimizes Z.
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Example 8: Graphical Method
Z = 10x1 + 15x2 (Profit)
Maximize the above objective function subjected to following constraints: 3x1 + 2x2 < 80 (Steel) 2x1 + 4x2 < 100 (Aluminum) x1 , x2
≥0
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Example: Graphical Solution Approach x2
3x1 + 2x2 < 80 (Steel) Potential Solution Point x1 = 0, x2 = 25, Z = 375.00
40 35
Optimal Solution Point x1 = 15, x2 = 17.5, Z = 412.50
30 25
2x1 + 4x2 < 100 (Aluminum)
20 15 10
Feasible Solution Space
MAX 10x1 + 15x2 (Profit) Potential Solution Point x1 = 26.67, x2 = 0, Z = 266.67
5 5
10
15
20
25
30
35
40
45
50
x1 31
Example 9: Graphical Method
Z = 6x1 + 8x2
Maximize the above objective function subjected to following constraints: 5x1 + 10x2 < 60 4x1 + 4x2 < 40 x1 , x2
≥0
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Example 10: Graphical Method
Z = 2x1 + 3x2
Minimize the above objective function subjected to following constraints: x1 + x2
≥
7x1 + x2 x1 , x2
6
≥
14
≥0
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Example 11: Graphical Method
Z = 100x1 + 80x2
Maximize the above objective function subjected to following constraints: 5x1 + 10x2 < 50 8x1 + 2x2 ≥ 16 3x1 - 2x2 ≥ 6 x1 , x2
≥0 34
