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A duality analysis of the transportation problem
Omega,
A DUALITY
V o l . 2, N o . 5
ANALYSIS
TRANSPORTATION
OF THE
PROBLEM
xq = amount of product shipped from The transportation problem, as a special the ith plant to the flh destination case of linear programming, has been a t = amount of supply at the ith plant analyzed in various forms in management bj = amount demanded at the jth destinascience literature to determine the optimal tion. solutions to many allocation problems. Most existing analyses deal with only one aspect of the optimization procedure known The dual problem is formulated as: Maximize as the primal problem. In addition to the n m primal aspect, every transportation problem F = Y~ bjzj -- ~, aiy t (3) has another related aspect, known as the j=l i=l dual problem. The dual of the transportation problem reveals some implicit relations. subject to Because of the unusual mathematical formulation, the dual aspect of the transz1 -- Yi -< cO (4) portation problem has not been given enough attention to many students in and management science. The purpose of this paper is to illustrate the dual formulation . . . . . m ; j = 1,2 . . . . . n) of a transportation problem and to interpret Yi, Z j > l O ( i = l , 2 the implicit relations contained in the dual where Yt and zj are the dual decision solution. variables representing the supply and demand requirement restrictions respectD u a l f o r m u l a t i o n and interpretation ively. The transportation problem may be Since, at the optimum, Z = F, then stated in the general form: Minimize m n n m m n Y~ Y. cij xij = ~E bj zj -- Y~ ai y t (5) . Z = Y~ Y ~ c o x o (1) i = l j = l j:l i=1 ' i=lj= 1 Substitution of a i and bj values from (2) subject to into (5) yields n
m
~x o~a. j=l m
n
X ai = Xbi
i=1
m
gx o/>b~ /=1
n
Y E xo'(co+ i=lj=l
(2)
j=l
and
Yt -- z~) = 0
(6)
Since xtj >1 0 and (c# + Yt - zj) >I 0, the following relations can b e found from Equation (6): zj = c o + Yi if x O > 0
xly >~ 0 ( i = 1,2 . . . . . m ; j = 1,2 . . . . . n) where c 0 = unit transportation cost from the ith plant to the jth destination
(7) zj ~ cq + Yi if xij = 0
Here, the dual variables Yi and zj may be interpreted as the cost of the product, Lo.b.
706
Omega, Vol. 2, No. 5 at the ith plant and its delivered value at: the jth destination respectively. Thus, the delivered..value at the jth destination equals its delivered cost if x U > 0 - d e l i v e r e d cost being equal to a cost at the ith plant plus transportation cost. The delivered value is less than the delivered cost if xq = 0~ therefore, the product is not shipped. Alternate optimal solutions exist if z~i : Cij @" Yi and Xij : 0 .
A concluding note The duality analysis of the transportation problem is equally applicable to assignment of the firm's accounting personnel to a variety of jobs (e.g., accounts receivable,
accounts payable, sales' audit, payroll etc.) when the problem is formulated in a transportation matrix. The qualifications of the accounting personnel can be implicitly rated for effective cost control and better decision making. N. K. Kwak (May 1974)
School o f Business Administration Saint Louis University 3674 Lindeil Boulevard Saint Louis, Missouri 63108, U.S.A
M A R G I N A L COSTS OF RESEARCH WORKERS FRITZ MACHLUP [2] applied utility theory to the analysis of marginal costs of research workers and gave the following illustration
S =
new total b i l l - o l d total bill number of recruits
[ll: There are 150,000 scientists with average income of 10,000 dollars each, i.e. total cost of 1500 million dollars. The target is to increase the number of scientists by 10 per cent, and this is achieved by the salaries of both old and new scientists increasing by 20 per cent--assuming the elasticity of supply is 0.5. The average annual income becomes 12,000 and the total bill 1980 million an increase of 480 million or 32 480 m per cent, which is -- 32,000 dollars 15,000 per new scientist, of which 12,000 is the new salary level and 20,000 is increased salary for the old scientists. By using such an example Machlup presents a somewhat biased viewpoint, particularly as it is not clear why an increase of 10 per cent in manpower should increase the average salary. A more general treatment of the problem would be as follows: Let N be the number of old research workers and ~tN the number of new recruits, increasing the population to N (1 + ~t). Let S be the old average income, S(I + f l 0 the average income of the old scientist after the recruiting campaign and S(1 +f12) the income of a recruit. The marginal cost of a new scientist is, therefore,
{N(I + f i t ) + ~N(1 +f12)-- N}S ~N
If S is the utility of an old scientist (his value to the employer, assumed here to match his salary), is the marginal cost S' larger or smaller than S? Clearly, S ' > S when fl~- + flz > 0 Gt
Note that this condition is met when f l > 0 , f12>~0 or fll ~>0, f z > 0 .
Special cases: (1)ill, f120> this is a generalization of Machlup's elasticity of supply, occurring when the supply of scientists is limited and when recruitment will increase salaries of old and new scientists. (2)fl1>0, f12=O when there is sufficient supply to maintain salaries of new recruits at the old level, but old scientists (or at least some) get compensated for supervising and managing new recruits.
707
A DUALITY
V o l . 2, N o . 5
ANALYSIS
TRANSPORTATION
OF THE
PROBLEM
xq = amount of product shipped from The transportation problem, as a special the ith plant to the flh destination case of linear programming, has been a t = amount of supply at the ith plant analyzed in various forms in management bj = amount demanded at the jth destinascience literature to determine the optimal tion. solutions to many allocation problems. Most existing analyses deal with only one aspect of the optimization procedure known The dual problem is formulated as: Maximize as the primal problem. In addition to the n m primal aspect, every transportation problem F = Y~ bjzj -- ~, aiy t (3) has another related aspect, known as the j=l i=l dual problem. The dual of the transportation problem reveals some implicit relations. subject to Because of the unusual mathematical formulation, the dual aspect of the transz1 -- Yi -< cO (4) portation problem has not been given enough attention to many students in and management science. The purpose of this paper is to illustrate the dual formulation . . . . . m ; j = 1,2 . . . . . n) of a transportation problem and to interpret Yi, Z j > l O ( i = l , 2 the implicit relations contained in the dual where Yt and zj are the dual decision solution. variables representing the supply and demand requirement restrictions respectD u a l f o r m u l a t i o n and interpretation ively. The transportation problem may be Since, at the optimum, Z = F, then stated in the general form: Minimize m n n m m n Y~ Y. cij xij = ~E bj zj -- Y~ ai y t (5) . Z = Y~ Y ~ c o x o (1) i = l j = l j:l i=1 ' i=lj= 1 Substitution of a i and bj values from (2) subject to into (5) yields n
m
~x o~a. j=l m
n
X ai = Xbi
i=1
m
gx o/>b~ /=1
n
Y E xo'(co+ i=lj=l
(2)
j=l
and
Yt -- z~) = 0
(6)
Since xtj >1 0 and (c# + Yt - zj) >I 0, the following relations can b e found from Equation (6): zj = c o + Yi if x O > 0
xly >~ 0 ( i = 1,2 . . . . . m ; j = 1,2 . . . . . n) where c 0 = unit transportation cost from the ith plant to the jth destination
(7) zj ~ cq + Yi if xij = 0
Here, the dual variables Yi and zj may be interpreted as the cost of the product, Lo.b.
706
Omega, Vol. 2, No. 5 at the ith plant and its delivered value at: the jth destination respectively. Thus, the delivered..value at the jth destination equals its delivered cost if x U > 0 - d e l i v e r e d cost being equal to a cost at the ith plant plus transportation cost. The delivered value is less than the delivered cost if xq = 0~ therefore, the product is not shipped. Alternate optimal solutions exist if z~i : Cij @" Yi and Xij : 0 .
A concluding note The duality analysis of the transportation problem is equally applicable to assignment of the firm's accounting personnel to a variety of jobs (e.g., accounts receivable,
accounts payable, sales' audit, payroll etc.) when the problem is formulated in a transportation matrix. The qualifications of the accounting personnel can be implicitly rated for effective cost control and better decision making. N. K. Kwak (May 1974)
School o f Business Administration Saint Louis University 3674 Lindeil Boulevard Saint Louis, Missouri 63108, U.S.A
M A R G I N A L COSTS OF RESEARCH WORKERS FRITZ MACHLUP [2] applied utility theory to the analysis of marginal costs of research workers and gave the following illustration
S =
new total b i l l - o l d total bill number of recruits
[ll: There are 150,000 scientists with average income of 10,000 dollars each, i.e. total cost of 1500 million dollars. The target is to increase the number of scientists by 10 per cent, and this is achieved by the salaries of both old and new scientists increasing by 20 per cent--assuming the elasticity of supply is 0.5. The average annual income becomes 12,000 and the total bill 1980 million an increase of 480 million or 32 480 m per cent, which is -- 32,000 dollars 15,000 per new scientist, of which 12,000 is the new salary level and 20,000 is increased salary for the old scientists. By using such an example Machlup presents a somewhat biased viewpoint, particularly as it is not clear why an increase of 10 per cent in manpower should increase the average salary. A more general treatment of the problem would be as follows: Let N be the number of old research workers and ~tN the number of new recruits, increasing the population to N (1 + ~t). Let S be the old average income, S(I + f l 0 the average income of the old scientist after the recruiting campaign and S(1 +f12) the income of a recruit. The marginal cost of a new scientist is, therefore,
{N(I + f i t ) + ~N(1 +f12)-- N}S ~N
If S is the utility of an old scientist (his value to the employer, assumed here to match his salary), is the marginal cost S' larger or smaller than S? Clearly, S ' > S when fl~- + flz > 0 Gt
Note that this condition is met when f l > 0 , f12>~0 or fll ~>0, f z > 0 .
Special cases: (1)ill, f120> this is a generalization of Machlup's elasticity of supply, occurring when the supply of scientists is limited and when recruitment will increase salaries of old and new scientists. (2)fl1>0, f12=O when there is sufficient supply to maintain salaries of new recruits at the old level, but old scientists (or at least some) get compensated for supervising and managing new recruits.
707