Lecture No. 12

 

Held on: Monday, September 4, 2000

 

Notes Prepared by: Rochan R. Upadhyay

 

 

Example of a production planning problem formulated as an LP (Transportation LP could handle only simple PP problems)

 

Assume that the best demand estimates are as follows

Jan    5300                          May  4100                          Sept   7300

Feb   5100                          June  4800                          Oct    7800

Mar  4400                           July  6000                          Nov   7600

Apr  2800                           Aug  7100                           Dec   6400

 

To adjust for fluctuations, management can:

(i)                  Change the workforce level by hiring or firing

(ii)                Cover temporary shortages by overtime work

(iii)               Store some of the present surplus to cover future shortages

Constraints

-         Change of workforce level is restricted to at most 40 workers per month (up or down)

-         It costs $300 to hire and $420 to fire a worker

-         Each worker produces 20 units per month on regular time and will produce no more than 6units per month on overtime.

-         Overtime production cost exceeds regular production cost by $20 per unit

-         It costs $8/unit/month to store each unit

-         At present there are 290 workers and no inventory

-         Long range planning dictates a zero inventory by next December

 

Solution :

 

Let d_j denote the demand in the j^th month.

Let x_j denote normal production in month j.

Let y_j denote overtime production in month j.

Let z_j denote inventory at the end of month j.

Upper bound on the monthly change of workforce requires that:

     |x_j/20 - x_j-1/20| <= 40

     |x_j - xj-1 | <= 800        j = 1,2,3……,12 ---------------- (1)

Hence there are x_j/20 workers in month j,

     y_j <= 6(x_j/20)

Or, y_j <= 0.3x_j                 j = 1,2,3,……,12 ----------------(2)

Additional physical constraint is

     Z _j-1+x_j+y_j = d_j+z_j     j = 1,2,3,…….,12 ----------------(3)

(This is the law of goods conservation)

 

Expenses during the j th month :

            Overtime cost = 20 y_j

            Storage cost = 8 z _j

            Hiring and firing cost t_j equals

            T_j = 15 ( x_j - x _j-1) if x_j >= x _j-1

                 = 21 (x _j-1 - x_j) if x _j-1 >= x _j

Hence the objective is to minimise

    42

S       (20y_j + 8z_j + t_j)

   j = 1

Boundary conditions are x₀ = 5800 , z₀ = z₁₂ = 0 ------------------(4)

Non-negativity constraints are

            x_j, y _j , z_j , t_j >= 0  j = 1,2,….,12----------------------(5)

A linear relationship is required between the tj s and the xj s

Note that t_j = max{15(x_j - x _j-1), 21(x _j-1 - x_j)}

This can be modeled as t_j >= 15(x_j - x _j-1 )

                                    & t_j >= 21(x _j-1 - x_j)  j = 1,2,3,…….,12------------------------(6)

Thus we now have a linear objective function

Min    n

          S ( 20y_j+ 8z_j +t_j )

      j = 1

subject to (1),(2),(3),(4),(5),(6)

x1,x2,x3,……..,x20 have to be multiples of 20 to have a feasible solution. However we will ignore this obstacle completely. Having obtained an optimal LP solution we simply round each xj to the nearest multiple of 20 and adjust the level of overtime and storage accordingly. The resulting solution is  quite close to the optimal one.

Month       Demand     Workforce         Normal     Overtime                 Inventory

                                                            Production            Production

Jan                  5300       265                        5300

Feb                 5100       255                        5100

Mar                4400        220                        4400

Apr                 2800        201                        4020                                                    1220

May                4100        201                        4020                                                    1140

Jun                  4800        241                        4820                                                    1160

Jul                   6000        281                        5620                                                    780

Aug                 7100        321                        6420                                                    100

Sept                 7300        361                        7220                                                     20

Oct                  7800        361                        7220                   560                       

Nov                 7600        360                        7200                   400           

Dec                 6400         320                        6400

 

During January to March there is no overtime or inventory.

From April to September there is a large initial inventory that gradually dwindles to zero. No overtime used.

From June to September, 40 new workers are hired monthly.

From October to December no inventory is created. Some overtime is required to cover the demand.

Drop between November and December is sharp leading to a situation in November where we have overtime and a worker is fired.

 

LP formulation of the generic Aggregate Production Planning (APP) problem

 

The decision variables are :

X_it = units of product i to be produced in period t.

W_t = regular workforce hours available in time t.

O_t = overtime workforce hours used in period t. It is constrained to be no more than a fraction p of  W _t .

H_t = hours of regular workforce hired in period t.

F_t = hours of regular workforce fired in period t.

I_it⁺ = units of inventory for product type i at the end of period t

I _it^- = units back ordered for product type i at the end of period t.

Each unit of X_it uses K_i hours of workforce either in regular time or overtime

Following costs are involved

Variable cost v_ti X_it

Inventory holding cost C_it I_it⁺

Back order of cost C_it I_it^-

Regular payroll cost X_tW_t

Overtime payroll cost o_t O_t  (o_t > r_t)