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**Linear Programming Problem**

Prepared by Sayed Mohibul Hossen Lecturer in Statistics

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Linear programming Linear programming is a method or technique of determining an optimum program of inter-dependent activities in a view of available resources. Linear programming is a mathematical technique for determining the optimal allocation of resources and obtaining a particular objective.

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Linear programming Mathematical programming is used to find the best or optimal solution to a problem that requires a decision about how best to use a set of limited resources to achieve a state goal of objectives.

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Linear programming Linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality or constraints. LP is a mathematical modeling technique used to determine a level of operational activity in order to achieve an objective, subject to restrictions called constraints.

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Linear programming In Linear Programming the term linear implies that all the mathematical relations used in the problem are linear and Programming refers to the method of determining a particular plan of action from amongst several alternatives.

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**Examples of LP Problems**

A Product Mix Problem A manufacturer has fixed amounts of different resources such as raw material, labor, and equipment. These resources can be combined to produce any one of several different products. The decision maker wishes to produce the combination of products that will maximize total income.

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**Examples of LP Problems**

A Production Scheduling Problem A manufacturer knows that he must supply a given number of items of a certain product each month for the next n months. They can be produced either in regular time, subject to a maximum each month, or in overtime. The cost of producing an item during overtime is greater than during regular time. A storage cost is associated with each item not sold at the end of the month. The problem is to determine the production schedule that minimizes the sum of production and storage costs.

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Examples Devising of a product schedule that could satisfy future demands ( seasonal or otherwise) for the firm’s product and at the same time minimize production costs. Selecting of production–mix to make the best use of machines, man-hours with a view to maximize profits. Selecting the advertising mix that will maximize the benefit subject to the total advertising budget.

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Example A manufacturer company produces two types of product. Following table shows the per unit labor, per unit clay and per unit profit is given. RESOURCE REQUIREMENTS Labor Clay profit PRODUCT (hr/unit) (lb/unit) ($/unit) Bowl Mug There are 40 hours of labor and 120 pounds of clay available each day. How much bowl and how much mug does the manufacturer to produce for maximizing the profit?

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**General LPP a11x1 + a12x2 (≤, =, ≥) b1 Constraints**

Max/min z = c1x1 + c2x objective function Where, x1 and x2 are decision variables. Subject to a11x1 + a12x2 (≤, =, ≥) b Constraints a21x1 + a22x2 (≤, =, ≥) b2 : x1 ≥ 0, x2 ≥ Non-negativity restriction

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Decision variables The decision variables are those quantities whose values are to be determined. Here x1, x2, x3… xn are the decision variables which optimize the objective function and also satisfy the constraints.

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Objective Function A linear function (z = c1x1 + c2x cnxn) which reflecting the objective of an operation and which has to be maximized or minimized is called a linear objective function. Objective function represents cost, profit, or some other quantity to be maximized or minimized subject to the constraints.

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Constraints The linear inequalities or equations or restrictions on the available resources (Labor, Capital, Materials and Machines) of a linear programming problem are called constraints.

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**Non-negative constraints**

Non-negative constraints included because variables are usually the number of items produced and one cannot produce a negative number of items. The smallest number of items one could produce is zero.

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Example A manufacturer company produces two types of product. Following table shows the per unit labor, per unit clay and per unit revenue is given. RESOURCE REQUIREMENTS Labor Clay profit PRODUCT (hr/unit) (Kg./unit) ($/unit) Bowl Mug There are 40 hours of labor and 120 Kg. of clay available each day. How much bowl and how much mug does the manufacturer to produce for maximizing the profit.

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**Formulation of Linear Programming Problem**

Step I: In every LPP, certain decisions are to be made. These decisions are represented by decision variables. These decision variables are those quantities whose values are to be determined. Identify the variables and denote them by x1, x2 and x3 or x, y and z etc. Step II: Identify the objective function and express it as a linear function of decision variables introduced in step I.

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**Formulation of Linear Programming Problem**

Step III: In a LPP, the objective function may be in the form of maximizing (profits) or minimizing (costs). So identify the type of optimization i.e., maximization or minimization. Step IV: Identify the set of constraints, stated in terms of decision variables and express them as linear inequalities or equalities according to the conditions. Step V: Add the non-negativity restrictions on the decision variables, as in the physical problems, negative values of decision variables have no valid interpretation.

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**Problem 1 Formulate the L.P.P for maximizing the profit.**

A company is manufacturing two products A and B. The manufacturing times required to make them, the profit and capacity available at each work centre are given below. Formulate the L.P.P for maximizing the profit. Work Centre Product Matching Fabrication Assembly Profit per unit (in Tk.) A B 1 hr 2 hrs 5 hrs 4 hrs 3 hrs 80 100 Total capacity 720 hrs 1800 hrs 900 hrs

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Problem 2 A firm makes two types of furniture: chairs and tables. The contribution for each product is Tk. 20 per chair and Tk. 30 Per table. Both products are processed on three machines M1, M2 and M3. The time required in hours by each product and total time available in hours per week on each machine are as follows. Formulate the L. P. P. How should the manufacturer schedule his product in order to maximize contribution? Machines Chair Table Available time M1 3 hrs 36 hours M2 5 hrs 2 hrs 50 hours M3 6 hrs 60 hours

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Problem 3 Food X contains 6 units of vitamin A per Kg. and 7 units of vitamin B per Kg. and cost Tk. 12 per Kg. Food Y contains 8 units of vitamin A per kg and 12 units of vitamin B and costs tk. 20 per Kg. The daily minimum requirements of vitamin A and vitamin B are 100 units and 120 units respectively. Formulate L. P. P. and find the minimum cost of product mix.

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Problem 4 A company produces two products A and B. There are two machines M1 and M2 through which the products are processed. The potential time capacity of machine M1 is 60 hours a week and that of machine M2 is 48 hours a week. To make one unit of A, it requires 4 hours in M1 and 2 hours in M2. To make one unit of B, it requires 2 hours in M1 and 4 hours in M2. If the profit per unit of A is Tk. 24 and the profit per unit of B is Tk. 18 can be expected. Formulate an LPP in order to maximize the profit.

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Problem 5 A dealer wish to buy some numbers of Cycles and Scooters. He has only Tk to invest and has a space for at most 60 items. One Scooter cost him Tk.2500 and a Cycle cost him Tk.500. His expectation is that he can sell a Scooter Tk.3000 and a Cycle Tk.650. Assuming that he can sell all the items that he can buy. Formulate L. P. P. How many Scooters and Cycles can he buy and sell in order to maximize his profit?

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Problem 1 A company produces two products A and B. There are two machines M1 and M2 through which the products are processed. The maximum time capacity of machine M1 is 36 hours a week and that of machine M2 is 42 hours a week. To make one unit of A, it requires 2 hours in M1 and 4 hours in M2. To make one unit of B, it requires 9 hours in M1 and 3 hours in M2. If the profit per unit of A is Tk. 10 and the profit per unit of B is Tk. 8 can be expected. Find out the number of units of A and B to be produced in order to maximize the profit.

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**Solution by Graphical method**

Step 1: Formulate the L. P. P. Step 2: Plot the constraints graphically. Step 3: Locate the feasible solution region. The feasible solution region is the graphical area which satisfies all the constraints at the same time. Step 4: Identify each of the corner points of the feasible region from the graph or by solving the simultaneous equations.

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**Solution by Graphical method**

Step 5: Calculate the objective function by putting the co-ordinates (x, y) of each corner point. Step 6: In a maximization problem, the optimal solution occurs at that corner point which gives the highest (profit) value of objective function. Step 7: In a minimization problem, the optimal solution occurs at that corner point which gives the lowest (cost) of the objective function.

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Problem 2 Food X contains 6 units of vitamin A per Kg. and 7 units of vitamin B per Kg. and cost Tk. 12 per Kg. Food Y contains 8 units of vitamin A per kg and 12 units of vitamin B and costs tk. 20 per Kg. The daily minimum requirements of vitamin A and vitamin B are 100 units and 120 units respectively. Find the minimum cost of product mix using graphical method.

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Problem 3 A company produces two products P and Q. There are two machines M1 and M2 through which the products are processed. The potential time capacity of machine M1 is 60 hours a week and that of machine M2 is 48 hours a week. To make one unit of P, it requires 4 hours in M1 and 2 hours in M2. To make one unit of Q, it requires 2 hours in M1 and 4 hours in M2. The profit per unit of P is Tk. 8 and the profit per unit of Q is Tk. 6. Find out the number of units of P and Q to be produced to maximize the profit. Also find the maximum profit.

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