Formulate the linear program based on the data above show

Question

A food company blames seasonality in production for difficulties that it has encountered in scheduling its activities efficiently. The company has to cope with three major difficulties:

I) Its food products are perishable. On the average, one unit spoils for every seven units kept in inventory from one month to another.

II) It is costly to change the level of the work force to coincide with requirements imposed by seasonal demands. It costs $750 to hire and train a new worker, and $500 to fire a worker.

III) On the average, one out of eight workers left idle in any month decides to leave the firm.

Because of the ever-increasing price of raw materials, the company feels that it should design a better scheduling plan to reduce production costs, rather than lose customers by increasing prices of its products. The task of the team hired to study this problem is made easier by the following operating characteristics of firm:

i) Practically, the firm has no problems procuring any raw materials that it requires;

ii) Storage capacity is practically unlimited at the current demand level; and

iii) The products are rather homogeneous, so that all output can be expressed in standard units (by using certain equivalence coefficients). The pertinent information for decision-making purposes is:

iv) The planning horizon has T = 12 months (one period = one month);

v) Demand D_i is known for each period (i = 1, 2, . . . , 12);

vi) Average productivity is 1100 units per worker per month;

vii) The level of the work force at the start of period 1 is L₁; S₀ units of the product are available in stock at the start of period 1;

viii) An employed worker is paid W_t as wages per month in period t;

ix) An idle worker is paid a minimum wage of M_t in month t, to be motivated not to leave;

x) It costs I dollars to keep one unit of the product in inventory for one month. With the above information, the company has decided to construct a pilot linear program to determine work-force level, hirings, firings, inventory levels, and idle workers.

a) Formulate the linear program based on the data above. Show that the model has a staircase structure.

b) Restate the constraints in terms of cumulative demand and work force; show that the model now has block triangular structure.