3. Transportation Problem (Part 1)

Transcription

1 3 Transportation Problem (Part 1) 31 Introduction to Transportation Problem 32 Mathematical Formulation and Tabular Representation 33 Some Basic Definitions 34 Transportation Algorithm 35 Methods for Initial Basic Feasible Solution 1 North-West Corner Rule 2 Lowest Cost Method 3 Vogel s Approximation Method 31 Introduction to Transportation Problem The transportation problem deals with the distribution of goods from several points of supply (sources) to a number of points of demand (destinations) Usually we are given the capacity (availability) of goods at each source and the requirements (demands) at each destination Typically the objective is to minimize total transportation and production costs The availability as well as the requirements is finite It is assumed that the cost of transportation is linear 32 Mathematical Formulation and Tabular Representation Let m denote number of supply centre (S1, S2 Sm) Let n denote number of demand centre (D1, D2 Dn) D S D1 D2 Dn Capacities (Availability) S1 c11 x11 c12 x12 c1n x1n a1 S2 c21 x21 c22 x22 c2n x2n a2 Sm Requirement (Demand) cm1 xm1 cm2 xm2 cmn xmn am b1 b2 bn Σai = Σbj wwwbeinggouravcom

2 The product cij xij gives the net cost of transporting units from the supply centre Si to demand centre Dj, where i = 1,2,m and j = 1,2,n Let ai = number of units available at i th source (supply centre) of a certain product Let bj = number of units demanded at j th destination (demand centre) of a certain product Let cij be the cost of transporting one unit from i th source to j th destination Let xij be the amount to be transported from i th source to j th destination The problem now, is to determine non-negative allocation xij satisfying both the availability constraints and minimize cost of transportation Min Z = c11x11 + c12x12 + +c1nx1n subject to + c21x21 + c22x22 + +c2nx2n + +cm1xm1 + cm2xm2 + +cmnxmn x11 + x12 + +x1n = a1 x21 + x22 + +x2n = a2 supply constraints xm1 + xm2 + +xmn = am x11 + x21 + +xm1 = b1 x12 + x22 + +xm2 = b2 demand constraints x1n + x2n + +xmn = bn We assumed that the total availabilities Σai satisfy the total requirements Σbj ie Σai = Σbj (i = 1, 2, 3 m and j = 1, 2, 3 n) Transportation Problem is special type of LPP wwwbeinggouravcom

3 33 Some Basic Definitions Balanced Transportation Problem If Total supply equals to total demand, the problem is said to be a balanced transportation problem ie Σai = Σbj (i = 1, 2, 3 m and j = 1, 2, 3 n) Unbalanced Transportation Problem If Total supply not equals to total demand, the problem is said to be an unbalanced transportation problem ie Σai Σbj (i = 1, 2, 3 m and j = 1, 2, 3 n) Occupied and Unoccupied Cell The positive allocated cell in the transportation table is called occupied cells, otherwise empty or unoccupied cell Feasible Solution A set of non-negative allocations xij 0 which satisfies the supply and demand limitations is called feasible solution to a transportation problem Basic Feasible Solution A feasible solution called basic if the number of positive allocations xij = m+n-1, where m = number of supply centre and n = number of demand centre If the number of positive allocations is less than m+n-1 then it is called as Degenerate Basic Feasible Solution If the number of positive allocations xij = m+n-1 and in independent position then it is called as Non - Degenerate Basic Feasible Solution Optimum Solution A feasible solution which minimizes the total transportation cost is called optimal solution 34 Transportation Algorithm 1 Check the problem is balanced or not If not balanced then make it balanced by adding either dummy row or dummy column as required wwwbeinggouravcom

4 2 Find Initial basic Feasible Solution (IBFS) by North-West Corner Rule (NWCR), Lowest Cost Method (LCM), Vogel s Approximation Method (VAM) 3 The initial solution obtained by any of the three methods must be feasible and non- degenerate(see definition above) 4 Test initial solution for optimality (here we discuss Modified Distribution method (MODI) to test optimality) 35 Methods for finding Initial Basic Feasible Solution(IBFS) IMPORTANT NOTE: For finding IBFS, first check Transportation Problem is balanced or not Here we shall discuss only three different methods to obtain the initial basic feasible solution They are 1 North-West Corner Rule (NWCR) This method advocates that allocation should be made on the basis of geographical location of the cells in the table In particular, the method attaches greater importance to the cell situated at the upper left hand corner of the table and makes as much as possible an allocation to the cell with both the supply restriction and demand constraint taking into consideration The algorithm for North-West Corner Rule is: Start allocation from upper left hand corner of the table (Adjust supply and demand accordingly) Exhaust the supply (source) capacity at each row before moving down to the next row Exhaust the demand (destination) requirements of each column before moving to the right of the next column Continue in the same manner until all supply has been exhausted and demand requirements have been met Find the initial basic feasible solution by NWCR Warehouse Factory W 1 W 2 W 3 W 4 Capacity F F F Requirement wwwbeinggouravcom

5 Initial basic feasible solution by using NWCR wwwbeinggouravcom

6 2 Least cost Method (LCM) This method advocates that allocation should be based on minimum cost of transportation It says that the First allocation must be made to the cell with the most minimum (least) cost of transportation per unit In other words, we look at the scheduled of the transport cell and identify the most minimum After identifying the cell with the least transportation cost, we next make maximum allocation to the cell without violating both supply and demand restriction Repeat above steps until all supply and demand are satisfied In case of tie among minimum cost, select the cell for allocation where maximum unit can be allocated Find the initial basic feasible solution by LCM (Least Cost Method) Warehouse Factory F F F Requirement W1 W2 W3 W4 Capacity wwwbeinggouravcom

7 Initial basic feasible solution by using Least Cost Method wwwbeinggouravcom

8 3 Vogel s Approximation Method (VAM) The Vogel s Approximation Method takes into account not only the least cost cij but also the cost that just exceeds cij The IBFS obtained by Vogel s method is either optimal or very close to the optimal solutions The steps of the method are given below Compute difference between the smallest and second smallest costs in each row and column These differences are called penalties or opportunity cost Identify the row or column with the largest difference among all the rows and columns Check the row or column with the largest difference and select minimum cost cell and allocate as much as possible to these cell Adjust the supply and demand constraints Again compute difference between the smallest and second smallest costs in each row and column of reduced transportation table and do the same as above until all the supply and demand constraints are satisfied If there is a tie among maximum difference then select row or column for allocation in which total cost is minimum And if there is tie among minimum total cost then select any row or column arbitrary Find the initial basic feasible solution by VAM (Vogel s method) Warehouse Factory F F F Requirement W1 W2 W3 W4 Capacity wwwbeinggouravcom

9 Initial basic feasible solution by using Vogel s Approximation Method wwwbeinggouravcom