19) (5 points) Suppoea plant can supplemen its capacity by subcontracting part o

Question

19) (5 points) Suppoea plant can supplemen its capacity by subcontracting part of or ll the production of certain parts. Show how to modify LP (16.28) (16.32) to include this option, where we define Vunits of product i received from a subcontractor in period t premium paid for subcontracting product i in period t (i.e., cost above variable cost of making it in-house) = minimum amount of product i that must be purchased in period t (e-g., specified as part of long-term contract with supplier) maximum amount of product i that can be purchased in period t (e.g., due to capacity constraints on supplier, as specified in long-term contract) LP (16.28-16.32) is given on the following page. are assumed fixed and the objective is to minimize the invent these demands. To do this, we introduce the following notation: tension of the previous single-product model in which demands ory carrying cost of meeting an index of product, i = 1 , . . . , m, so m represents total number of products an index of workstation, j-1, . . . , n, so n represents total number of J workstations t = an index of period, t-1, . . . , , so t represents planning horizon dy = maximum demand for product i in period t dr = minimum sales allows of product i in period t = time required on workstation j to produce one unit of product i cj capacity of workstation j in period t in units consistent with those used to define ay = net profit from one unit of product i hi cost® to hold one unit of product i for one period t Xit amount of product i produced in period t Sir = amount of product i sold in period t ltinventory of product i at end of periodr (io is given as data) Again, Xt, Sit, and Iu are decision variables, while the other symbols are constants representing input data. We can give a linear program formulation of the problem to maximize net profit minus inventory carrying cost subject to upper and lower bounds on sales and capacity constraints as (16.28) 5--hdit Maximize Subject to: di, S Si S d or alli (16.29) ajXi 3Cj for all j, (16.30) (16.31) (16.32) for all i, t lit = 111-1 + Xit-Sit Xl for all ,

Explanation / Answer

The revised LP model is as follows

Maximize riSit - hiIit - kitVit

s.t.

dit <= Sit <= dit

aijXit <= cjt

Iit =  Iit-1  + Xit + Vit - Sit  

vit <= Vit <= vit

Xit , Vit , Sit , Iit >= 0

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